IB grade 9 Math book-Chapter3
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CHAPTER
3
Number Sense and Algebra 2
Substitute into and evaluate algebraic expressions involving exponents.
2
Describe the relationship between the algebraic and geometric representations of a single-variable term up to degree three.
2
Derive, through the investigation and examination of patterns, the exponent rules for multiplying and dividing monomials, and apply these rules in expressions involving one and two variables with positive exponents.
2
Relate understanding of inverse operations to squaring and taking the square root.
2
Extend the multiplication rule to derive and understand the power of a power rule, and apply it to simplify expressions involving one and two variables with positive exponents.
2
Add and subtract polynomials with up to two variables.
2
Multiply a polynomial by a monomial involving the same variable.
2
Expand and simplify polynomial expressions involving one variable.
100
Polynomials As science becomes more complex, it becomes more important to communicate mathematical ideas clearly. Imagine if Einstein always had to write out his famous equation E = mc 2 as “energy equals mass times the square of the speed of light.” Using letters or symbols to represent unknown amounts is called algebra. Algebra is a basic building block of advanced mathematical and scientific thinking. How can you use algebra to communicate your mathematical ideas?
Vocabulary variable algebraic expression term polynomial
degree of a term degree of a polynomial like terms distributive property
Chapter Problem Alysia has selected the letter E to design the logo for her school team, the Eagles. The design will be used to make differentsized crests for clothing such as jackets, sweaters, and baseball caps. How can Alysia make sure that, when the crest is made larger or smaller, the proportions do not change?
101
Add and Subtract Integers To add integers, you can use a number line. • Start at the first integer. • Add the second integer by drawing an arrow. • The arrow points to the right if the second integer is positive, and to the left if it is negative. • The answer is at the tip of the arrow. 2
(1) (2) 3
3 2 1
0
Opposite integers add to zero. For example, (4) (4) 0. Subtracting an integer is the same as adding the opposite. 5 (2) 5 (2) 7
2 is the opposite of 2.
(3) (1) (3) (1) 4
1 is the opposite of 1.
⫺3
2 (3) 1
⫺2 ⫺1
0
1
2
1. Add or subtract.
2. Evaluate.
a) 7 5
b) 10 3
a) (2) (2) 1
b) 10 (3)
c) 5 (9)
d) 5 (4)
c) 5 (7) 7
d) 4 (3) (2)
e) (4) 6
f) 7 9
e) (9) 6
f) 1 (1)
g) (3) (11)
h) (4) (8)
g) (5) 8
h) (8) 9 (2)
Multiply and Divide Integers The product or quotient of two integers of the same sign gives a positive result. The product or quotient of two integers of opposite signs gives a negative result. 5 3 15 14 (7) 2
3 (4) 12 8 1 8
3. Multiply.
5 6 30 15 (5) 3
4. Divide.
a) 3 (8)
b) (4) (6)
a) (8) 4
b) 9 (3)
c) (8) 4
d) (5)(6)
e) 12(5)
f) 2(20)
c)
16 8 e) 25 (5)
6 6 f) 36 (4)
102 MHR • Chapter 3
d)
Multiply Rational Numbers To multiply fractions, multiply the numerators together and multiply the denominators together. When a numerator and a denominator share a common factor, you can divide it out before multiplying. 2 3 3 4 1
1
1
2
2 3 3 4
3 –– 4 2 –– 3
1 2
When negative fractions or decimals occur, apply the same rules as for products of integers. 1 3 a b 4 5 3 1 a b 4 5
3 20
When I multiply a negative by a positive, I get a negative result.
My scientific calculator may need different keystrokes. I’ll check the manual.
Check: Using a scientific calculator:
0.25 0.5 0.125 Using a scientific calculator:
ç3 \∫4 *1 ∫5 =
ç0.25\*0.5 =
Using a graphing calculator: –3 ÷4 *1 ÷5 k1 e
Using a graphing calculator: –0.25 * 0.5 e
5. Multiply.
6. Multiply.
1 1 3 4
a)
4 1 5 3
b)
c)
3 2 a b 5 9
d) a b a b
e)
3 4 8 5
3 4
f)
a) a b
2 5
1 5
1 4
c) 0.6 (0.95) e) 2.5 (3.2)
b) a
5 3 b a b 10 6
d) (0.3)(0.4) f) 8(3.8)
1 1 a1 b 2 2
Get Ready • MHR 103
3.1
Build Algebraic Models Using Concrete Materials Petra likes to run at the track to keep in shape. This year, to motivate herself, she will record her training progress visually. What are some ways that she can do this?
Investigate 䊏
algebra tiles
How can you model length, area, and volume using concrete materials?
䊏
linking cubes
Part A: Model Length
Tools
Petra’s running record for the first 2 weeks of the year is shown below. Petra used algebra tiles to model each distance. She used the side length of a unit tile to represent 1 km. Algebra tiles are tools that can be used to model measured quantities. What other concrete materials could I use to represent one unit?
A unit tile is a square tile that measures 1 unit by 1 unit. It can be used as a counter. On July 14, Petra ran 4 km. Date
Distance (km)
January 1
1
January 3
2
January 6
2
January 8
3
January 10
3
January 14
4
Progress
A tile model is a good way of tracking Petra’s progress visually.
104 MHR • Chapter 3
unit tile
In this book, positive algebra tiles are green and negative algebra tiles are white.
One weekend, Petra cross-country skiied around a lake three times. She did not know the distance around the lake, so she used a variable , x, to represent it. An x-tile is a rectangular tile that is used to represent the variable x. It has the same width as the side length of a unit tile. You can use an x-tile to describe any unknown value.
variable 䊏
a quantity whose value can change (or vary)
䊏
usually represented by a letter
The total distance can be modelled using three x-tiles: x-tile
This means that on the weekend, Petra skiied a total of 3x kilometres, where x is the distance around the lake. 1. Use tiles to represent each length. a) 6 km b) an unknown distance c) an unknown distance and back d) 3 km plus an unknown distance
algebraic expression
2. Use tiles to model each algebraic expression . a) 7
b)
4x
c)
x3
d)
3x 2
䊏
a mathematical phrase made up of numbers and variables, connected by addition or subtraction operators
䊏
can be used to model real-life situations
䊏
2x 3 is an algebraic expression
3. Write an algebraic expression for each model. a)
b)
c)
d)
4. Create an algebraic expression of your own and build a tile
model to represent it. Record the expression and the model. Part B: Model Area
You can use an x2-tile to represent an unknown area, measured in square units, if you let the side length of each tile represent x units. Note that the side length of an x2-tile is equal to the length of an x-tile.
x-tile
1
4 9 (square units)
Literac
onnections
Expressions are sometimes confused with equations. For example, 2x 3 is an expression, but 2x 3 1 is an equation. Equations contain an equal sign, expressions do not.
x 2-tile
x 2-tile x-tile
3.1 Build Algebraic Models Using Concrete Materials • MHR 105
This model shows that Petra mowed one square lawn 6 times over a summer. The total area mowed, in square metres, is 6x2, where x is the side length, in metres, of the square lawn.
5. Use algebra tiles to represent each area. a) 16 square units b) 25 square units c) 3x2 d) 5x2 6. Use algebra tiles to represent each expression. a) x2 2 b) 2x2 x c) x2 3x 2 d) 3x2 5x 1 Part C: Model Volume 7. A cube is a rectangular prism with length, width, and height all
equal. a) Use linking cubes to build a model of a cube that has a side
length of 3 cm. Sketch your model. b) What is the volume, in cubic centimetres, of this cube? c) Express the volume of the cube as a power. 8. Repeat step 7 for a cube with a side length of 5 cm. 9. Describe other concrete materials you could use to build an
algebraic model of volume. 10. Suppose you do not know the side length of a cube. You can use
the variable x to represent the side length. a) Sketch the cube and label its length, width, and height. b) Write an algebraic expression for the area of one face of
the cube. c) Write an algebraic expression for the volume of the cube. 11. Reflect Describe how concrete materials can be used to build
algebraic models of length, area, and volume. Use words, expressions, and diagrams to support your explanation.
106 MHR • Chapter 3
Key Concepts 䊏
䊏
You can build length models with algebra tiles.
䊏
You can build area models with algebra tiles.
䊏
⫹1
Concrete materials, such as algebra tiles and linking cubes, can be used to build algebraic models.
unit tile
⫺1 negative unit tile
x-tile
⫺x-tile
x 2-tile
⫺x 2-tile
You can build volume models with linking cubes.
Communicate Your Understanding C1
State the length and width of each. a) a unit tile b) an x-tile c) an x2-tile
C2
a) How are the length and width of a unit tile and an x-tile related? b) How are the length and width of an x-tile and an x2-tile related?
C3
a) Suggest two other objects that could be used to model length. b) Suggest two other objects that could be used to model area. c) For your answers to parts a) and b), identify any advantages or
disadvantages of each object. C4
Explain how concrete materials can be used to model each type of measurement. Include a diagram to support each explanation. a) length
b) area
c) volume
Practise 1. Which expression is represented by the algebra
tile model? A 4x2 2x 5
B 4x2 2x 5
C 4x2 2x 5
D 4x2 2x 5
3.1 Build Algebraic Models Using Concrete Materials • MHR 107
Did You Know ? The Greek mathematician Diophantus (about 250-275) was the first person to use a letter to represent an unknown.
2. Use tiles to model each algebraic expression. a) x2 3x b) 2x2 5 c) 3x2 x 2 d) x2 2x 4 3. Write the algebraic expression represented by each model. a) b)
c)
d)
Connect and Apply 4. Each unit tile represents 1 km that Miko rode her bicycle.
Find each distance. a) b) c) d) 5. Create an algebraic expression of your own, using x2-tiles, x-tiles,
and unit tiles, and build a tile model to represent it. Record the expression and the model. 6. a) Build a volume model to represent a cube with length, width,
and height all equal to 4 cm. Sketch the model and label the length, width, and height. b) What is the volume? Write this as a power. c) Write an expression for the area of one face as a power.
Evaluate the area of one face.
108 MHR • Chapter 3
7. A cube has a volume of 216 cm3. a) What side length of the cube would give this volume? b) Determine the area of one face of the cube. 8. The area of one face of a cube is 49 m2. a) What side length of the cube would give this area? b) Determine the volume of the cube.
Extend 9. Build an area model using tiles that have length and
width as indicated. a) length x 3, width x b) length x 4, width x 1 10. A cube has a volume of 8 cm3. Find the total surface area of
all six faces. 11. Math Contest Mersenne numbers are numbers of the form 2n 1.
Father Marin Mersenne (15881648) was especially interested in prime numbers of this form. One conjecture about Mersenne numbers is that numbers of the form 2p 1 are prime if p is prime. Investigate this conjecture and write a brief report of your findings. 12. Math Contest Find the smallest possible value of ab cd e f if a, b,
c, d, e, and f are all different and are chosen from the values 1, 2, 3, 4, 5, and 6. 13. Math Contest When 3040 4030 is written in expanded form, the
number of zeros at the end of the number is A 30 B 40 C 70 D 120 E 1200 n
14. Math Contest Fermat numbers are numbers of the form 22 1.
Pierre de Fermat (16011665) conjectured that all numbers of this form are primes. Investigate this conjecture. Write a brief report of your findings.
3.1 Build Algebraic Models Using Concrete Materials • MHR 109
3.2
Work With Exponents
Suppose the trend in the cartoon continues: every day each new customer tells two new friends at school about the Barney Burger. How many new customers will Barney get each day?
Day 1
Day 2 friend
friend Sammy friend
friend friend friend
110 MHR • Chapter 3
Investigate How can you use exponential models to describe growth patterns? 1. Copy and complete the table. In the last column, write the number
of new customers as a power of 2. New Customers After Sammy Tells His Two Friends Day
New Customers
Expanded Form
Power
1
2
2
21
2
4
22
22
3 4
2. Barney’s is open 7 days a week. Use this model to determine how
many new customers Barney should expect on Day 7. Explain how you found your answer. 3. Use this model to determine how many new customers Barney
should expect on Day 14. Is this answer realistic? Why or why not? 4. Estimate the number of students at your school. How long would
it take for everyone at your school to find out about Barney’s? Describe how you found your answer, and identify any assumptions you made. 5. Suppose that each new customer told three friends about
Barney’s, instead of two, and that this trend continued. Use exponents to help explain your answers to the following. a) How many new customers should Barney expect after 2 days? b) How many new customers should Barney expect after 4 days? c) How much more quickly would word reach all the students
at your school? Explain. 6. Reflect Explain how exponents are useful in describing growth
patterns.
A power is a product of identical factors and consists of two parts: a base and an exponent. 24 base
is a power
exponent
The base is the identical factor, and the exponent tells how many factors there are. 24 2 2 2 2 exponential form
expanded form
Powers are useful for expressing repeated multiplication. 3.2 Work With Exponents • MHR 111
Example 1 Evaluate Powers Write in expanded form, and then evaluate. a) 25
b) (3)3
c) (3)4
d) 34
e) 3.53
f) a b
2 3
3
Solution a) 25 2 2 2 2 2
32
There is an odd number of negative factors. The answer is negative.
b) (3)3 (3) (3) (3)
27 There is an even number of negative factors. The answer is positive.
c) (3)4 (3) (3) (3) (3)
81 d) 34 (3 3 3 3)
The base of this power is 3, not 3. The negative sign in front makes the result negative.
81 e) 3.53 3.5 3.5 3.5
42.875 3
ç3.5 Y3 =
f) a b a ba ba b
2 3
2 3
2 3
2 3
222 333 8 27
To multiply fractions, I multiply numerators together and multiply denominators together.
Example 2 Apply Exponents to Solve Problems 10.4 cm
Popcorn
9.0 cm
9.2 cm
9.2 cm 9.2 cm
Mega-Box
Jumbo Drum
Which container holds more popcorn? How much more? Assume that each container is filled just to the top. Round your answer to the nearest cubic centimetre.
112 MHR • Chapter 3
Solution Mega-Box
Reasoning and Proving Representing
The Mega-Box is in the shape of a cube. Apply the formula for the volume of a cube.
Selecting Tools
Problem Solving Connecting
V s3 9.23 778.688
s is the side length of the cube.
Reflecting Communicating
ç 9.2 Y 3 =
The Mega-Box holds about 779 cm3 of popcorn. Jumbo Drum The Jumbo Drum is in the shape of a cylinder. Apply the formula for the volume of a cylinder. V r 2h (5.2)2(9.0) 764.54
r is the radius of the base and h is the height of the cylinder. The radius is half the diameter: 10.4 2 5.2 ç π * 5.2 x * 9 =
The Jumbo Drum holds about 765 cm3 of popcorn. The Mega-Box holds 14 cm3 more popcorn than the Jumbo Drum.
779 765 14
Key Concepts 䊏
Powers are a useful way to express repeated multiplication. For example, 4 4 4 43
䊏
A power consists of a base and an exponent, e.g., 43.
• The base is the identical factor. • The exponent tells how many factors there are. 䊏
Powers sometimes appear in formulas. When evaluating expressions involving powers, follow the correct order of operations.
Communicate Your Understanding C1
Identify the base and the exponent of each power. a) 34
C2
b) a b
1 2
4
c) (2)6
d) 26
e) 1.22
a) Evaluate each power in question 1. b) Explain why the answers to parts c) and d) are different.
3.2 Work With Exponents • MHR 113
C3
C4
The first step in evaluating the volume of a cylinder is to substitute the known values for r and h into the formula for the volume of a cylinder: V r 2h. Describe the next step.
r
h
Which expressions would you evaluate using a calculator? Explain. a) 23
b) (4)2
c) (1.25)4
d) 82
e) 76
f) (0.1)3
Practise For help with questions 1 to 5, see Example 1. 1. Which is 6 6 6 6 written as a power? A 64
B 64
C 46
D 1296
2. Which is 35 written in expanded form? A 35
B 555
C 33333
D 243
3. Write each expression as a power. a) (5) (5) (5) b) 1.05 1.05 1.05 1.05 1.05 1.05 c)
3 3 3 a b a b a b 5 5 5
4. Write each power in expanded form. Then, evaluate the expression. a) (4)3
3 4 a b 4
b) 0.82
c)
b) (7)2
c) 24
5. Evaluate. a) 93 d) a b
5 6
3
g) 18
e) a b
2 3
4
h) (1)55
f) 1.22 i) 0.53
6. Evaluate. Remember to use the correct order of operations.
114 MHR • Chapter 3
a) 25 42
b)
53 52
c)
13 16 12
d) (32 42) (34 43)
e)
2 3 3 2 a b a b 3 4
f)
500(1.08)5
7. Substitute the given values into each expression. Then, evaluate
the expression. Round your answers to one decimal place where necessary. s5
a) 6s2
Technology Tip
b)
r 2
c)
a2
d)
r 2h
r 2.3, h 5.2
e)
4 3 r 3
r 1.5
r 2.5
If your calculator does not have a π key, use 3.14 as an approximate value for .
a 3, b 4
b2
f) x 2 2x 24
x 6
Connect and Apply 8. a) Evaluate each power.
(2)2
(2)3
(2)4
(2)5
b) Examine the signs of your answers. What pattern do you notice? c) Explain how you can tell the sign of the answer when a power
has a negative base. Create and use examples of your own to illustrate your explanation. 9. Listeria is a type of bacteria that can cause dangerous
health problems. It doubles every hour. The initial population of a sample of Listeria is 800.
Time (min)
a) Copy and complete this table, which shows the
Population of Listeria
0
800
60
1600
120
population of Listeria over time.
180
b) Construct a graph of population versus time. Use a
smooth curve to connect the points. Describe the shape of the graph.
240
c) What will the population be after
• 1 day?
• 2 days?
d) The symptoms of food poisoning can start as quickly as 4 h after
eating contaminated food or as long as 24 h later. Discuss why some types of food poisoning begin quickly and others much more slowly.
3.2 Work With Exponents • MHR 115
Did You Know ? Food and water contaminated with E. coli (escherichia coli) can be very dangerous, but infections are easily treatable with certain antibiotics.
10. E. coli is a type of bacteria that lives in our intestines and is
necessary for digestion. It doubles in population every 20 min. The initial population is 10. a) Copy and complete the table. Refer to your table from question 9
to complete the second column. Time (min)
Population of Listeria
Population of E. Coli
0 20 40 60 80 100 120
b) When will the population of E. coli overtake the population of
Listeria? c) What population will the two cultures have when they are equal? 11. The durations (lengths of time) of musical notes are related by
powers of
1 , beginning with a whole note. Copy and complete 2
the table. Reasoning and Proving Representing
Selecting Tools
Problem Solving Connecting
Note
Symbol
Duration (in beats)
Power Form
whole
1
half
1 — 2
(—12)
1
quarter
1 — 4
(—12)
2
Reflecting Communicating
eighth
sixteenth
thirty-second
12. Refer to question 11. Look at the pattern in the last column. Extend
this pattern backward to write the power form for a whole note. Does this answer make sense? Use a calculator to evaluate this power. Describe what you observe.
116 MHR • Chapter 3
13. Chapter Problem Alysia has selected the letter E to design the logo
for her school team, the Eagles. The design will be used to make different-sized crests for clothing such as jackets, sweaters, and baseball caps. The height of the crest is twice the width. How can Alysia make sure that, when the crest is made larger or smaller, the proportions will not change? a) Find an expression for the area of the crest in terms of the width. b) Determine the area of a crest with a width of 8 cm. c) Determine the height of a crest with an area of 72 cm2. 14. Uranium is a radioactive material that emits energy when it changes
into another substance. Uranium comes in different forms, called isotopes. One isotope, U-235, has a half-life of 23 min, which means that it takes 23 min for a sample to decay to half its original amount. a) Suppose you started with a 100-mg sample of U-235. Copy and
complete the table. Number of Half-Life Periods
Time (min)
Amount of U-235 Remaining (mg)
0
0
100
1
23
50
2
46
3
Did You Know ? Expression
1 1 100 — 2
() 1 100(—) 2
2
Uranium is used as a fuel source in nuclear fission reactors, which provide 48% of Ontario’s electrical power. CANDU (Canada Deuterium Uranium) reactors are among the safest nuclear power generators in the world.
4
b) Construct a graph of the amount, in milligrams,
of U-235 remaining versus time, in minutes. Describe the shape of the graph. c) Approximately how much U-235 will remain
after 2 h? d) How long will it take until only 1 mg of U-235
remains? e) Use the pattern in the table to write an
1 , for the 2 original amount of U-235. Does this make sense? expression, using powers of
3.2 Work With Exponents • MHR 117
Literac
onnections
Scientific notation is a convenient way to write very large or very small numbers. In scientific notation, the value is expressed as the product of a number between 1 and 10 and a power of 10. For example, 56 000 000 000 5.6 1010 0.000 342 3.42 104 Scientific calculators express numbers in scientific notation when there are too many digits for the display. Multiply 1234 by 1000. Repeat until the output appears in scientific notation. You can enter a number in scientific notation into a scientific calculator. For example, to enter 5.6 1010, press EE 10 = ç 5.6 Í
Extend 15. Uranium-233 is another isotope that is used in nuclear power
generation. 1 kg of U-233 can provide about the same amount of electrical power as 3 000 000 kg of coal. This number can be written in scientific notation as 3 106. a) Another isotope of uranium, U-238, has a half-life of
4 500 000 000 years. Write this number in scientific notation. b) What is the half-life of U-238, in seconds? Write your answer
in scientific notation. c) The number 6.022 1023 is a very important number in
chemistry. It is called “one mole.” One mole is the amount of a substance that contains as many atoms, molecules, ions, or other elementary units as the number of atoms in 12 g of carbon-12. Carbon-12 is the basic building block of living things. Write one mole in standard notation. d) Describe any advantages you see to using scientific notation. 16. Refer to the cartoon at the beginning of the section. Suppose that
every new customer returns to Barney’s every day for lunch, in addition to recruiting two new customers. a) How many customers in total will Barney have
• 2 days after Sammy’s first visit? • 5 days after Sammy’s first visit? b) On which day will Barney’s reach 500 new customers for lunch?
To enter 3.42 104, press EE 4 \ = ç 3.42 Í
c) Write an expression that gives the total number of new lunch
customers n days after Sammy’s first visit. d) Describe any assumptions you must make in finding your
answers. Not all calculators show scientific notation, or let you enter such numbers, in exactly the same way. Experiment or refer to the user’s manual for your calculator.
17. Math Contest Determine the last digit of the number 31234 when
written in expanded form. Justify your answer. 18. Math Contest If 3x 729, the value of x is A 3
B 5
C 6
D 7
E 8
19. Math Contest Numbers are called perfect powers if they can be
written in the form x y for positive integer values of x and y. Find all perfect powers less than 1000. 20. Math Contest x x is always greater than y y as long as x > y. For what
whole-number values of x and y is x y > y x ? Justify your answer.
118 MHR • Chapter 3
3.3
Discover the Exponent Laws The 100-m dash is one of the most exciting events in track and field. If you ran this race, how many centimetres would you run? How many millimetres is this?
Investigate A How can you simplify expressions involving products and quotients of powers? Part A: Patterns Involving Powers of 10
In the metric system, length measures are related by powers of 10. For example, there are 10 mm in 1 cm. This makes it easy to convert one unit of length to another. Note that hectometres and decametres are uncommon units. Did You Know ? A Canadian named Donovan Bailey set the men’s world record for the 100-m dash at the 1996 Olympics. He ran the race in 9.84 s, and was considered at that time to be the world’s fastest human.
kilometre (km) 1000 m
To convert to a smaller unit, move the decimal point to the right.
hectometre (hm) 100 m decametre (dam) 10 m metre (m) decimetre (dm) 0.1 m
centimetre (cm) 0.01 m
To convert to a larger unit, move the decimal point to the left.
millimetre (mm) 0.001 m
1. How many metres are in 1 km? Write I can use the metric ladder to help. I count the steps on the ladder to find the exponent.
this as a power of 10.
kilometre (km) 1000 m
hectometre (hm) 100 m decametre (dam) 10 m metre (m) decimetre (dm) 0.1 m centimetre (cm) 0.01 m millimetre (mm) 0.001 m
2. Copy and complete the table. Unit decimetre centimetre
Number of these in 1 m
Power of 10
10
101
100
millimetre
3.3 Discover the Exponent Laws • MHR 119
3. a) Multiply the number of centimetres in 1 m by the number
of metres in 1 km. What does this answer give you? b) Write the product in part a) using powers of 10. Write the
answer as a power of 10. 4. Repeat step 3 for millimetres instead of centimetres. 5. Reflect Look at the exponents in the powers of 10 in your answers
to steps 3b) and 4b). Describe how these numbers are related. Part B: Products of Powers How can you simplify expressions containing products of powers with the same base? 6. Copy and complete the table, including an example of your own. Quotient
Expanded Form
Single Power
32 34
(3 3) (3 3 3 3)
36
43 43 64 61 24 22 23 k3 k5 Create your own example
7. What do you notice about the bases of the powers in each product
in the first column? 8. Look at the exponents in the first column for each product. How
does the sum of the exponents compare to the exponent in the last column? 9. Reflect Explain how you can write a product of powers using a
single power. Use your example to illustrate your explanation. 10. Write a rule for finding the product of powers by copying and
completing the equation xa xb 䊏. Part C: Quotients of Powers How can you simplify expressions containing quotients of powers with the same base? 11. Copy and complete the table, including an example of your own.
I can reduce common factors.
Quotient
Expanded Form
Single Power
55 53
55555 555
52
74 71 106 104 27 26 p8 p5 Create your own example
120 MHR • Chapter 3
12. What do you notice about the bases of the powers in each quotient
in the first column? 13. Look at the exponents in the first column for each quotient. How
do they relate to the exponent of the single power in the last column? 14. Reflect Explain how you can write a quotient of powers using a
single power. Use your example to illustrate your explanation. 15. Write a rule for finding the quotient of powers by copying and
completing the equation x a x b 䊏 .
The patterns in the activity above illustrate two exponent laws. The exponent laws are a set of rules that allow you to simplify expressions involving powers with the same base. Product Rule When multiplying powers with the same base, add the exponents to write the product as a single power: xa xb xa b Quotient Rule When dividing powers with the same base, subtract the exponents to write the quotient as a single power: xa xb xa b
Example 1 Apply the Product Rule Write each product as a single power. Then, evaluate the power. a) 32 33
b) 52 5 52
c) (2)4 (2)3
d) a b a b
1 2
3
1 2
2
e) 0.14 0.12
Solution a)
32 33 32 3 35 243
The bases are the same, so I can add the exponents. ç3 Y5=
3.3 Discover the Exponent Laws • MHR 121
b)
c)
52 5 52 52 51 52 55 3125
When no exponent appears, I know that it is 1. 52 5 52 52 51 52 Now, I can add the exponents: 2125
(2)4 (2)3 (2)4 3 (2)7 128 1 3 1 2 a b a b 2 2
d)
1 32 a b 2 1 5 a b 2 1 1 1 1 1 a ba ba ba ba b 2 2 2 2 2 11111 22222 1 32
e)
0.14 0.12 0.14 2 0.16 0.000 001
To find the sixth power of 0.1, I need to multiply 0.1 by itself six times. In the product, there will be six digits after the decimal point.
Example 2 Apply the Quotient Rule Write each product as a single power. Then, evaluate the power. a) 87 85 b) 47 4 43 3 3 3 2 a b a b (0.5)6 4 4 c) d) (0.5)3 3 5 a b 4
Solution a)
122 MHR • Chapter 3
87 85 87 5 82 64
The bases are the same, so I can subtract the exponents.
b)
c)
47 4 43 47 41 43 47 1 43 46 43 46 3 43 64
Divide in order from left to right.
(0.5)6 (0.5)3 (0.5)6 3 (0.5)3 0.125
d)
3 3 3 2 a b a b 4 4 3 5 a b 4 3 32 a b 4 3 5 a b 4 3 5 a b 4 3 5 a b 4
Apply the product rule first to simplify the numerator. What if I use the quotient rule? 3 —) (————— 3 4 — ( ) 3 (—4) 4 3 (—) 4 5
Anything divided by itself equals 1.
5—5
5
0
I know the answer is 1. I wonder if an exponent of 0 always gives an answer of 1?
1
Investigate B How can you simplify expressions involving powers of powers? 1. Copy and complete the table, including an example of
your own. Power of a Power
Expanded Form
Single Power
(22)3
(22) (22) (22) (2 2) (2 2) (2 2)
26
(53)4 (104)2 Create your own example
3.3 Discover the Exponent Laws • MHR 123
2. Look at the exponents in the first column for each case. How do
they relate to the exponent of the single power in the last column? 3. Reflect Explain how you can write a power of a power using
a single power. Use your example to illustrate your explanation. 4. Write a rule for finding the power of a power by copying and
completing the equation (xa)b 䊏.
The patterns in Investigate B illustrate another exponent law. Power of a Power Rule A power of a power can be written as a single power by multiplying the exponents. (xa)b xa b
Example 3 Apply the Power of a Power Rule Write each as a single power. Then, evaluate the power. a) (32)4 b) [(2)3]4 2 2
c) c a b d
2 3
d) (0.23)2
Solution a) (32)4 32 4
b) [(2)3]4 (2)3 4
6561
(2)12 4096
38
2 2
c) c a b d
2 3
2 2:2 a b 3 2 4 a b 3
124 MHR • Chapter 3
2 2 2 2 3 3 3 3
16 81
d) (0.23)2 0.23 2
0.26 0.000 064
Example 4 Simplify Algebraic Expressions Simplify each algebraic expression by applying the exponent laws. a) y 3 y 5
b) 6p7 3p3
c) a2b3 a6b4
d)
2uv 3 8u2v 2 (4uv 2)2 ˛˛˛
˛˛˛
˛˛˛
Solution a) y 3 y 5 y 3 + 5
Apply the product rule.
y8 b) 6p7 3p3 2p 7 3
Divide the numeric factors, 6 3 = 2.
2p4 c) a2b3 a6b1 a8b4
2uv 3 8u2v 2 (4uv 2)2 ˛˛˛
d)
Simplify numerator and denominator first. The exponent in the denominator applies to all the factors inside the brackets.
˛˛˛
˛˛˛
Exponent laws only apply to powers with the same base. • First, I add exponents of a: a2 6 a8 • Then, I add exponents of b : b3 1 b4
(2) 8 u u2 v 3 v 2 ˛˛
˛˛˛
42u2(v 2)2 ˛˛˛
16 u1 2 v 3 2 16u2v 2 2
16u v 16u2v 4
˛˛
˛˛˛˛
3 5 ˛˛
Divide.
˛˛
u3 2v 5 4 uv
Key Concepts 䊏
The exponent laws are a way to simplify expressions involving powers with the same base.
䊏
When multiplying powers with the same base, add the exponents: xa x b xa b
䊏
When dividing powers with the same base, subtract the exponents: xa x b xa b
䊏
When finding the power of a power, multiply the exponents: (xa)b xa b
3.3 Discover the Exponent Laws • MHR 125
Communicate Your Understanding C1
Identify which exponent law you can apply to simplify each expression. If no exponent law can be used, explain why not. a) 63 62 b) (m2)3 c) 34 43 d) a2b a3b4 e)
f)
p3q2 pq u4v 5 w 2x 3 ˛˛
˛˛˛
C2
˛˛
Create an example involving powers where you can a) add exponents b) multiply exponents c) subtract exponents
C3
Look at part d) of Example 4. Suppose that u 3 and v 2: Original expression
Simplified expression
2uv 8u v (4uv 2)2
uv
3
2 2
˛˛
˛˛˛
˛˛˛
a) Which expression would you rather substitute into to evaluate
the expression, and why? b) What is the value of the expression after substituting the given
values?
Practise For help with questions 1 and 2, see Example 1. 1. Which is 73 72 expressed as a single power? A 76
B 75
C 79
D 496
2. Apply the product rule to write each as a single power.
Then, evaluate the expression.
126 MHR • Chapter 3
a) 34 37
b) 24 2 23
c) (1)5 (1)6
d) a b a b
2 5
3
2 5
3
For help with questions 3 and 4, see Example 2. 3. Which is 117 115 expressed as a single power? A 1112
B
111.4
12
C
112
D
Did You Know ?
4. Apply the quotient rule to write each as a single power.
Then, evaluate the expression. a) 128 122 c)
3 4 3 a b a b 4 4
b) (6)5 (6)2 (6)2 d)
0.16 0.14 0.12
For help with questions 5 and 6, see Example 3. 5. Which is (54)2 expressed as a single power? A 58 C 256
B 56 D 258
6. Apply the power of a power rule to write each as a single power.
The notation that we use for powers, with a raised number for the exponent, was invented by Réné Descartes (1596-1650). Descartes used this notation in his text Géométrie, published in 1637. In this famous text, Descartes connected algebra and geometry, starting the branch of mathematics called Cartesian geometry.
Then, evaluate the expression. a) (42)2
b) [(3)3]2
c) [(0.1)4]2
d) c a b d
3 2
3 2
7. Simplify using the exponent laws. Then, evaluate. a) 52 53 54
b) 37 35 3
3 4
c)
(0.5 )
0.56 0.54
d) (2)4 (2)5 [(2)3]3
For help with questions 8 and 9, see Example 4. 8. Simplify. a) y 4 y 2 d) (c3)4 g) m2n mn2
b) m8 m5 e) a2b2 a3b h) h2k3 hk
c) k 2 k 3 k 5 f) (2uv2)3 i) (a3b)2
b) 8a5 (2a3)2
c) (x2)3 (3x2)2
Connect and Apply 9. Simplify. a) 12k 2m8 4km5 d)
g)
4d4w3 6dw4 3d3w 8dw2
e)
5c 3d 4c2d2 (2c 2d)2
h)
˛˛˛
˛˛˛˛˛
˛
3f 4g3 8fg4 ˛˛˛
(6f 2g3)2 ˛˛˛˛˛
f) (3a2b)2 (ab)2
˛
(3xy2)3 (4x2y) 2 2 2
(2x y ) ˛˛˛˛˛
˛
30g 2h (2gh)2 ˛˛˛
i)
5gh 2 6gh ˛˛˛˛˛
3.3 Discover the Exponent Laws • MHR 127
5xy 2 2x 2y ˛˛˛
10. Consider the expression
˛˛
2
(2xy)
.
a) Substitute x 3 and y 1 into the expression. Then, evaluate
the expression. b) Simplify the original expression using the exponent laws.
Then, substitute the given values and evaluate the expression. c) Describe the advantages and disadvantages of each method.
Literac
11. A crawlspace in a space station has the shape of a rectangular prism.
onnections
Rectangular prism is the mathematical name for a box.
It is about 100 cm high, 10 m wide, and 1 km long. What is the volume enclosed by the crawlspace? 1 , because 2 it is one of two possible outcomes. The probability of tossing three 1 3 1 heads in a row is a b or . 2 8
12. The probability of tossing heads with a standard coin is
a) What is the probability of tossing
• six heads in a row? • 12 heads in a row? b) Write each answer in part a) as a power of a power. 13. a) What is the probability of rolling a 6 with a standard
number cube?
2 3 6
b) What is the probability of rolling four 6s in a row? c) What is the probability of rolling a perfect square with a
number cube? d) What is the probability of rolling eight perfect squares in a row? e) Write each answer in parts b) and d) as a power of a power.
Achievement Check 14. Consider the expression
3m2n 4m3n2 . (2m2n)2 3mn
a) Substitute m 4 and n 3 into the expression and evaluate it. b) Simplify the original expression using the exponent laws. c) Substitute m 4 and n 3 into the simplified expression and
evaluate it. d) What did you notice? What are the advantages and disadvantages
of the two methods? e) Josie made two errors in copying the above expression. She wrote
3m2n 4mn2 , but she still got the correct answer. (2mn)2 3mn Explain how this is possible.
128 MHR • Chapter 3
Extend 15. You can multiply and divide numbers in scientific notation by
applying the exponent laws. For example, (3 105) (2 104) 3 2 105 104 6 109
(9 108) (3 105) 9 108 3 105 3
9 108 3 105 1
3 103 Evaluate each of the following. Express each answer in scientific notation and then in standard notation. a) 4 102 2 103
b) 1.5 104 6 106
c) (8 107) (2 105)
d) (3.9 1012) (3 108)
16. a) Predict the screen output of your scientific or graphing calculator
when you enter the following calculation: (3 105) (2 106). b) Is the answer what you predicted? Explain the answer that the
calculator has provided. 17. a) Predict the screen output of your scientific or graphing calculator
when you enter the following calculation: (3 1018) (6 102). b) Is the answer what you predicted? Explain the answer that the
calculator has provided. 18. a) Evaluate (2 105)3. Express your answer in both scientific and
standard notation. b) Explain how you can evaluate a power of a number expressed in
scientific notation. Create an example of your own to help illustrate your explanation. 19. Math Contest Copy and complete the table to make the square a
multiplicative magic square (the product of every row, column, and diagonal is equal). a19b16 a9b6
a15b12 a13b10 a21b18
a7b4
20. Math Contest If x
1 , place the following values in order from 9
least to greatest: 1 x, x2, x3, 2x , x 3.3 Discover the Exponent Laws • MHR 129
3.4
Communicate With Algebra
You have seen how algebra tiles can be used to model algebraic expressions. The model below shows that Petra mowed a square lawn of unknown area once a week for 4 weeks.
The algebraic representation is 4x2. Look at each part of this expression. What does the 4 represent? What does the x2 represent? term 䊏
an expression formed by the product of numbers and/or variables
The expression 4x2 is called a term . A term consists of two parts: 4 coefficient
x2 variable
When you represent an algebraic expression using algebra tiles, the variable in the expression tells you which type of tile to use. To represent x2, use an x2-tile. x 2-tile
The number of tiles corresponds to the coefficient. Since the expression is 4x2, there are four x2-tiles.
130 MHR • Chapter 3
Example 1 Identify Coefficients and Variables
Literac
Identify the coefficient and the variable of each term.
The coefficient is also called the numerical coefficient. It is a number only.
a) Jim earns $7 per hour at his part-time job. If he works for x hours,
his earnings, in dollars, are 7x. b) The depth, in metres, of a falling stone in a well after t seconds is
4.9t2. c) The area of a triangle with base b and height h is
1 bh. 2
d) The area of a square with side length k is k2. e) Amir walks 6 blocks to school.
Solution Expression
Coefficient
Variable
a)
7x
7
x
b)
4.9t 2
4.9
t2
The negative sign is included with the coefficient.
c)
1 —bh 2
1 — 2
bh
The variable can consist of more than one letter or symbol.
d)
k2
1
k2
When the coefficient is not shown, it is 1.
e)
6
6
none
onnections
The variable is also called the literal coefficient. It consists of one or more variables and their exponents, if they exist. Exponents on the variables belong to the literal coefficient, because they represent a product of variables: x 2 x x.
Comments
A term with no variable is called a constant term, or simply a constant.
polynomial 䊏
A polynomial can be classified by the number of terms it has. Type of Polynomial
Number of Terms
Examples
monomial
1
x, 3y, 4a 2, 5
binomial
2
2x 3, ab 2a,
trinomial
3
2x 2
0.4x 2
an algebraic expression consisting of one or more terms connected by addition or subtraction operators
Literac
onnections
The prefixes of the polynomial names have the following meanings: • mono means 1 • bi means 2 • tri means 3
x
3x 1, a 2b c
Example 2 Classify Polynomials by Name Classify each polynomial by the number of terms it has. a) 3x2 2x
b) 2m
c) 4x2 3xy y 2
d) a 2b c 3
3x 2 2x has two terms, 3x 2 and 2x.
Solution Polynomial
Number of Terms
Type of Polynomial
a)
3x 2 2x
2
binomial
b)
2m
1
monomial
c)
4x 2 3xy y 2
3
trinomial
d)
a 2b c 3
4
four-term polynomial
I can find the number of terms by looking for the addition and subtraction operators that separate the terms: 4x 2 3xy y 2 Two operators separate the three terms in this trinomial.
3.4 Communicate With Algebra • MHR 131
Example 3 Classify Terms by Degree degree of a term 䊏
the sum of the exponents on the variables in a term
Find the degree of each term . a) x2
b) 3y 4
c) 0.7u
d) 2a2b
2 e) xy 3
f) 5
Solution Look at the exponents of the variables. Add them if there is more than one. Remember, when no exponent appears on a variable, the value of the exponent is 1. For example, 4u 4u1.
If the term has no variable at all, then the degree is 0.
Term
Sum of Exponents
Degree
a)
x2
2
2
b)
3y 4
4
4
c)
0.7u1
1
1
d)
2a 2b1
213
3
e)
2 —x 1y 1 3
112
2
f)
5
0
0
Example 4 Classify Polynomials by Degree degree of a polynomial 䊏
the degree of the highest-degree term
Find the degree of each polynomial . a) x 3 b) 5x 2 2x c) 3y 3 0.2y 1 d) 7x2y 4 x 6y
Solution The degree of the first term is 1. The degree of the second term is 0. The highest degree is 1.
The degree of the first term is 2 4. The degree of the second term is 6 1.
132 MHR • Chapter 3
Polynomial
Term With Highest Degree
Degree of Term in Column 2
Degree of Polynomial
a)
x3
x
1
first
b)
5x 2 2x
5x 2
2
second
c)
3y 3 0.2y 1
3y 3
3
third
d)
7x 2y 4 x 6y
x 6y
7
seventh
Example 5 Use an Algebraic Model to Solve a Problem Cheryl works part-time as a ski instructor. She earns $125 for the season, plus $20 for each children’s lesson and $30 for each adult lesson that she gives. a) Write an expression that describes Cheryl’s total
earnings for the season. Identify the variable and the coefficient of each term and explain what they mean. b) One winter, Cheryl gave eight children’s lessons and
six adult lessons. What were her total earnings?
Solution a) Cheryl’s total earnings can be described by the
polynomial expression 20c 30a 125. Term
Variable
Meaning of Coefficient
20c
c represents the number of children’s lessons.
20 represents the earnings per children’s lesson.
30a
a represents the number of adult lessons.
30 represents the earnings per adult lesson.
125
There is no variable.
Cheryl has fixed earnings of $125 per season.
b) Substitute c 8 and a 6, and evaluate the expression.
20c 30a 125 20(8) 30(6) 125 160 180 125 465 Cheryl’s total earnings for this season were $465.
Key Concepts 䊏
Algebraic expressions can be used to communicate mathematical ideas.
䊏
A term is the product of a coefficient and variable part.
䊏
A polynomial can be a single term or a combination of terms using addition or subtraction operators.
䊏
A polynomials can be classified
• by the number of terms it has • by its degree
3.4 Communicate With Algebra • MHR 133
Communicate Your Understanding C1
Create two examples of each. a) monomial
b) binomial
c) trinomial
d) four-term polynomial
C2
Julio says that the term x2 has a coefficient of 2 and a variable x. Is Julio correct? Explain.
C3
a) Are these expressions equivalent? Explain.
2w 1t
2w t
b) Are these expressions equivalent? Explain.
3x 1
3x
c) Explain when you must write the number 1, and when you
do not need to.
Practise For help with question 1, see Example 1. 1. Identify the coefficient and the variable part of each term. a) 2y d)
1 2 x 2
b) 3x
c) mn
e) w 2
f) 0.4gh3
For help with questions 2 and 3, see Example 2. 2. 7x2 3xy 4y 2 is a A monomial
B binomial
C trinomial
D term
3. Classify each polynomial by the number of terms. a) 2x
b) 6y 2 2y 1
1 b 2 1 e) 3k2 k 2 c) a
d) 3u2 uv 2v 2 f) m 0.2n 0.3 mn
For help with questions 4 and 5, see Example 3. 4. The degree of 4u 5u2 9 is A 1
B 2
C 3
D 0
5. State the degree of each term.
134 MHR • Chapter 3
a) 5x2
b) 6y
d) u2v 4
e)
1 2 3 xy 3
c) 3 f) 0.2a2b
For help with question 6, see Example 4. 6. State the degree of each polynomial. a) 3x 4
b) y2 3y 1
d) a3b2 8a2b5
e) 2x2y 4
c) m 2m3
2 3 xy 5
For help with questions 7 and 8, see Example 4. 7. In a TV trivia show, a contestant receives 500 points for a correct
answer and loses 200 points for an incorrect answer. Let c represent the number of correct answers and i represent the number of incorrect answers. Which expression describes a contestant’s total points? A 500c 200i
B 500c 200i
C 500i 200c
D 500i 200c
8. A hockey team earns 2 points for a win and 1 point for a tie. Let
w represent the number of wins and t represent the number of ties. a) Which expression can be used to describe the total points? A 2w 1
B wt
C 2w 1t
D 2w t
b) Is there more than one correct answer? Justify your answer. 9. Substitute the given values and evaluate each expression. a) 3x 5
x2
b) 4y 4
y 2
2b 7
c)
a2
d)
2m2
3n 8
a 4, b 1 m 2, n 5
Connect and Apply 10. The students at Prince Albert Public School sell magazine
subscriptions to raise money. The school receives 37% of the money paid for the subscriptions. a) Choose a variable to represent the money paid for the
subscriptions. b) Using your variable from part a), write an expression for the
amount of money the school will receive. c) Tyler sold one magazine subscription to his aunt for $25.99.
Calculate the amount the school receives on this sale. d) The sum of all the subscription orders was $4257.49. Use your
formula to calculate how much the school will receive for this fundraiser.
3.4 Communicate With Algebra • MHR 135
11. Meredith has a summer job at a fitness club. She earns a $5 bonus
for each student membership and a $7 bonus for each adult membership she sells. a) Write a polynomial expression that describes Meredith’s total
bonus. b) Identify the variable and the coefficient of each term and explain
what they mean. c) How much will Meredith’s bonus be if she sells 12 student
memberships and 10 adult memberships? 12. An arena charges $25 for gold seats, $18 for red seats, and $15 for
blue seats. a) Write an expression that describes the total earnings from seat
sales. b) Identify the variable and the coefficient of each term and explain
what they mean. c) How much will the arena earn if it sells 100 gold seats, 200 blue
seats, and 250 red seats? 13. On a multiple-choice test, you earn 2 points for
each correct answer and lose 1 point for each incorrect answer. a) Write an expression for a student’s total score. b) Maria answered 15 questions correctly and 3 incorrectly.
Find Maria’s total score. 14. a) Describe a situation that can be modelled by an algebraic
expression. b) Select variables and write the expression. c) Illustrate your expression using algebra tiles or a diagram. 15. Write a response to this e-mail from a classmate. FROM: TO:
ManuelS JillP
SUBJECT: changing a decimal to a percent Hi Jill, Too bad I missed today’s class. I copied your notes and I get most of it, I gues s. I’m just a little confused about the differenc e between a term and a polynomial. Can you please help me out? Thanks, Manuel
136 MHR • Chapter 3
16. Chapter Problem Alysia is designing a logo for her school team, the
Eagles. The design will be used to make different-sized crests for clothing such as jackets, sweaters, and baseball caps. How can Alysia make sure that, when the crest is made larger or smaller, the shape will not change? The height will always be double the width. a) If w represents the width, what expression represents the height? b) How high will a crest that is 5 cm wide be? c) How wide will a crest that is 25 cm high be?
Achievement Check 17. In a soccer league, teams receive 3 points for a win, 2 points for a
loss, and 1 point for a tie. a) Write an algebraic expression to represent a team’s total points. b) What variables did you choose? Identify what each variable
represents. c) The Falcons’ record for the season was 5 wins, 2 losses, and
3 ties. Use your expression to find the Falcons’ total points. d) The 10-game season ended with the Falcons tied for second place
with the same number of points as the Eagles. The Eagles had a different record than the Falcons. How is this possible?
Extend 18. Alberto is training for a triathlon, where athletes swim, cycle, and
run. During his training program, he has found that he can swim at 1.2 km/h, cycle at 25 km/h, and run at 10 km/h. To estimate his time for an upcoming race, Alberto rearranges the formula distance distance speed time to find that: time . speed a) Choose a variable to represent the distance travelled for each
part of the race. For example, choose s for the swim. b) Copy and complete the table. The first row is done for you. Part of the Race
Speed (km/h)
Distance (km)
Time (h)
swim
1.2
s
s ——— 1.2
cycle run
c) Write a trinomial to model Alberto’s total time. d) A triathlon is advertised in Kingston. Participants have to swim
1.5 km, cycle 40 km, and run 10 km. Using your expression from part c), calculate how long it will take Alberto to finish the race. e) Is your answer a reasonable estimate of Alberto’s triathlon time?
Explain. 3.4 Communicate With Algebra • MHR 137
19. Ashleigh can walk 2 m/s and swim I can use the formula distance ————— to time —— speed calculate Ashleigh’s travel time.
1 m/s. What is the quickest way for Ashleigh to get from one corner of her pool to the opposite corner?
10 m
25 m
a) Predict whether it is faster for
Ashleigh to walk or swim. b) Ashleigh can walk at a speed of 2 m/s. The time, in seconds, for
w , where w is the distance, in metres, she 2 walks. Use this relationship to find the travel time if Ashleigh walks around the pool. Ashleigh to walk is
Path 1: Walk the entire distance. 10 m 25 m
c) Write a similar expression to represent the time taken for
Ashleigh to swim a distance s. Her swimming speed is 1 m/s. Use this relationship to find the travel time if Ashleigh swims straight across. Path 2: Swim the entire distance.
d) Which route is faster, and by how much? 20. Refer to question 19. a) Do you think it will be faster for Ashleigh to walk half the length
and then swim? Explain your reasoning. Did You Know ? Sixteen year-old Marilyn Bell of Toronto was the first person to swim across Lake Ontario. In September 1954 Marilyn swam from Youngstown, NY, to the CNE grounds in Toronto. The 51.5 km distance took her 21 h.
138 MHR • Chapter 3
Path 3: Walk half the length, then swim.
b) Find the travel time for this path. Compare this with your
answers to question 19. c) Do you think this is the fastest possible path? Find the fastest
path and the minimum time required to cross the pool, corner to opposite corner. Describe how you solved this.
21. Some algebraic expressions involve more than one variable. You can
model these using an expanded set of algebra tiles.
unit tile
x-tile
x2
y-tile
y 2-tile
Fly By Night Aero Insurance company charges $500 for liability, plus 10% of the value of the plane, plus $300 per seat. Let v represent the value of the plane and s represent the number of seats. The cost of the insurance is modelled by C 500 0.1v 300s. A four-seat Piper Cherokee valued at $30 000 would cost 500 0.1(30 000) 300(4) or $4700 per year to insure. a) Explain how you would use tiles to
represent 0.1v. b) Explain how you would use tiles to
represent 300s. c) Build an algebraic model to describe
the cost for airplane insurance for the Piper Cherokee, using algebra tiles, diagrams, or virtual algebra tiles. d) Find the cost of insurance for a
50-seat plane, valued at $500 000. 22. Math Contest If 3m 5 23
and 2n 7 21, the value of 3m 2n is A 20
B 44
C 46
D 54
E 56
23. Math Contest If ax b y cz 18 144 and a, b, and c are all prime
numbers, the value of x y z is A 7
B 10
D 12
E 20
C 11
24. Math Contest Find value(s) of m for which a b
1 2
2m
1 m a b . Is there 4
more than one possible value of m? Explain.
3.4 Communicate With Algebra • MHR 139
Use Technology
Virtual Algebra Tiles With The Geometer’s Sketchpad® You can create and manipulate virtual algebra tiles using computer software such as The Geometer’s Sketchpad®. You can build algebraic models, plus you can change the length of the variable tiles.
Investigate Tools 䊏
䊏
computer equipped with The Geometer’s Sketchpad® Algebra Tiles.gsp
How can you build algebraic models using virtual algebra tiles? 1. a) Start The Geometer’s Sketchpad® and open the sketch
Algebra Tiles.gsp. b) Read the instructions and click on the Algebra Tiles page button. 2. Explore the pre-made tiles. a) Click on the Show example tiles button.
b) Click and drag the top of the x-slider. Which tiles are affected
by the x-slider? c) Repeat part b) for the y-slider. d) The unit tile seems to have a slider next to the x-slider. Try to
change its length, and describe what happens. Why can you not change the dimensions of the unit tile? 140 MHR • Chapter 3
For the rest of this activity, you will work only with • unit tiles
• x-tiles (horizontal and vertical)
• x2-tiles
3.
Clear the workspace and bring out only the algebra tiles you need: • Click on Hide example tiles.
Technology Tip
• Click and hold the Custom Tool icon at the left side of the screen.
If you do not see the list of tools when you click on the Custom Tool icon, try holding the left mouse button for a couple of seconds, and they will appear.
Technology Tip
• Select 1 (the unit tile) and place a unit tile somewhere on the workspace. • Click and hold the Custom Tool again and select x (horizontal) and place a tile on the workspace.
You can add more tiles by using the Custom Tool or by using the Copy and Paste commands in the Edit menu. To copy an entire tile, use the Selection Arrow Tool to click and drag a dashed box around the tile you want to copy. Then, paste it and move it wherever you like.
• Repeat for the x (vertical) and x^2 (x2) tiles.
Use Technology: Virtual Algebra Tiles With The Geometer’s Sketchpad® • MHR 141
Use Technology
Once you have all four tiles, choose the Selection Arrow Tool and click somewhere in the white space to deselect the last object you created.
4. Explore the relationships between the tiles. Arrange the tiles as
shown, by clicking and dragging them one at a time.
a) Move the x-slider and describe what happens to the length and
width of each tile. Tile
Length (changes/does not change)
Width (changes/does not change)
unit x (horizontal) x (vertical) x2
b) Describe how the length of the x-tile is related to the length and
width of the x2-tile. Why is this so? c) Describe how the width of the x-tile is related to the length and
width of the unit tile. Why is this so? 142 MHR • Chapter 3
5. a) Why is it that you can change the length of an x-tile, but not a
unit tile? b) What advantage does this give to virtual algebra tiles over
physical algebra tiles? 6. Build each algebraic model using virtual algebra tiles. a) 3x 2
b)
x 2 2x 5
7. Build each algebraic model and try to put the parts together to form a
rectangle. You may have to change horizontal x-tiles to vertical ones, or vice versa. Once you have built the rectangle, write expressions for its length and width. The first one is shown as an example. a) x 2 5x 6
The length is x 3 and the width is x 2. Now try these. b) x 2 2x 1 c) x 2 7x 12 8. a) Create your own example like the ones in step 7. Not all
expressions will work. Make sure that you can create a rectangle to model your expression. b) Trade expressions with a classmate and try to build each other’s
algebraic models. 9. Describe at least one advantage and one disadvantage of using
virtual algebra tiles to build algebraic models. Use Technology: Virtual Algebra Tiles With The Geometer’s Sketchpad® • MHR 143
3.5
Collect Like Terms
Recreational vehicles can be a lot of fun. However, they sometimes require costly repairs. Suppose you are faced with a repair bill involving parts and labour charges. How can you use algebra to simplify the information?
Example 1 Build and Simplify an Algebra Tile Model Crystal bought a used personal watercraft with her summer earnings. Unfortunately, it needs some repairs before she can use it. a) Model each repair using
algebra tiles and write an expression to describe it. b) Find a simplified expression
Item
Alternator Fuel Pump
Cost for Parts($) Labour (h) 200 100
4 3
for both repairs. c) Calculate the total repair cost
if the mechanic charges $50/h for labour.
Solution a) Let each unit tile represent $100 and each x-tile represent an
Makin
onnections
You will study relations such as this in 5.2 Partial Variation.
Literac
hour’s labour charge. Alternator
Fuel Pump
200 4x expresses the cost to repair the alternator.
100 3x expresses the cost to repair the fuel pump.
b) Find a simplified expression for
onnections
To simplify an expression is to find an equivalent, simpler, and shorter expression. Original expression: 200 4x 100 3x Simplified expression: 300 7x
144 MHR • Chapter 3
both repairs by adding these expressions.
200 + 4x
100 + 3x
300 + 7x
c) The mechanic charges $50/h. Substitute 50 into the expression
and evaluate it. 300 7x 300 7(50) 300 350 650 The total repair cost for Crystal’s personal watercraft is $650.
In Example 1 you added fixed costs—the cost of the parts: 200 100 300 You also added variable costs—the cost of the labour: 4x 3x 7x When you add all the fixed costs, or add all the variable costs, you are adding like terms .
like terms 䊏
Example 2 Identify Like and Unlike Terms
terms that have identical variables
Identify the like terms in each group. a) 2x, 2, 5x, 3xy, 3x b) 4a2, 2a, a2, a3, a2b c) 3u2, 4uv, 6, 2u2,
1 v, 5 2
Solution a) Like terms have identical variable parts.
2x , 2, 5x , 3xy, 3x 2x, 5x, and 3x all have x as the variable. They are like terms.
2x and 2 are unlike terms because the second term has no variable. 2x and 3xy are unlike terms because the second term includes a y.
b) 4a2 , 2a, –a2 , a3, a2b
4a2 and a2 have identical variables: a2. They are like terms. The other terms all have different variables. 1 v, –5 2 3u2 and 2u2 are like terms. 6 and 5 are also like terms.
The exponents on all variables must be identical for terms to be like.
c) 3u2 , 4uv, 6 , 2u2 ,
1 v and 4uv are unlike any of the other terms. 2
3.5 Collect Like Terms • MHR 145
To simplify a polynomial expression, add or subtract like terms.
Example 3 Add Like Terms Add. a) 4x 3x b) 4x 3 2x 5
Solution a)
4x 3x 7x
b)
4x 3 2x 5 4x 2x 3 5 6x 8 6x 8
Example 4 Subtract Like Terms Subtract. a) 8x 3x b) 2x 5x
Solution a) 8x 3x Method 1: Take Away
Start with 8 x-tiles. Remove 3 of them. 8x 3x 5x
146 MHR • Chapter 3
Method 2: Add the Opposite
Start with 8 x-tiles. Add 3 negative x-tiles. I use a different colour of tile to represent negative values.
Remove 3 zero pairs.
3x
⫺3x
3x and 3x are opposites. Their sum is 0, leaving 5x as the result.
8x 3x 5x
b) Start with 2x-tiles.
You cannot remove 5 x-tiles, because there are only 2 of them. You must add 5 x-tiles.
Remove 2 zero pairs. 2x 5x 3x
Adding and subtracting to simplify algebraic expressions is called collecting like terms.
3.5 Collect Like Terms • MHR 147
Example 5 Collect Like Terms With Concrete Materials Use algebra tiles to simplify each expression. a) 5k 3k 6 3 b) 3c2 c 2 5c 4c2
Solution a)
5k 3k 6 3
5k 3k 6 3
I can rearrange the tiles to group zero pairs, then remove them.
2k 3 b) 3c2 c 2 5c 4c2
Rearrange the tiles to group like terms.
148 MHR • Chapter 3
Remove zero pairs.
3c2 4c2 c 5c 2
When the coefficient is 1 or 1, you can omit the 1. But you must keep the minus sign. 3c 2 4c 2 1c 2 c 2
Algebraic expressions are usually written in order of degree, with the greatest-degree term first and the least-degree term last.
c2 4c 2
Example 6: Collect Like Terms Without Concrete Materials Simplify. a) 6x 4 8x 3 b) 2b b 7 8 c) 3r 2 2 6r 9r 3r 2 d) 0.5m 4.5n 0.7m 4.5n 1.5
Solution When collecting like terms, • group or identify like terms • add or subtract like terms only • apply integer rules to the coefficients of like terms • do not change the variable parts a) 6x 4 8x 3
Identify like terms first. 6x and 8x are like terms. 4 and 3 are also like terms.
Method 1: Align Like Terms Vertically
Method 2: Collect Like Terms
6x 4 8x 3 14x 7
6x 4 8x 3 6x 8x 4 3
Once you have had some practice, you may be able to skip this step.
14x 7
3.5 Collect Like Terms • MHR 149
Literac
b) 2b b 7 8
onnections
When reading coefficients, always include the sign on the left. For example, the coefficient of b is 1. Then, add the coefficients of like terms using integer rules. It is important that when you collect terms, you include the sign to the left of them:
2b and b are like terms. 7 and 8 are also like terms.
Method 1: Align Like Terms Vertically
2b 7 b 8 b1 Method 2: Collect Like Terms
Apply integer rules to the coefficients of like terms: 2 (1) 1 7 (8) 1
2b b 7 8 b1
2b b 7 8
c)
3r 2 2 6r 9r 3r 2 3r 2 3r 2 6r 9r 2 3r 2
d)
0.5m 4.5n 0.7m 4.5n 1.5 0.5m 0.7m 4.5n 4.5n 1.5 1.2m 1.5
4.5n and 4.5n are opposites. They add to 0.
Key Concepts 䊏
Like terms have identical variables.
䊏
You can simplify a polynomial by adding or subtracting like terms. This is called collecting like terms.
䊏
It is helpful to think of only adding terms. When a subtraction sign appears, think of adding a negative term.
Communicate Your Understanding C1
a) Explain what is meant by like terms and unlike terms. b) Provide two examples of each.
C2
Explain why the two sides are not equal. a) 2x x 3x 5x b) y y y y 3 c) 5m 2m 3m d) 2x 2x 0 e) x x x2 f) 3ab 2b a
150 MHR • Chapter 3
Practise For help with questions 1 to 4, see Example 2. 1. Which polynomial contains no like terms? A 2x 5 3x 2xy
B 3x2 3xy 3
C 4 9x 9y 3
D 4a3 5b 2a2 7b
2. Classify each pair of terms as either like or unlike.
1 2 x 2
a) 2x and 5x
b) 3y and 3z
c) x2 and
d) 4a2 and 3a3
e) 2ab and 3a2
f) 5x2y and 2xy2
g) 3uv and 2vu
h) 9p2q3 and 4q3p2
3. Copy the two columns of terms into your notebook. Connect each
term in the first column with the like term in the second column. 3x2
–4xy
–x
y2
xy
1 y 2
2y2
7x
5x2y
4x2
–7y
3x2y
4. Write two like terms for each. a) 5m
b) x
c) 2y 2
d) 3ab
For help with question 5, see Example 3. 5. Simplify where possible. If it is not possible to simplify,
explain why. a) 3x 6x
b) 2m 5n
c) 5h 8h 2h
d) 7u 4u u
For help with question 6, see Example 4. 6. Simplify. If it is not possible to simplify, explain why. a) 4k 2k
b) 8n n
c) 3z 7z
d) p 6
For help with questions 7 to 9, see Example 5. 7. Simplify by collecting like terms. a) 3x 5 2x 1
b) 3 7y 8 y
c) 2k 3m 4m 6k
d) 7u v v u
e) 8n 5 3n 2
f)
4p 7q 3q p
3.5 Collect Like Terms • MHR 151
Makin
8. Simplify.
onnections
Algebraic expressions are used in work with spreadsheets. You use formulas in cells to tell the computer to make specified calculations. For example, the formula in cell D2 might be 2 B2 2 C2. You will have the opportunity to apply this connection in Chapter 9 Optimizing Measurement.
a) 3x 8 4x 3 b) y 9 7 6y c) 2x2 7x 4x2 x d) 7m 6m2 2m m2 e) 3k 5 8 k 1 4k f) 3u 2 u2 5 3u 2u2 3 9. Simplify. a) 2a2 3ab 6 4b2 7 5ab 3b 2a2 b) 3mn 6m2 n2 3 m2 3mn 2n2 4
Connect and Apply 10. Claudette, Johanna, and Ming all have summer jobs at a clothing
store. They earn $7/h, plus a bonus every week that depends on how well they do that week. The table shows how many hours they work and their bonuses one week this summer. Employee
Number of Hours
Bonus
Claudette
20
$100
Johanna
25
$125
Ming
27
$110
a) Using t to represent the time, in hours, write a binomial to
represent the amount each student makes that week. b) Substitute the time, in hours, into each expression, and find
how much each student makes. c) The three students decide to put their earnings for that week
together to go for an end-of-summer trip. Write a binomial to represent the total amount they earn that week. d) Substitute the total number of hours and find the total amount
the students make that week. Compare your answer with what you found in part b). 11. Yannick simplified the following expression:
3x 4 6x 13x a) Describe the error Yannick made. b) How can you convince Yannick that these two expressions are
not equal? c) Simplify the expression properly. How can you convince
Yannick that your answer is correct?
152 MHR • Chapter 3
12. The length of a rectangular field is three times its width. w
a) Write an expression for the perimeter of the field. b) Find the perimeter if the field is 300 m wide.
3w
c) Find the length and width of the field if the perimeter is 1600 m. 13. Use algebra tiles, virtual algebra tiles, or a diagram to model and
simplify each expression. a) 3x 1 5x 4 b) 4y 3 y 2 c) x 5y 8 x 2y 2 14. Chapter Problem The white bars from the right of the crest into
the centre run halfway across the width. a) Find an expression for the
• perimeter • total length of white trim needed b) What length of trim will be needed for a crest that is 10 cm wide?
Extend 15. a) An equilateral triangle has an unknown side length, x. Write a
simplified expression for its perimeter. b) A right isosceles triangle has two sides equal to x. Which triangle,
the equilateral triangle in part a) or the right isosceles triangle, has the greater perimeter? Use algebraic reasoning to justify your answer. 16. Which triangle in question 15 has the greater area? Use algebraic
reasoning to justify your answer. 17. Math Contest What is the value of the 100th term in the sequence
3x 2y, 5x2 5y 3, 7x3 8y 5, … ? A 199x100 296y197 B 200x100 300y199 C 200x100 300y197 D 201x100 299y199 E 203x100 299y199 18. Math Contest The last digit of the number 22020 when written in
expanded form is A 2
B 4
D 8
E 0
C 6
3.5 Collect Like Terms • MHR 153
3.6
Add and Subtract Polynomials
Have you ever wondered about who writes textbooks like this one? When do they write these books? How do they get paid? Experienced mathematics teachers write most math textbooks. They usually write when they are not teaching. They are paid in different ways, depending on their contracts: • a fixed amount, or flat rate • a royalty, which depends on the number of books sold • a combination of fixed amount plus royalty Why do you think that they are paid in these ways?
Example 1 Add Polynomials Simplify each expression. a) (4x 3) (7x 2)
b) (2p 2) (4p 7)
c) (0.5v 2 2v) (2.4v 2 3v)
Solution a)
b)
(4x 3) (7x 2) 4x 3 7x 2 4x 7x 3 2 11x 5
First, remove the brackets, and then collect like terms.
(2p 2) (4p 7) 2p 2 4p 7 2p 4p 2 7 6p 9
I can rearrange terms to group like terms.
I use integer rules to collect like terms.
c) Method 1: Remove Brackets and Collect Like Terms
(0.5v 2 2v) (2.4v 2 3v) 0.5v 2 2v 2.4 v 2 3v
Apply the integer rules when you remove brackets.
0.5v 2 2.4v 2 2v 3v 1.9v 2 v Method 2: Write the Addition Vertically
0.5v 2 2v 2.4v 2 3v
Add the numerical coefficients of the like terms vertically. 0.5 (2.4) 1.9 and 2 (3) 1.
1.9v 2 v (0.5v 2 2v) (2.4v 2 3v) 1.9v 2 v 154 MHR • Chapter 3
Example 2 Opposite Polynomials State the opposite of each expression. b) 2x
a) 7 d) 5y 2
e)
x2
c) 4x 1
3x 7
Solution Opposites add to give 0. a) The opposite of 7 is 7. b) The opposite of 2x is 2x.
c) To find the opposite of a polynomial,
find the opposite of each term. The opposite of 4x 1 is 4x 1. d) The opposite of 5y 2 is 5y 2.
e) The opposite of x2 3x 7 is
x2 3x 7.
Example 3 Subtract Polynomials Simplify. a) (3y 5) (7y 4)
b) (a2 2a 1) (a2 2a 5)
Solution a)
b)
(3y 5) (7y 4) (3y 5) (7y 4) 3y 5 7y 4 3y 7y 5 4 4y 9
To subtract a polynomial, add its opposite.
(a2 2a 1) (a2 2a 5) (a2 2a 1) (a2 2a 5) a2 2a 1 a2 2a 5 2a2 6
The opposite of 7y 4 is 7y 4.
2a and 2a are opposites. They add to give 0. There is no a term in the final expression.
3.6 Add and Subtract Polynomials • MHR 155
Example 4 Apply an Algebraic Model to Solve a Problem Four authors team up to write a textbook. The publisher negotiates the following contracts with them. Author
Fixed Rate ($)
Royalty ($ per n books sold)
Lita
2000
—
Jamal
1000
2n
Vera
1500
n
—
3n
Fleming
a) Write a simplified expression for the total payout to the authors. b) Determine the total amount paid to the authors for 1200 books sold.
Solution a) Write an expression for the total amount paid to each author.
Then, add these expressions. Lita: Jamal: Vera: Fleming:
2000 1000 2n 1500 n 3n
Total payout: 2000 (1000 2n) (1500 n) 3n 2000 1000 2n 1500 n 3n 4500 6n The publisher will pay 4500 6n to the authors, where n is the number of books sold. b) Substitute n 1200 and evaluate the expression.
4500 6n 4500 6(1200) 4500 7200 11 700 If 1200 books are sold, the publisher will pay $11 700 to the authors.
Key Concepts
156 MHR • Chapter 3
䊏
To add polynomials, remove brackets and collect like terms.
䊏
To subtract a polynomial, add the opposite polynomial.
Communicate Your Understanding C1
a) Describe the process of adding one polynomial to another.
Use an example to support your explanation. b) Describe the process of subtracting one polynomial from
another. Use an example to support your explanation. C2
Each of the following solutions contains an error. Describe the error and suggest how it should be corrected. a)
(2x 3) (6x 2) 2x 3 6x 2 8x 1
b)
(4y 7) (2y 5) 4y 7 2y 5 2y 12
Practise For help with questions 1 and 2, see Example 1. 1. (2x 7) (3x 8) simplified is A 5x 15
B 5x 1
C 5x 1
D 6x 56
2. Simplify by removing brackets and collecting like terms. a) (3x 4) (7x 5)
b) (y 2) (3 6y)
c) (4m 1) (3m 8)
d) (5 3d) (d 6)
e) (4k 3) (5 k) (5k 3) f) (6r 1) (3r 2) (6r 1)
For help with questions 3 and 4, see Examples 2 and 3. 3. (3x 5) (x 4) simplified is A 2x 1
B 2x 1
C 2x 9
D 2x 9
4. Simplify by adding the opposite polynomial. a) (2x 3) (x 6)
b) (8s 5) (s 5)
c) (6m 4) (2m 1)
d) (4v 9) (8 3v)
e) (9 6w) (6w 8)
f) (5h 9) (5h 6)
5. Simplify. a) (7x 9) (x 4)
b) (3y 8) (y 5)
c) (8c 6) (c 7)
d) (k 2) (3k 2)
8p 1)
e)
(3p2
f)
(5xy 2
(9p2
4p 1)
6x 7y) (3xy 2 6x 7y)
g) (4x 3) (x 8) (2x 5) h) (2uv 2 3v) (v 3u) (4uv 2 9u)
3.6 Add and Subtract Polynomials • MHR 157
Connect and Apply For help with questions 6 and 7, see Example 4. 6. A group of musicians who made a CD are paid according to the
following breakdown, where n is the number of CDs sold. Musician
Fixed Rate ($)
Royalty ($ for n CDs sold)
2000
0.1n
—
0.3n
Nigel
1500
0.2n
Tulia
5000
—
Ling Fredrick
a) Find a simplified expression for the total amount paid
to the group. b) The table shows sales achievement levels for the Canadian
recording industry. Status
Number of CDs Sold
Gold
50 000
Platinum
100 000
Diamond
1 000 000
Find the total amount paid to the group if their CD • sells 100 copies • reaches gold status • reaches diamond status c) Which musician makes the most money at each level
in the table in part b)? d) Describe the advantages and disadvantages of being paid
• by a fixed rate • by royalty • by a combination of fixed rate and royalty Use mathematical reasoning to support your answers. 7. A women’s basketball team gives players a bonus of $100 on top of
their base salary for every 3-point basket. Data for some of the team’s players are given. Player
Base Salary ($1000s)
3-Point Baskets
Gomez
50
25
Henreid
40
20
100
44
Jones
a) Find a simplified expression for the total earnings for these
three players. b) Find the total earnings for these players.
158 MHR • Chapter 3
8. Use algebra tiles, virtual algebra tiles, or a diagram to model and
simplify each expression. a) (2x 5) (3x 2) b) (y 2 3y 1) (y 2 2y 2) c) (2x2 3x 4y 2) (2x2 x 2y 2) 9. A swimming pool manufacturer installs rectangular pools whose
length is twice the width, plus 5 m.
Reasoning and Proving Representing
a) Draw a diagram of the pool and label the width and length
with algebraic expressions.
Problem Solving Connecting
b) The entire outer edge of the pool must be fitted with coping,
Selecting Tools
Reflecting Communicating
which is a cap used to join the wall of the pool and the deck. Find a simplified algebraic expression that describes the total length of coping needed. c) How much coping is required if the width of the pool is 6 m? 10. Refer to question 9. a) Predict how the amount of coping will change if you double
the width of the pool. b) Calculate the new amount required and compare this with
your prediction. Explain the results.
Extend 11. Refer to question 9. If you use the same expressions for the length
and width of the pool as in question 9, do the proportions of the pool change when you change the width? Describe the relationships between width, length, and perimeter as you change the width. Use diagrams, words, and algebraic expressions to support your explanations. 12. Math Contest a) Copy and complete the addition cascade. 5k 7p 8k 4p
3k 11p
4p 3k k 2p
b) Build your own addition cascade. Have another student complete
it, and then verify each other’s work.
3.6 Add and Subtract Polynomials • MHR 159
3.7
The Distributive Property
Any successful sports team must practise. Often a practice begins with a warm-up. Suppose the coach tells her players to run the perimeter of the field. How can you use algebra to write an expression for the total distance? Is there more than one way to do this? James claims, “I can calculate the perimeter of the field by using the formula P 2(l w).” Sylvie replies, “That’s not right, James. The correct formula is P 2l 2w.” Who is correct, James or Sylvie? Can they both be right?
Investigate Tools 䊏
algebra tiles
How can you simplify algebraic expressions involving multiplication? The area of a rectangle can be expressed as the product of its width and length: w
Awl 2
A=6
A = 2x + 2
3
x+1
A=23 =6
A = 2(x + 1) = 2x + 2
2
l
The dimensions of algebra tiles: 1 1 x
x
x
160 MHR • Chapter 3
1
1. Copy and complete the table. Use algebra tiles to help. Rectangle
Width
Length
Area
Equation wlA
2
x1
2x 2
2 (x 1) 2x 2
x
2x
x2 2x
2x 3
Create your own
2. Look at the last column. Describe how you can start with the
expression on the left side of the equation and get the expression on the right side. 3. Use the technique from step 2 to multiply the following. a) 4(x 3) b) x(2x 7) c) 3x(x 2) 4. Reflect Create an example and use it to explain how you can
multiply a monomial by a polynomial.
3.7 The Distributive Property • MHR 161
Suppose that you have three dimes and three nickels.
The total amount can be expressed different ways: 3(10) 3(5) 30 15 45 3(10 5) 3(15) 45 Whether you use 3(10) 3(5) or 3(10 5), the result is 45¢. What does this tell you about these two numeric expressions? Some algebraic expressions cannot be simplified using the second method. Reflect on the situations when you cannot simplify an expression. Suppose you add two nickels to a pile of nickels. The value of the nickels can be modelled by 5(x 2), where x is the number of nickels in the original pile. 5(x 2) You cannot add x 2 because they are not like terms. However, you can multiply the 5 by both terms in the binomial: 5(x 2) 5(x) 5(2) 5x 10 distributive property
This is called the distributive property .
• a (x y ) ax ay
Example 1 Apply the Distributive Property Expand. Literac
onnections
When you apply the distributive property, you are expanding an expression.
162 MHR • Chapter 3
a) 3(x 2)
b)
5(4m 3)
c) (2y 5)(4)
d)
2(5a2 7a 2)
Solution a)
3(x 2) 3(x) 3(2) 3x 6
To apply the distributive property, I need to multiply both terms in the brackets by 3.
b) Method 1: Show the Distributive Step
5(4m 3) 5(4m (3)) (5)(4m) (5)(3)
When I distribute a negative term, I must follow the integer rules for multiplication.
Remember to keep the negative sign when you are distributing.
20m 15 Method 2: Apply the Distributive Step Mentally
5(4m 3) 20m 15
(5) 4m 20m (5) (3) 15
c) The monomial is usually written before the polynomial, but
sometimes it appears after. Apply the distributive property in the same way. (2y 5)(4) 2y (4) 5(4) 8y (20) 8y 20 d) You can also apply the distributive property to trinomials.
2(5a2 7a 2) 10a2 14a 4
The distributive property can be applied to polynomials with any number of terms.
Example 2 Distribute Variables Simplify using the distributive property. a) x(x 3)
b) p(p2 2p 1)
Solution a)
x(x 3) x(x) x(3) x2 3x
To multiply variables, I need to use the exponent laws. The product rule is x a x b x a b.
b)
p(p2 2p 1) p3 2p2 p
I need to use the product rule. p p 2 p (1 2) p3
The distributive property allows you to simplify complicated expressions.
3.7 The Distributive Property • MHR 163
Example 3 Expand and Simplify Expressions Expand and simplify. a) 4(x 3) 2(2x 1) b) 3m(m 5) (2m2 m) 1 2 c) (2w 6) (9w 6) 2 3
Solution a)
4(x 3) 2(2x 1) 4(x) 4(3) 2(2x) 2(1) 4x 12 4x 2 4x 4x 12 2 14
Apply the distributive property first to remove brackets. Then, collect like terms.
b) Method 1: Add the Opposite Polynomial
3m(m 5) (2m2 m) 3m(m 5) (2m2 m) 3m2 15m 2m2 m 3m2 2m2 15m m m2 14m
2m 2 m is the opposite of 2m 2 m.
Method 2: Distribute 1
3m(m 5) (2m2 m) 3m(m 5) (1)(2m2 m) 3m2 15m 2m2 m m2 14m c)
2 1 (2w 6) (9w 6) 2 3 1 1 1 3 2 3 2 2 (2w) (6) (9w) (6) 2 2 3 3 1
1
1
1
1(1w) 1(3) 2(3w) 2(2) w 3 6w 4 5w 1
164 MHR • Chapter 3
Multiplying a polynomial by 1 produces the opposite polynomial: (1)(2m 2 m ) 2m 2 m
Example 4 Nested Brackets Expand and simplify. 3[2 5(2k 1)]
Solution When simplifying expressions involving nested brackets, begin with the inner brackets and work your way outward. 3[2 5(2k 1)] 3(2 10k 5) 3(10k 3) 30k 9
Simplify the inner brackets before distributing the 3.
Key Concepts 䊏
Literac
The distributive property allows you to expand algebraic expressions: a(x y) ax ay
onnections
When you distribute a handout in class, you give one copy to each student. Similarly, when you use the distributive property, the term in front of the brackets is multiplied by, or given to, each term inside the brackets..
• When distributing, multiply the monomial by each term in the polynomial. • Multiply numerical coefficients. • Apply exponent laws to variables.
Communicate Your Understanding C1
Consider the expression 3(x 5). Why do you need to use the distributive property? Why can you not just add the terms in the brackets first?
C2
Explain the distributive property. Create an example to support your explanation.
C3
Dmitri simplified the following expression, as shown: 3x(x 2) 3x 2x 5x a) Explain the errors in Dmitri’s solution and how to correct them. b) How can you prove that Dmitri’s answer is not correct?
C4
Explain why the following simplification is incorrect. (x2 3x 1)(5) x2 3x 4
C5
Describe the steps you must follow to simplify the expression 3[x 2(x 1)]. 3.7 The Distributive Property • MHR 165
Practise For help with questions 1 to 3, see Example 1. 1. Which expression shows 3(x 5) expanded? A 3x 15
B 3x 5
C 3x 8
D 3x 15
2. Use algebra tiles, virtual algebra tiles, or a diagram to model and
simplify each expression. a) 2(x 3)
b) x(x 4)
c) 3(x 1) 2(x 5) 3. Expand, using the distributive property. a) 4(x 2)
b) 5(k 3)
c) 2(y 1)
d) 8(2 d)
e) 5(2t 3)
f) (4y 5)
For questions 4 and 5, see Example 2. 4. Expand. a) y(y 4)
b)
r(r 5)
c) x(2x 5)
d)
q(4q 8)
e) z(3z 2)
f)
m(m 5)
5. Expand. a) 2b(3b 5)
b) 3v(8v 7)
c) 4w(3w 1)
d) 6m(m 5)
e) 2q(4q 3)
f) 3d(d 2)
6. Expand using the distributive property. a) (n 5) 4
b) (2p 4) 9
c) (7m 6) (4)
d) (7 c)(3c)
e) (3 6w)(2)
f) (4k 7)(3k)
7. Expand.
166 MHR • Chapter 3
a) 2(a2 5a 3)
b) 3(2n2 8n 5)
c) 4k(k2 k 3)
d) 5h(3h2 7h 2)
e) (x2 5x 2)(3)
f) (2y2 3y 1)(4y)
For help with question 8, see Example 3. 8. Expand and simplify. a) 3(x 2) 4(x 5)
b) 4(y 1) 2(2y 3)
c) 2(u v) 3(u v)
d) 4(w 2) 2(2w 7)
e) 3(a b) 2(a b)
f) 2(p q) 2(p q)
For help with question 9, see Example 4. 9. Expand and simplify. a) 3[x 2(x 4)]
b) 5[2(y 1) 3]
c) 3[2k (2 k)]
d) 4[3(r 5) 2r]
e) 2[h 2(h 1)]
f) 3[2(p 2) 3p]
Connect and Apply 10. A computer repair technician charges $50 per visit plus $30/h for
house calls. a) Write an algebraic expression that describes the service charge for
one household visit. b) Use your expression to find the total service charge for a 2.5-h
repair job. c) Suppose all charges are doubled for evenings, weekends, and
holidays. Write a simplified expression for these service charges. d) Use your simplified expression from part c) to calculate the cost
for a 2.5-h repair job on a holiday. Does this answer make sense? Explain. 11. Who is right? Use examples to explain your answer.
3.7 The Distributive Property • MHR 167
a
12. The formula for the area of a trapezoid is
h
A b
1 (a b)h 2
Apply the distributive property twice to write this formula in another way. 13. A garden has dimensions as shown. 3x + 1
2x
a) Find a simplified expression for the perimeter. b) Find a simplified expression for the area. c) Repeat parts a) and b) if both the length and width are tripled. d) Has this tripled the perimeter? Justify your answer. e) Has this tripled the area? Justify your answer. 14. Is the distributive property true for numerical expressions
Reasoning and Proving Representing
Selecting Tools
Problem Solving Connecting
Reflecting Communicating
(expressions without variables)? Make a prediction. Then, create some numerical expressions and explore this. Describe any conclusions that you reach. 15. Expand and simplify. a) 2(x 3) 3(x 5) b) 3(k 4) 2(k 1) c) 0.2p(p 5) 0.4p(3p 2) d) 4h(h 2) h(2h 3) e) 5j(j 3) (2j2 3j) f) 0.7w(2w 3) 0.6w(w 3) g) 3(y 2) 2(4 2y) (6 7y) h) 4k(k 3) 2(k2 3k 4) (k2 5) 16. Expand and simplify. a)
1 1 (3a 2) (4a 2) 3 4
b)
1 1 (x 2y) (3y 2x) 2 3
c)
2 3 (3m 2) (8m 2) 3 4 1 4
d) (4u 3v)
168 MHR • Chapter 3
3 (6u 10v) 5
Achievement Check 17. a) Find a simplified expression for the perimeter of each figure.
Use algebra tiles if you wish. 2x 1
x
2x 1
x 2x + 1
2x + 1
3x + 1
3w 2
20 4x
2x + 3
2w +3
b) A rectangle has length 2x 1 and width 8 2x. What is unusual
about the perimeter? c) For what value of x is the rectangle in part b) also a square?
Extend 18. Expand and simplify. a) 2m[m 3(m 2)] b) 3p[2p(p 4) p2] c) 2[3 2(x 6)] 3[2(x 5) 8] d) 3[2 (y 5)] 4[3(y 1) (6 y)(2)] 19. How do you distribute a binomial? Consider the following product
of two binomials: (x 3)(2x 2) Distribute the (x 3) to each term in the second binomial to form an expression consisting of two products:
Makin
onnections
You will multiply binomials in grade 10 and beyond as a method of simplifying polynomial expressions.
(x 3)(2x) (x 3)(2) a) Simplify this expression. Check to see that your result appears to
be equivalent to the original expression, by substituting values into both expressions. b) Make up your own question similar to this and simplify it. c) Can you see an easier way to expand the product of two
binomials? Find and explain an easier method. 20. Expand and simplify. Check to see if your result is correct.
If you need a hint, look at the method from question 19. (k 1)(k2 2k 1) x
21. Math Contest If a b 0.015 625, determine the value of x.
1 2
3.7 The Distributive Property • MHR 169
Use Technology
Computer Algebra Systems A computer algebra system (CAS) is a type of calculator or computer software that can perform algebraic functions. They are sometimes called symbolic manipulators. You can investigate many of the skills that you have learned in this chapter using CAS systems.
Investigate A Tools 䊏
TI-89 calculators
How can you learn about a CAS?
Technology Tip Before you start doing calculations, clear the calculator’s memory. Press n¡ for [F6] to display the Clean Up menu. Select 2:NewProb. Press e.
function keys N key b key
Technology Tip
variable keys
Note that you use a different button for leading negative signs –, as in the 5 in question 2 part c), than the subtraction button -. o key
170 MHR • Chapter 3
1. Press the o key to turn
the calculator on. If you do not see the Home screen, as shown, press HOME key. the Í
2. Perform some basic calculations to get familiar with the
keyboard and display. Follow the keystrokes shown. a) 3(6 4)5 3 *(6 - 4 )^5 e
b) (2.5)2 n^*2.5 ^2 e
c) 5 29 –5 -n*9 )e
3. Explore the Function keys at the top of the keypad. a) Press ¡ to see the TI-89 Tools menu. Identify at
least three commands that you recognize from working on a computer. Use the up and down cursor keys to see more commands. b) Press ™ to see the algebra functions that the TI-89
can perform. Identify two commands that you recognize from this chapter and two that you do not recognize. c) You can also use the left and right cursor keys to see the
different function menus. Scroll over to see the menus of F3 and F4. You will learn about these functions in future mathematics courses. d) Clean up the home screen:
• Press n [F6] to display the Clean Up menu. • Select 2:NewProb. • Press e.
Use Technology: Computer Algebra Systems • MHR 171
Use Technology
Investigate B How can you use a CAS to simplify and expand expressions? 1. Collect like terms for the
expression 2x 3y x 5y. • Type the expression 2x 3y x 5y. • Press e. 2. Use the CAS to collect like
terms for each expression. a) 3y 2x 5x 3y b) x2 3x 2 6x 3x2 c) Create your own expression.
Technology Tip Note the nested brackets required in this command line. If you get an error message, check that you have the same number of open and close brackets, and that they are placed correctly.
3. Expand 4(2x 3) using the
distributive property. • Press ™. • Select 3:expand(. • Type the expression. • Press ). • Press e.
4. Use a CAS to simplify using the distributive property. a) 2z(3z 5)
b) 3(4x 7y) 2(3x 2y)
c) Create your own expression.
Investigate C How can you verify the exponent laws? 1. a) Use a CAS to illustrate the
product rule of exponents by entering the expression y 2 y 5. b) Create two of your own
examples to illustrate the product rule using a CAS. 2. You can also evaluate expressions involving powers, such as
y2 y3 when y 3. 172 MHR • Chapter 3
Press
Y Y | Y Í ^2 * Í ^3Í Í = 3.
Use this feature to evaluate a) y 3 y 4
x x x5 4
c)
7
when y 4 when x
b)
2 3
z8 z5
when z 5
3. Design your own method for verifying either the quotient rule or
the power of a power rule for exponents. Describe your steps, keystrokes, and screen outputs.
Investigate D How can you explore an unknown function? Many of the functions of the CAS will be unknown to you at this level. Several of them you will learn about in future mathematics courses, but in this section you will have the opportunity to experiment and learn a little about some of these “mystery” functions. 1. a) Clear the home screen. Press n¡ for [F6] to display the
Clean Up menu. Select 2:NewProb and press
e.
b) Use the Expand command to simplify the expression 3(x 4). c) Factor the result:
• Press ™. • Select 2:Factor(. • Type 3x 12. • Press ). • Press e.
What does the Factor function do?
d) Write down the result. What do you notice? 2. Repeat step 1 for each expression. a) y(2y 3) b) 5z(2z 1)
Makin
3. Explain what the Factor command seems to do. How does it seem
You will learn about factoring in grade 10.
to be related to the Expand command?
onnections
4. a) Repeat step 1 for the expression 2(2x 6). b) Is the factored result what you expected? Explain. c) Create a few examples of your own. Describe what you discover. 5. Explore some of the unknown features of the TI-89. Try to figure
out what one of them does. Write a brief report of your findings.
Use Technology: Computer Algebra Systems • MHR 173
Chapter 3 Review 3.1 Build Algebraic Models Using Concrete Materials, pages 104109 1. Use algebra tiles to build a model for each
situation. Write an algebraic expression to represent the model. a) Jeanne ran 4 km. b) Klaus drove an unknown distance, twice. c) Evelyn ran 3 km plus an unknown
distance.
6. The half-life of carbon-14 (C-14) is
5700 years. a) Copy and complete the table for
a 50-g sample of C-14. Number of Half-Life Periods
Years
Amount of C-14 Expression Remaining (g)
0
0
50
1
5 700
25
2
11 400
d) Suki painted her house with two coats of
paint. 2. a) Build a volume model to represent a
cube with side length 3 cm. Sketch the model and label the length, width, and height. b) What is the volume of the cube? Write
this as a power. c) Write an expression for the area of one
face of the cube as a power. Evaluate the area of one face.
1 1 50 — 2
() 1 50(—) 2
2
3 4
b) Construct a graph of the amount of C-14
remaining versus time, in years. Describe the shape of the graph. c) Approximately how much C-14 will
remain after 20 000 years? d) How long will it take until only 1 g of
C-14 remains? 3.2 Work With Exponents, pages 110118 3. Evaluate. a) 45 b) (3)4 c)
2 3 a b 5
d) 1.058 4. $100 is put into a bank account that pays
interest so that the amount in the account grows according to the expression 100(1.06)n, where n is the number of years. Find the amount in the account after a) 5 years b) 10 years 5. Find the annual interest rate of the account
in question 4.
174 MHR • Chapter 3
Did You Know ? Carbon dating is a method of determining the age of fossilized creatures, such as dinosaur remains. Scientists compare the ratio of two carbon isotopes in the sample: C-14, which is radioactive, and C-12, which is stable. Because everything on Earth has about the same ratio of C-14 to C-12 at any given time, measuring this ratio tells when the creature lived.
3.3 Discover the Exponent Laws, pages 119129 7. Write as a single power. Then, evaluate the
expression. a) 23 22 24
b) 67 62 63
c) [(4)2]3
d)
74 75 (74)2
8. Simplify.
15. Simplify by collecting like terms.
a)
n5 n3 n4
c)
2ab2 3a3b3 (4ab2)2
b) cd3 c4d2
d) 3x2 4xy 5y 2 6 3x2 4xy 2
9. Identify the coefficient and the variable part
of each term. b) uv
d) de2f
e) 8
b) 7k 5m k 6m c) 6a2 5a 3 3a2 5a 4
3.4 Communicate With Algebra, pages 130139
a) 5y
a) 4x 3 6x 5
c)
1 2 ab 2
10. Classify each polynomial by the number
of terms. a) x2 3x 5
b) 24xy
c) a 2b c 3
d)
2 3
3.6 Add and Subtract Polynomials, pages 154159 16. Simplify. a) (4x 3) (3x 2) b) (5k 2) (3k 5) c) (6u 1) (2u 5) d) (y 2 3y) (2y 2 5y) e) (2a2 4a 2) (a2 4a 2) f) (3v 2) (v 3) (2v 7) 17. A rectangular window frame has
e) 16u2 7v 2 11. In a hockey tournament, teams are awarded
3 points for a win, 2 points for an overtime win, and 1 point for an overtime loss. a) Write an expression that describes the
number of points a team has. b) Use your expression to find the number of
points earned by a team that has 4 wins, 1 overtime win, and 2 overtime losses. 12. State the degree of each term. a) 3x2
b) 6n4
c) 17
d) abc2
13. State the degree of each polynomial. a) 3y 5
b) 2d2 d
c) 3w 6w 2 4
d) 3x3 5x2 x
3.5 Collect Like Terms, pages 144153 14. Identify the like terms in each set. a) 2p, 3q, 2, p, 3q2 b) 5x2, 5x, x5, 5x2, 3x2
dimensions expressed by 3x and 2x 5. Find a simplified expression for its perimeter. 3.7 The Distributive Property, pages 160169 18. Expand. a) 3(y 7) b) 2(x 3) c) m(5m 3) d) 4k(2k 6) e) 5(p2 3p 1) f) 4b(b2 2b 5) 19. Expand and simplify. a) 2(q 5) 4(3q 2) b) 5x(2x 4) 3(2x2 8) c) 3(2m 6) (8 6m) d) 4(2d 5) 3(d2 3d) 2d(d 1) 20. Simplify. a) 2[4 3(x 5)] b) 3[9 2(k 3) 5k]
Chapter 3 Review • MHR 175
Chapter 3 Practice Test Multiple Choice
9. What does 2m(3m 1) simplify to?
For questions 1 to 9, select the correct answer. 3
1. What is the value of a b ?
1 3
A 1 C
B
1 9
D
1 3 1 27
2. What is the simplified form of the
expression w 2 w 4 w? B w7
C w8
D w9
C 6m2 2m
D 6m2 2m
Short Response Show all steps in your solutions. 10. Write as a single power. Then, evaluate. a) (2)2 (2)3 (2)
35 33 (32)3
11. Simplify. a) k2n3 kn4
45 3. What does 3 equal? 4
b) 6p5 3p2
A 4
B 8
C 16
D 32
c) (3g2h)3 12. Simplify. a) (5x 3) (2x 7)
4. What does (23)2 equal? A 10
B 12
C 32
D 64
5. Which of the following shows three like
terms? A 3x
2x
x2
B 4a
4b
4c
C 3y
y
8y
2mn2
4mn
D
B 6m2 1
b)
A w6
3m2n
A 6m2 2m
6. 2k2 3k is a A monomial
B binomial
C trinomial
D term
7. What is the degree of 2u3v? A 1
B 2
C 3
D 4
8. 3x 5 2x 3 can be simplified to A 3x
B 5x 2
C x2
D 5x 2
b) (3u 4) (5u 1) 13. Expand and simplify. a) 3(y 4) 6(y 2) b) 4(6b 3) (3b 5) 14. James and Sylvie need to paint the line
around the outside of a soccer field. To know how much paint to buy, they must find the perimeter of the field. James claims, “I can calculate the perimeter of the field by using the formula P 2(l w).” Sylvie replies, “That’s not right, James. The correct formula is P 2l 2w.” a) Who is correct? Is it possible for both to
be right? Provide evidence to support your answer. b) Find two other ways to write a valid
formula for the perimeter of a rectangle and explain why they work. c) Even though the students accurately
calculated the perimeter, they ran out of paint. Discuss possible reasons for this situation. 176 MHR • Chapter 3
Extended Response
a) Find a simplified expression that
represents the total that the publisher must pay the writing team.
Provide complete solutions. 15. Vanessa, the student council president,
needed to get a message to the whole school, but she did not have time to e-mail every student. So she set up an e-mail tree. She sent the message to her two vice-presidents, and asked them each to forward it to two students. Suppose that this pattern is repeated and assume that no one receives the e-mail more than once.
b) Determine the total payout if the book
sells • 200 copies • 5000 copies 17. Suppose you place a penny on the first
square of a checkerboard.
a) How many people will receive the e-mail
on the seventh mailing? b) On which mailing will the e-mail be sent
to at least 500 people? Explain how you found this answer. c) At this point, how many people would
have received the e-mail, in total? d) There are 500 students in the school.
How many mailings would it take to reach all the students? 16. Three authors team up to write a children’s
Now, suppose you double the amount and place this on the second square. If you repeat this for the entire board, how much will be on the a) 64th square? b) entire board?
book. The publisher pays them according to the following contracts. Author
Fixed Rate ($) Royalty ($ for n books sold)
Latoya
3000
—
Sven
1500
2n
Michelle
1000
3n
Chapter Problem Wrap-Up a) Design a logo for your school team, or a
team of your choice. You can pick your favourite professional team and create a new logo for them, or make up your own fantasy team. Your design should include one or more polygons. b) Use one or more variables to represent
the dimensions in your design and label them on your design.
c) Find a simplified expression for the
perimeter of your logo. d) Find a simplified expression for the area
of your logo. e) Create two problems involving your logo.
Solve the problems. f) Trade problems with a classmate. Solve
each other’s problems and check your solutions.
Chapter 3 Practice Test • MHR 177
Chapters 1 to 3 Review Chapter 1 Mathematical Processes 1. Find the next three terms in each sequence.
Describe how to find successive terms.
9.
Can you find five odd numbers that add up to 50? What about six numbers? Explain.
10. Give a counter-example to prove that each
a) 1, 2, 4, 7
b) 1, 4, 9, 16
statement is false.
c) 17, 12, 7, 2
d) 2, 6, 12, 20
a) The sum of two square numbers is
2. Use the clues to find the value of E.
Describe your strategy.
always a square number. b) The sum of two acute angles is an obtuse
angle.
A B 80 A C 200 B D 36 D E 18 C 100
c) The sum of two prime numbers is a
prime number.
Chapter 2 Relations
3. You have three $5 bills, a $10 bill, and two
$20 bills. How many different sums of money can you make? 4. The area of this figure is
11. a) Make a hypothesis about the percent of
students in your school that have computers in their home. b) Describe how you could collect data to
test your hypothesis. Is this primary or secondary data?
400 cm2. What is its perimeter?
12. Describe how you could choose a random 5. Evaluate.
sample to determine
2 3 1 a) 3 4 2
2 3 1 b) a b 3 4 2
3 2 1 c) 4 3 2
3 2 1 d) a b 4 3 2
6. The daily high temperatures during one
week in February were 6°C, 2°C, 8°C, 5°C, 4°C, 1°C, and 9°C. What was the mean high temperature that week? 7. A number is considered a perfect number if
its factors (not including itself) add up to the number. For example, 6 is a perfect number because 1, 2, and 3 are factors of 6 and 1 2 3 6. Find the next perfect number. Describe the strategy that you used. 8. How many breaths do you take in 1 year?
Describe your strategy.
178 MHR • Chapter 3
a) the favourite TV show of grade 9 girls b) the percent of students in your school
who have their own cell phone 13. The table shows the
heights and shoe sizes of 10 grade 9 girls. a) Make a scatter plot of
the data. b) Describe the
relationship between a student’s height and her shoe size. c) Identify any outliers.
Should you discard them? Explain.
Height (cm)
Shoe Size
157
5.5
153
5
165
8
160
6
175
9
162
7
164
6.5
155
7
168
7.5
162
7
Chapter 3 Polynomials
14. A store has 30 female employees and 20
male employees. The manager wants to select 10 employees to help choose a new uniform.
17. The area of one face of a cube is 64 cm2. a) What is the side length of the cube? b) Determine the volume of the cube.
a) Identify the population. b) Describe how the manager can choose a
stratified random sample.
18. Evaluate. a) 32 23
15. The table shows the numbers of storeys and
c) (43 33) (25 42)
heights of six Canadian buildings. Number of Height Storeys (m)
Building
19. Simplify. a) n2 n3
b) d8 d2 d) 3m2n 4mn3
First Canadian Place, Toronto
72
290
c) (a3)4
Manulife Place, Toronto
36
146
e) 24k 5q3 (2k 2q)2
Petro-Canada Centre, West Tower, Calgary
52
210
Place de Ville, Ottawa
29
112
Royal Centre Tower, Vancouver
36
140
Toronto Dominion Centre, Winnipeg
33
126
a) Graph the data.
20. In a quiz show, contestants receive 10 points
for each correct answer but lose 5 points for each incorrect answer. a) Write an expression for a contestant’s
total score. b) Theo answered 12 questions correctly
and 5 questions incorrectly. Find his total score.
b) Draw a line of best fit. c) Describe the relationship between the
number of storeys and the height of the building. d) Use your graph to predict the height of a
new 40-storey office tower. 16. Claire is training for a half-marathon. The
21. Simplify. a) 5m 8 3m 10 b) 3x2 6x 3 x2 5x 1 c) (h 5) (3h 8) d) (4t 5w) (t 2w) (3t 4w) 22. Expand and simplify. a) 5(x 3)
b) k(2k 1)
c) 4(3y 2) 3(2y 7)
d 7 6 5 4 3 2 1
d)
2 1 (3a 1) (4a 1) 3 2
23. a) Find a simplifed expression
for the perimeter of the triangle. 9: 15 9: 30 9: 4 10 5 :0 10 0 :1 10 5 :3 10 0 :4 11 5 :0 11 0 :1 11 5 :3 11 0 :4 5
Distance (km)
graph shows how her distance from home changed with time on a 14-km run. Write a description of her run.
0
52 62 22 3 2 9 d) a b 5 10 b)
5n – 8
t
4n – 16
Distance (km)
4n + 3 b) Determine the perimeter when n 5.
Chapters 1 to 3 Review • MHR 179
Tasks Task Electricity and Gas Costs Tico keeps records of the amount of electricity, in kilowatt hours (kWh), and natural gas, in cubic metres, used by his family business. He set up a table to compare monthly hydro and natural gas costs.
Cost of Gas
Amount of Gas Used (m3)
Cost of Electricity
Amount of Electricity Used (kWh)
Average Monthly Temperature
January
$174.60
949
$165.75
1950
5°C
February
$177.30
965
$159.38
1875
6°C
March
$158.40
861
$150.11
1766
1°C
April
$135.00
733
$139.57
1642
5°C
May
$76.50
416
$106.00
1247
15°C
June
$35.10
192
$88.57
1042
20°C
July
$32.40
176
$90.10
1060
24°C
August
$31.50
171
$84.83
998
25°C
September
$34.20
187
$95.54
1124
21°C
October
$70.20
381
$108.20
1273
15°C
November
$126.90
691
$129.88
1528
8°C
December
$167.40
912
$161.08
1895
0°C
Month
a) Use a graphing calculator or graphing software to
draw a scatter plot of the relationship between the amount of gas used each month and the cost. b) Use the scatter plot to describe the relationship
between these two variables. Is the relationship linear or non-linear? Justify your response. c) Find another set of data involving two variables
that have a relationship similar to the one you found in part b). Construct a scatter plot to check your conjecture and explain your conclusions. d) Find a set of data involving two variables that
have a different relationship. Explain the relationship.
180 MHR • Chapter 3
Perimeters and Areas x+2
a) Write an expression, in simplified form, for the perimeter
of this rectangle. x
b) Write an expression for the area of the rectangle. c) Suppose 16 of these rectangles are put together to form a larger
rectangle. Show all the possible ways of combining them. Write an expression for the perimeter in each case. d) For each of the values x 1, x 2, and x 3, which combination of
small rectangles from part c) has the least perimeter? Is this more than one combination? Justify your answer. e) Find the area of each combined rectangle in part c).
Mind Reader Part 1
Part 2
Aidan’s cousin Heather found this Mind Reader challenge in a games magazine. She asked Aidan to explain how it works.
Aidan found another Mind Reader in the magazine. It involves a person’s birth date. In this challenge, 15 April 1993 is written numerically as 150493.
• Think of a number. • Add 20. • Multiply the answer by 5. • Double the result. • Subtract 200. • Divide by the number you picked.
• Write down the day of the month that you were born. • Multiply the number by 5. • Add 4. • Multiply by 20. • Add the number of the month you were born. • Multiply by 100. • Add the last two digits of the year you were born. • Subtract 8000.
The answer will always be the same. What is it? a) Heather thought of the number 5.
Show how the steps work for this number. b) Explain how this challenge
works, using words, numbers, and/or symbols.
a) What is the result? What does it mean? b) Explain, using symbols, how this Mind
Reader works.
Chapter 3 Tasks • MHR 181
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