A. Unknown *An unknown is a quantity whose value has not been determined. *Letters can be used to represent unknowns or objects. Example Exercise 1. The teacher gives some pencils 1. I bought some books to the students Solution : …………………………… Solution : The unknown is the number of pencils 2. There are many monkeys in the garden. Solution : ……………………………. 2. There are x students in m class Solution : x is unknown 3. Azman bought y durian in z shop m is object yesterday. Solution : unknown…………….. object ………………. 4. Mr a sold his car for k ringgit Solution : unknown ……………….. object :………………….. B Algebraic Terms i) Algebraic Term with one unknown - is the product of an unknown and a number. Example : 4y is called an algebraic term 4y Number
4y = 4 x y = y + y + y + y
unknown
* Identify coefficients in given algebraic term - Coefficient is the number that multiply the unknown Example 1) 7m : coefficient of m is 7 2)
r 1 : coefficient of r is 4 4
3) – y : coefficient of y is -1
Exercise 1) -3z : coefficient of z is…………… 2)
2 x : coefficient of x is ………….. 5
3) 0.7 h : coefficient of h is ………… 4) p : coefficient of p is …………..
Algebraic Expressions
77
Module PMR
(ii) Like and Unlike Algebraic Terms * Like term : terms with the same unknowns * Unlike terms : terms with different unknowns. Example 1. 3m and -4m 2. 4x and ¼ x 3. 0.9z and 5z 1. 2w and 8h 2, -5f and ½g 3. 1.2q and 3.5g
Exercise Determine whether each of the Like term following pairs of algebraic terms are (same unknown) like term / unlike term 1. 6s , - t : …………………… y 2. , 8y : …………………….. 4 3. 19 d , 19e : …………………… 2 4. , 4e : ……………………….. e
unlike term (different unknown)
C Algebraic Expressions An algebraic expression is a combination of two or more algebraic terms by addition, subtraction or both Examples : 2x + 4y , 6r – 3s + 6z (i)
Number of terms in a given algebraic expression
Example Determine the number of terms in the algebraic expressions below :
Exercise Determine the number of terms in the algebraic expressions below :
1. 3x + 6y : 2 terms
1. 6m + 8n – 9 : …………………..
2. 7p + 5q – 9 : 3 terms
2. 3b + 2e – 10b -5e : …………….
3. w – 2z – 8y + 1 : 4 terms
3. 2s – 4s + 5s + 3 – w :………….
(ii) Simplifying Algebraic Expressions - Group all the like terms together - Add / subtract the coefficient of the terms - unlike terms cannot be simplified
D. Algebraic Terms in two or more unknowns Is the multiplying factors of the term Examples : 3xy , ½abc, 0.8 def * Identifying the coefficient of an unknown Example In the term 8xy2 * 8y(xy) the coefficient of xy is 8y * 8x(y2) the coefficient of y2 is 8x * 8y2 (x) the coefficient of x is 8y2 * 8(xy2) the coefficient of xy2 is 8 Algebraic Expressions
Exercise 1. in the term -3ab2c * the coefficient of abc =……………….. * the coefficient of ab2 = ……………….. * the coefficient of ab2c =…………….. * the coefficient of ac = ………………. 79
Module PMR
E. Multiplication & Division of 2 or more terms (i) Finding the product of 2 algebraic terms - collect all numbers and similar unknowns together - then multiply the numbers and the unknown separately. Example 1. 2ab x 4b2c =2xaxbx4xbxbxc =2x4xaxbxbxbxc = 8 x a x b2 x c = 8ab3c
Exercise 1. ab x a2b =
2. 3xy x (-2 yz) 2
2
2. 4m x ½ mn =4x½xmxmxmxnxn = 2 x m 3 x n2 = 2 m3 n2 Exercise 4 1 pqr × (− pqr ) 6. 5 2 =
7. (-3m2hk3)x (-7m2hk2) =
=
3.. 6ab2c x (½ bc3) =
4. (-8p3qr) x ( -7pqr2) =
5. − =
Algebraic Expressions
80
2 2 w z × ( −9 wz 3 ) 3
Module PMR
(ii) Finding the quotient of two algebraic terms - Express the division in fraction form - cancel similar unknowns that are found in both numerator and denominator Example 14 xyz 1. 7 xz 14 2 × x × y × z = 7× x× z = 2y 2. 12m2n ÷ 3mn 12 4 × m × m × n = 3× m × n = 4m 3. -5cd2e ÷ 15c2de (−5) × c × d × d × e = 3 15 × c × c × d × e d = − 3c
Exercise 1. 24pq2z ÷ 8qr =
− 16 x 3 y 2 2. − 4x 2 y 3 =
3. 12abc ÷ (-18cd) =
4. (- 18sr3t2) ÷ 6sr2t =
iii) Multiplication and Division involving algebraic terms Example 1. 4p x 6q2 ÷ 3pq 4 × p × 62 × q × q = 3× p × q = 8q − 4c 2 de × 9cde 2 6ce 2 − 4 2 × c × c × d × e × 93 × c × d × e × e = 6 31 ×c × e × e
2.
= -2 x c x d x 3 x c x d x e = -2 x 3 x c x c x d x d x e = -6c2d2e Algebraic Expressions
Exercise 1. 6p2qr ÷ 3pq x 8pr =
2.
− 12 gh × (−9kh) 6 gh 2 =
3. 10a2b3 x (-2b2c) ÷ 5abc 81
Module PMR
F) Computations involving Algebraic Expressions * In Multiplication / division of algebraic expressions by a number, every term In the expression is multiply/ divide by the same number Example Exercise 1. 3 ( 2a –b) 1. 8 ( 5m -2) = 3 x 2a – 3 x b = = 6a – 3b 2
1 − (5t + 10 s ) 5 1 1 = − × 5t − × 10 2 s 5 5 = −t − 2 s
3. h – 9(h – 2) = h – 9h + 18 = -8h +18
4. 2 (4e +y) – 5( 2e – 3y) = 2 x 4e + 2 x y – 5 x 2e + 5 x 3y = 8e + 2y – 10e + 15y = 8e -10e +2y + 15y = -2e + 17y 5. (6ab – 4bc) ÷ 2b = 6ab ÷ 2b – 4bc ÷ 2b = 3a – 2c 6.
2. - ½ ( 4a + 12b) = 3. – p – 7 ( p -3 ) =
4. – 5 ( t -2) + 8t = 5. 3 ( 2s -7) – 4( s + 3) =
6. . ( -12pq + 8qr – 4pqr) ÷ 4 =
7.
1 3 (4 p 2 − 8 pq) − (4 p 2 − 12 pq ) 2 4 2 = 2 p − 4 pq − 3 p 2 + 9 pq = 2 p 2 − 3 p 2 − 4 pq + 9 pq = − p 2 + 5 pq 12 x − 4 4 = 10x -5 – (3x - 1) = 10x -5 -3x + 1 = 10x -3x -5 + 1
8.
3x − 9 − 5y 3 =
1 2 (5 p 2 − 10 pq) − (6 p 2 − 3 pq) 5 3 =
7. 5(2x – 1) -
Algebraic Expressions
9. 82
18 − 24u 2 + 5(2u 2 − 3) 6
Module PMR
= 7x -4
=
Common Errors Errors 1. 7pq x 3pq = 21 pq
Correct Steps 1. 7pq x 3pq =7x3xpxpxqxq = 21 p2q2
2. 2 ( 4e – 3 d) = 8e – 3d
2. 2( 4e – 3d) = 2 x 4e – 2 x 3d = 8e – 6d
3. (6de2 – 4ef) ÷ 2e = 3de – 4ef
3. (6de2 – 4ef) ÷ 2e = 6de2 ÷ 2e – 4ef ÷ 2e = 3de – 2f
4. (x – 4y) – ( 2x + y) = x – 4y – 2x + y = x -2x -4y + y = -x-3y
4. (x – 4y) – ( 2x + y) = x – 4y – 2x - y = x -2y -4y - y = - x - 5y
5. -2p ( pq – 3) = - 2pq – 6p
5. -2p ( pq – 3) = - 2p2q + 6p
6. 10abc – 4 abc =6
6. 10abc – 4 abc = 6abc
7. 3a +6b – 8a – 3b = 3a + 8a - 6b -3b = 11a – 9b
7. 3a +6b – 8a – 3b = 3a - 8a + 6b -3b = -5a +3b
8. ( - 4rs2t) x 5r3st2 = (-4) x 5 x r x r3 x s2 x s x t x t2 = 20r3s2t2
8. ( - 4rs2t) x 5r3st2 = (-4) x 5 x r x r3 x s2 x s x t x t2 = -20r4s3t3
9. -5s - ( 3t – 2)
9. -5s - ( 3t – 2) = -5s -3t +2
= +15st -10s Algebraic Expressions
83
Module PMR
(G) Expanding single Brackets * Expanding algebraic expressions by multiplying each term inside the bracket by the number or term outside * p(q + r)= p × q + p × r = pq + pr Example 1. 2p (p – 3q) = 2p x p – 2p x 3q = 2p2 – 6pq 2. -4b(2a + b) = -4b x 2a -4b x b = -8ab – 4b2 3.
2 b (6a – 9c) 3 3 2 2 2 = b x 6 a - b x 9c 3 3 = 4ab -6bc
Exercise 1. y ( w + y) =
2. -5e ( 3f + 2g) =
3.
r (12 − 9 s + 3t ) 3 =
4. xy ( 4z – 2w + xy) =
5.
2 x( xy − 5 yz 2 + 10 z ) 5 =
6. - 7ab(2a – 4b + c) = Algebraic Expressions
84
Module PMR
(H) Expanding double brackets * Expanding algebraic Expressions by multiplying each term within the first pair of brackets by every term within the second pair of brackets ( a + b)(x + y) = a( x +y) + b( x+y) = ax + ay + bx + by Example 1. (x -3)(y+5) = x (y + 5) – 3(y + 5) = xy + 5x – 3y – 15 2. (2k -1)(k – 3) = 2k(k -3) – 1(k -3) = 2k2- 6k – k + 3 = 2k2 -7k + 3 3. (p – 3q)2 = (p – 3q)(p -3q) = p(p-3q) – 3q (p-3q) = p2 – 3pq – 3pq + 9q2 = p2 – 6pq + 9q2 4. (2a +b)2 = (2a+b)(2a+b) = 2a(2a+b) + b(2a+b) = 4a2 +2ab +2ab + b2 = 4a2 +4ab +b2
Exercise 1. (a -2)(b +1) =
2. (m +3)( 3m – n) =
3. (-2s -5)( 3t + 4) =
4. ( a -3)2 =
5. (3m –n)2 =
6. ( 5x +2)2 =
Algebraic Expressions
85
Module PMR
7. (y + 4d)2 =
Common Errors Errors 1. 2x(x-3)
Correct Steps 1. 2x(x-3)
= 2x2 -3 2. (a+b)2
= 2x2 – 6 2. (a+b)2
= a2 + b2 3. ( a – b)2
= a2 +2ab + b2 3.. ( a – b)2
= a2 – b2 4
2m2(3m2n – 4 mn3)
= a2 – 2ab + b2 4.
= 6m4n – 4m2n3 5. -4 ( 3de – 2rst2)
= 6m4n – 8m3n3 5. -4 ( 3de – 2rst2)
= - 12de – 2rst2 6. ( x -3)2
= - 12de + 8rst2 6. ( x -3)2
= x2 – 9 7. 4a2 –(a + b)2
2m2(3m2n – 4 mn3)
= x2 – 6x + 9 7. 4a2 –(a + b)2
= 4a 2 –a2 + b2
= 4a 2 –( a2 +2ab + b2 )
= 3a2 + b2
= 4a2 – a2 - 2ab - b2 = 3a2 -2ab –b2
8. (2x -3)(x + 4)
8. (2x -3)(x + 4)
= 2x( x+4) – 3 (x + 4) Algebraic Expressions
= 2x( x+4) – 3 (x + 4) 86
Module PMR
= 2x2 + 8x – 3x + 12
= 2x2 + 8x – 3x - 12
= 2x2 +5x + 12
= 2x2 +5x – 12
(I ) Factorization * Process of writing an expression as a product of two or more factors. -
List out common factors for each alg. term , determine the HCF of the terms . Write as the product of 2 factors
Example 1. st – sr = s( t – r )
ab – ac = a ( b – c) a = common factor Exercise 1. 6a – 24c =
2. 4m + 12mn – 16m2 2. 4m3 – 6m2 = 4 xm + 4 x 3x mxn- 4 x 4 x m x m = = 4m ( 1 + 3n – 4m) 3. 6d2 – 3d = 3 x 2 x d xd – 3 x d = 3d ( 2d – 1)
3. 8ax + 4bx – 2cx =
4. 10mn – 15m2 =5x2xmxn–5x3xmxm = 5m ( 2n – 3m )
4. x2yz – xy2z =
5. 3st2 – 15 stw =
6. 2yz – 4yz2 + 6xyz = Algebraic Expressions
87
Module PMR
*Factorize an expression by using the difference between 2 squares i) expressions which consist of 2 terms : a2 – b2 = ( a – b)( a + b) Example 1 9 – a2 = 32 – a2 = ( 3-a)(3+a)
J) Factorizing & Simplifying Algebraic Expressions * Algebraic Fractions are fractions with either its numerator or denominator or both having algebraic expressions Examples : Algebraic Expressions
x p − q 2a , , y + 1 x 2 4 − 3b 90
Module PMR
I) Simplifying algebraic Expressions * divide the numerator and denominator by their common factors. * factorizing the numerator or denominator or both and then divide the numerator and denominator by their common factors. Example 4a 2 b 1. 2ab 2 4× a× a×b = 2× a×b×b 2 × a 2a = = b b
Exercise 5rs 1. 10r 2 s =
− 12m 3 n 2. 20m 2 n 2 − 4 × 3× m × m × m × n = 5× 4× m× m× n × n − 3× m 3m =− = 5× n 5n
2.
3.
b+c b2 − c2 (b + c) 1 = = (b − c)(b + c ) b − c
x2 − 9 4. 2x 2 − 6x x 2 − 32 = 2× x × x − 2× 3× x ( x − 3)( x + 3) = 2 x ( x − 3) Algebraic Expressions
3.
21c 2 de 3 14cde 2 =
3−d 9−d2 =
e2 − f 2 4. 3e − 3 f =
91
Module PMR
=
x+3 2x
ii) Addition & Subtraction of Algebraic Expressions a) Algebraic Fractions with same denominator Example Exercise 2x 4 rt 5s − − 1. 1. y y 9k 9k = 2x − 4 = y 2.
iii)Multiplication and Division of Algebraic Expressions a) Multiplication 2 algebraic fractions involving 2 types : * Denominator with one term Example 1. 2 p 3q × 7 5a 2 p × 3q = 7 × 5a 6 pq = 35a 2.
6m 2 4n 2 × 2n 3m 2 × 3× m × m 2 × 2 × n × n × = 2× n 3× m 2× m× 2× n = 1 = 4mn
Algebraic Expressions
Exercise 7 2 × 1. 3r 9 y =
2.
3.
3x 18 y × 2 6 yz 15 x 2 =
p+r 3 × 12 x q
93
Module PMR
* denominator with two terms 1.
Example x2 − y2 x × 2x x− y ( x − y )( x + y ) x . = × 2x ( x − y) x+ y = 2
2.
2x 2 + x x 2 − y 2 × x+ y 2 xy + y x(2 x + 1) ( x − y )( x + y ) × = x+ y y (2 x + 1) x( x − y ) = y
Exercise 3x x−3 × 1. 2 x − 9 6x 2 =
2.
2x − 2 x + 3 × x 2 − 9 4x − 4 =
.
Division of Algebraic Fractions * Denominator with one term Example a 2c a e ae ÷ = × = 1. d e d 2c 2cd
2.
3.
m − n 2( m − n ) ÷ 4c 2c ( m − n) 2c × = 2(m − n) 2 4c 1 = 4 x 2 − 16 y 2 x + 4 y ÷ xy 2x 2 2 2 x − (4 y ) x + 4y ÷ = 2 xy 2x ( x − 4 y )( x + 4 y ) x× y × = 2× x× x ( x + 4 y)
Algebraic Expressions
Exercise 5 10 z 1. ÷ 2 x x =
2.
3.
94
a + b 2a + 2b ÷ cd 3c 2 d =
3b − 3a a 2 − b 2 ÷ 2d 4d 2 =
Module PMR
=
y( x − 4 y) 2x
.
m − 5 m 2 − 25 ÷ 4. 4 pq 6q
*Denominator with 2 terms Example 1. 2p p ÷ 2 q+2 q −4 2 p q 2 − 22 × q+2 p 2p (q − 2)(q + 2) = × (q + 2) p = 2(q − 2) =
2.
ab 2 2a ÷ 2 ( a − b) ( a − b 2 ) ab 2 2a ÷ (a − b) (a − b)(a + b) ab 2 (a − b)(a + b) × = ( a − b) 2a 2 b ( a + b) = 2
Exercise 5 10m ÷ 1. n − 3 3n − 9 =
3g 2 27 gh ÷ 2. 2 2 y−x y −x =
=
Algebraic Expressions
3.
95
s 2 + st rs + rt ÷ 2 w − 3 12 − 8w =
Module PMR
3,
7 x 2 − 14 x 2 x − 4 ÷ 2x + 2 x +1 1 7 x ( x − 2) ( x + 1)1 × = 2( x + 1)1 2( x − 2)1 7x ×1 = 2× 2 7x = 4
2 p − 3q 9q − 6 p (2 p − 3q )1 1 = = 3(3q − 2 p)1 3
Algebraic Expressions
3.
96
1 2 + x 3x 1× 3 2 + = 3x 3 x 3+ 2 5 = = 3x 3x 2 p − 3q 9q − 6 p ( 2 p − 3q )1 = − 3(2 p − 3q )1
=-
1 3
Module PMR
4.
5.
1 h −1 − hk h 1 (h − 1)(×k ) = − hk h(×k ) 1 − kh − k = hk
4.
2x ÷ xy y 2x = × xy y
1 h −1 − hk h 1 (h − 1)(×k ) = − hk h(×k ) 1 − kh + k = hk 2x ÷ xy y 2x 1 2 = × = 2 y xy y
5
= 2x 2 6.
m + 4 2x − m m 4 2x = − m m 4 − 2x 2x = = m m
6
m + 4 2x − m m m + 4 − 2x = m
Questions based on PMR Format (A) Simplify each of the following expressions : 1) 4a – (a – 5) 11. (a – 3)2
2) 10q + ( -6q) -5
12 ( 3x + 2)2
3) 6p – ( -3p) – 2p
13. (5d – t)2
4) 4a – a( b+4)
14. (x – 2)2 – x( x -6)
Algebraic Expressions
97
Module PMR
5) -5m – 4(m – 2)
15. ( 2y + 3)2 – ( 5y - 2)
6) 6b – (b +3)
16. ( 3w –z)2 + z(2w –z)
7) 5x – 3(2 - x)
17) (k-2)2- 8 + 3k
8) 4k(k – 3m) – 3m(m – 4k)
18) ( 6s -1)2 – ( 4s + 1)
9) 3(x –y ) – 2 ( y – x)
19) 2 ( 3y- 4) + ( y -5)2
10) -3 ( c – d) + 2 ( 4c -2d)
20) (2p +q)2 - q(4p – 2q)
(B) Factorise completely each of the following expressions : 1. 12xy – 4x2 11. 4x -3y –xy – 12
Algebraic Expressions
98
Module PMR
2. 6e – 18ef
12, a2 b2 + a2b + b + 1
3. 4x2 -100
13. 2m2 –m + 2mn
4, 75 – 3m2
14 9c2 – 100d2
5. 3y + 12
15 uv + wv –ux –wx
6. 20 – 5x2
16 ab+ bc –ad-cd
7. 3st – 15st2u
17. k2-14k + 49
8. 36x2 – 81y2
18. g2 -12g + 36
9. m3 – 9m
19. 3x-4y-6wx+8wy
10. 4p2 -1
20. 2pq- 6pz – 3rq + 9rz
Algebraic Expressions
99
Module PMR
(C) Express each of the following expressions as a single fraction in its simplest form 2 x p 1− p − − 1. 9. 3 x 12 3m m
2.
4q 3 − p pq
10
2m − 3 m + 2 − 4 6
3.
4 1− s − s 2s 2
11
2 m−3 − 3m 12m 2
4.
5 x −1 − 2w t
12.
8 1 − 4w − 5n 10n
5.
x+3 2− y − 2e 4
13.
3( p + 1) 6 − n − 3np n
6.
p 1− p + 3f f
14.
7 m−4 − 12m 4m 2
Algebraic Expressions
100
Module PMR
7
1 2z − 1 − 4n 8n 2
15.
8.
5p p2 + 6 − 6q 12 pq
16.
9 y − y ( y + 3) y + 3
2 4−n − 3n 9n 2
(D) Expand each of the following expressions 1. 2 (m+1) 10. (p + 2z )( p – x)
2. 3b (b – 3)
11 (n -7)2
3 -2a ( x – 4)
12. ( r – t)2 -4rt
4. 2k2 ( k – 7)
13 (4m -2)2 + 7m
5. – 5x ( x – 2y)
14 (a+ 2d)( a+ 2d)
Algebraic Expressions
101
Module PMR
6. 2e( 4e – f + 7)
7
15 (3ª +b)(2a- 2c)
16
2 s (6 x − 12 y + 9) 3
( x – 3y)( x + 3y)
8. -6pq(2pq + 4p – 3q)
17 . (2a + 1)( b- 3)
9 4 ( - 3s + 5h)
18. (x -2)( y + 3)
PMR past year questions 2004 1. Simplify (3x-1)2 –(7x + 4)
(2 marks)
2. Factorise completely a) 9xy -3x2
b) p2 – 6(p+1) – (8 –p)
( 3 marks)
Algebraic Expressions
102
Module PMR
1 1− p 3 2 − 3. Express as a single fraction in its simplest form 2m mp ( 3 marks)
2005 4. Simplify (2p- q)2 + q(4p –q)
5.
( 2marks)
Factorise completely each of the following expressions : (a) 4e – 12ef b) 3x 2 - 48 ( 3 marks)
5. Express
1 m+2 − as a single fraction in its simplest form. 2m 6m 2 ( 3 marks)
2006 6. Factorise completely 50 – 2m2
Algebraic Expressions
103
(2 marks)
Module PMR
7. Simplify 3 (2p -5) + (p – 3)2
8. Express
( 2 marks)
1 5 − 2v − as a single fraction in its simplest form 5m 15mv (3 marks)
2007 9. Factorise completely each of the following expressions : b) 12 – 3x2
a) 2y + 6
( 3 marks)
10. Expand each of the following expressions : (a) q(2 + p) (b) ( 3m –n)2 ( 3 marks)
11. Express
Algebraic Expressions
5 2 − 3w + as a single fraction in its simplest form 3n 6n (3 marks)
104
Module PMR
2008 12. Simplify 2p – 3q – (p + 5q)
( 2 marks)
13. Expand each of the following expressions : (a) 2g ( 5 –k)
(b) ( h – 5)(3h + 2) ( 3 marks)
14. Express
1 n−4 − as a single fraction in its simplest form n 3n ( 3marks)
CHAPTER 7 : ALGEBRAIC EXPRESSIONS ANSWERS A unknown 1. Number of books 2. number of monkeys 3. unknown : y Object : z 4. unknown : k Object : a
3) 0.7 4) 1 (ii) 1) unlike term 2) like term 3) unlike term 4) unlike term C) Algebraic Expressions i) Number of term 1) 3 2) 4 3) 5 ii) simplify Algebraic Exp. 1) 7x – 8y 2) 3z – 11w
ii) expressions which consist of 3 terms 1) (a + b)2 2) (2x – 5)2 3) ( 3e – 2f)2 iii) expressions which consist of 4 terms 1) (q +s)(p + r) 2) (b+ 3d)(2a+c) 3) (de – 7e)(1- e) 4) (5 –b)(2 – 3a ) J) Factorising & simplifying Alg Expressions. i) Simplifying alg Expressions 1 1. 2r 3ce 2 2 1 3. 3+ d 106
Module PMR
e+ f 3 ii) Addition & Subtraction of alg Exp. a) Alg Fraction with same denominator rt − 5s 1. 9k 3− x 2 3 xy b)Alg Fraction with different denominator 10 x + y 1. 5 8bq − 5aq 2. 2a 2 b 2 9abs + 9b − 4 + 2a 3. 36rs 4
iii) Mul & division of Alg Exp a) Multiplication of 2 alg fractions * Denominator with one term 14 1. 27 ry 3 2. 5 xz 2 p+r 3. 4 xq * denominator with 2 terms 1 1. 2 x( x + 3) 1 2 2( x + 3) b) Division of Alg Fractions * Denominator with one term x 1. 2z 3c 2. 2 3 3. − 2d ( a + b) Algebraic Expressions
4.
3 2 p (m + 5)
* Denominator with 2 terms 3 1. 2m g 2. 9h( y + x ) 4s 3. − r d +e 4 − d −e b 2 ( a + b) 5. 2
C) Express expressions as a single fraction in its simplest form 8 − x2 1. . 12 x 4q 2 − 3 2. pq 9s − 1 3. 2s 2 5t − 2 wx + 2 w 4. 2 wt 2 x + 6 − 2e + ey 5. 4e p + 3 − 3p 6. 3f 2n − 2 x + 1 7. 8n 2 9 p2 − 6 8. 12 pq PMR Past Year Questions
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