Tutorial polynomials .Doc

CHAPTER 6: POLYNOMIALS TUTORIAL POLYNOMIALS

1.

Find the quotient and remainder by using long division process when a)

x 3 − 2 x 2 − 6 x + 5 is divided by x − 3

b)

4 x 3 + 3 x 2 + 2 x − 4 is divided by x 2 − x + 1

c)

3 x 3 + 2 x 2 − 5 is divided by x 2 + 2

2.

3 2 Find the remainder when 2 x − 5 x − 28 x + 15 is divided by x − 2

3.

Determine if

4.

3 2 Find the value of r if P ( x) = 2 x − 9 x + 3 x + r leaves a remainder -54 when divided by

( x − 1)

3 2 is a factor of P ( x) = 12 x − 22 x + 2 x + 8

( x − 2)

5.

3 2 P( x) = 0 Factorise P ( x) = x − 3 x − x + 3 and hence solve the equation

(SCIENCE ONLY) 6.

4 2 Factorise the polynomial completely x − 15 x − 10 x + 24

(SCIENCE ONLY) 7.

Express a polynomial h(x) of degree 4, with leading coefficient 3 and -9, -6, 1 and 5 as its zeroes.

(SCIENCE ONLY) 8.

4 3 2 2 When x − ax + bx + 15 x is divided by x − 4 x + 3 , the remainder is 5 x + 3 .Find the

values of a and b.

MATHEMATICS UNIT

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CHAPTER 6: POLYNOMIALS

(SCIENCE ONLY) 9.

4 3 2 2 a, b and c Given that P ( x) = 2 x + 3 x − 10 x − 12 x + 8 = (2 x − a )( x − b)( x + c) , where are

positive integers (a)

10.

11.

(b)

State the zeroes of P (x)

(c)

2x + 1 ( x − 1) 2 ( x + 1)

(d)

x +1 x3 −1

Express the following in partial fractions

(a)

7 x − 12 ( x − 1)( x − 2 )

(b)

x+4 ( x − 5 )( 2 x 2 + 1)

Express the following in partial fractions (a)

(b)

12.

Determine the values of a, b and c

3 x 3 + 12 ( x − 1)( x 2 + 2)

x2 + 2x + 4 ( x − 1)( x − 2 )

(c)

x2 + x +1 x 2 + 2x + 1

x3 x 2 − 3x + 2

Polynomial

P (x)

(d)

3 2 is defined by P ( x) = x − 4 x − 7 x + 10.

(a)

Using long division, show that ( x + 2) is a factor of P (x)

(b)

Find all the zeroes of P (x)

(c)

Find the remainder when P (x) is divided by ( x − 3).

Hence, express P (x) in the form of ( x − 3)Q( x) + R ( x) when Q(x) is the quotient and R (x) is the remainder

13.

2n 2 Given that P ( x) = x − (k + 3)x + kx + 2 where k and n are positive integers

(a)

Show that ( x − 1) is a factor of P (x)

MATHEMATICS UNIT

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CHAPTER 6: POLYNOMIALS

14.

15.

(b)

Find the value of n when k = 3, ( x − 2) is a factor of P (x)

(c)

Hence, factorise P (x) completely

3 2 Given that P ( x) = 2 x − x − 5 x − 2

(a)

If ( x − a ) is a factor of P (x) where a is a positive integer, find the value of a

(b)

Obtain the roots of the equation P ( x) = 0

(c)

20 x 2 + 10 Express ( x + 1) P ( x ) in partial fractions

Given that ( x + 3) and (2 x − 1) are factors of the polynomial P ( x) = 2 x 4 + ax 3 + bx 2 + 11x − 12 (a)

By using the factor theorem, find the values of the constants a and b.

(b)

Factorise P (x) completely and show that the quadratic factor of P (x) is always positive for all real values of x.

(c)

Find the set of values x which satisfies the inequality P ( x) > 0

PAST YEAR EXAMINATION QUESTIONS

2004/2005 16.

Given

( x + 3)

2 3 P (x) is one factor of P ( x) = 9 − 12 x − 11x − 2 x . Factorise completely,

13 x + 18 and express P ( x) as a sum of partial fractions

17.

[8 marks]

3 2 A polynomial has the form P ( x) = 2 x − 3 x − px + q , with x real and p, q constants.

(a)

When P (x) is divided by ( x − 1) the remainder is (2 − 4 x) . Find the values of p and q, and factorise P (x) completely if 2 is one of the roots

MATHEMATICS UNIT

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[7 marks]

CHAPTER 6: POLYNOMIALS

2005/2006 18.

(a)

3 2 ( x + 1) Polynomial P ( x) = 2 x + ax − x + b has as a factor and leaves a remainder

12 when divided by ( x − 3) . Determine the values of a and b .

19.

[6 marks]

3 2 ( x + 1) ( x − 2) Two factors of the polynomial P ( x) = x + ax + bx − 6 are and . Determine

the values of a and b and find the third factor of the polynomial. Hence, express 2 x 2 − 5 x − 13 P( x) as a sum of partial fractions

[13 marks]

2006/2007

20.

4 x 3 − 3 x 2 + 6 x − 27 x 4 + 9x 2 Find the values of A, B, C and D for the expression in the form of A B Cx + D + 2+ 2 x + 9 where A, B, C and D are constants partial fractions x x

21.

(a)

Show that

( x − 3)

3 2 is a factor of the polynomial P ( x) = x − 2 x − 5 x + 6 . Hence,

factorise P (x) completely (b)

[5 marks]

[4 marks]

2 ( x − 1) If f ( x) = ax + bx + c leaves remainder 1, 25 and 1 on division by ,

( x + 1) and ( x − 2) respectively, find the values of a, b and c. Hence show that f (x) has two equal real roots.

[9 marks]

22.

2x + 1 2 Express ( x + 2)( x − 2 x + 4) in partial fractions

[6 marks]

23.

(a)

2007/2008

Find a cubic polynomial Q( x) = ( x + a )( x + b)( x + c) satisfying the following 3 conditions : the coefficient of x is 1, Q(-1) = 0 , Q(2) = 0 and Q(3) = -8

[4 marks] MATHEMATICS UNIT

Page 55

CHAPTER 6: POLYNOMIALS (b)

3 2 ( x + 2) A polynomial P( x) = ax − 4 x + bx + 18 has a factor and a remainder

(2 x + 18) when divided by ( x + 1) . Find the value of a and b. Hence, factorise P (x) completely

[8 marks]

2008/2009

24.

5 x 2 + 3x + 8 2 Express (1 − x )(1 + x) in partial fractions

25.

3 2 2 Polynomial P ( x) = mx − 8 x + nx + 6 can be divided exactly by ( x − 2 x − 3) . Find the

[5 marks]

values of m and n. Using these values of m and n, factorise the polynomial completely. 4 3 2 Hence, solve the equation 3 x − 14 x + 11x + +16 x − 12 = 0 using the polynomial P (x)

[13 marks]

2009/2010 26.

3 2 ( x + 2) ( x − 5) Given a polynomial P ( x) = 2 x + ax + bx − 30 has factors and .

(a) Find the value of the constants a and b.

[6 marks]

(b) Factorize P (x ) completely.

[3 marks]

(c) Obtain the solution set for P ( x ) < 0

[3 marks]

QS016 2010/2011 27.

2 ( x + 1) ( x − 1) Dividing M ( x) = x + ax + b by and give a remainder of -12 and -16

respectively. Determine the values of

a and b.

QA016 2010/2011 MATHEMATICS UNIT

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[6 marks]

CHAPTER 6: POLYNOMIALS 28.

By using the partial fraction method, show that 1 1 1 1  =  −  x −4 4 x−2 x+2 2

Hence , find

x2 +1 ∫ x 2 − 4 dx

[6 marks]

QS015 2011/2012 29.

3 2 a b The polynomial p ( x) = x − 2 x + ax + b, where and are constants, has a factor of

( x − 2) and leaves a remainder of a 3 when it is divided by ( x − a ).

(a) Find the values of a and b.

[6 marks]

(b) Factorize p (x ) completely by using the values of a and b obtained from part (a ).

a Hence, find the real roots of p ( x ) = 0, where and b are not equal to zero. [6 marks]

QA016 2011/2012 30.

3 2 2 8x − 1 When 3 x + px − qx − 3 is divided by x + x − 2 , the remainder is . By using the Remainder Theorem, find the values of p and q . Hence, obtain the quotient.

[7 marks]

SUGGESTED ANSWERS POLYNOMIALS

1.

2 (a) Q ( x ) = x + x − 3, R ( x ) = −4

(b) Q ( x ) = 4 x + 7,

R ( x ) = 5 x − 11

(c) Q( x) = 3 x + 2,

R( x) = −6 x − 9

2. 3.

-45 ( x − 1) is a factor of P(x)

4.

r = -40

MATHEMATICS UNIT

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CHAPTER 6: POLYNOMIALS 5.

x = −1 , x = 1 , x = 3

6.

f ( x) = ( x − 1)( x + 2)( x + 3)( x − 4)

7.

h( x) = 3 x 4 + 27 x 3 − 93 x 2 − 747 x + 810

8.

a = 2, b = -6 a = 1, b = 2, c = 2

9.

(a)

1 ,2,−2,−2 (b) 2

10.

5 2 + (a) x − 1 x − 2

1 3 1 + − 2 4( x + 1) (c) 4( x − 1) 2( x − 1)

3 6 x + 13 − 2 (b) 17( x − 5) 17(2 x + 1)

2 2x + 1 − 2 (d) 3( x − 1) 3 x + x + 1

1− 11.

(a) 1− (b)

12.

7 12 + x −1 x − 2 1 1 + x + 1 (x + 1)2

(

3+ (c)

)

5 2x + 8 − 2 ( x − 1) ( x + 2)

x +3− (d)

1 8 + x −1 x − 2

(a) ( x + 2) is a factor of P(x) (b) -2, 1, 5 2 (c) R( x) = −20, P( x) = ( x − 3)( x − x − 10) − 20

13.

2 (c) P ( x) = ( x − 1)( x − 2)( x + 3 x + 1)

(b) 2

14.

(a) 2

15.

(a)

1 2,− ,−1 2 (b)

a = 11, b = 20

10 10 2 24 + + − 2 x − 2 2x + 1 (c) x + 1 ( x + 1) 2 (b) P ( x) = ( x + 3)(2 x − 1)( x + 3 x + 4)

1 (−∞,−3) ∪ ( , ∞) 2 (c) P ( x) = ( x + 3) 2 (1 − 2 x) 16. 17.

2 1 3 + − 1 − 2 x x + 3 ( x + 3) 2

(a) p = 3, q = 2, P ( x) = ( x − 2)(2 x − 1)( x + 1)

MATHEMATICS UNIT

Page 58

CHAPTER 6: POLYNOMIALS 18.

(a) a = -5, b = 6 ( x + 3)

19.

a = 2, b = -5,

20.

2 10 A = , B = −3, C = , D = 0 3 3

21.

(a) ( x − 1)( x + 2)( x − 3) −

22. 23.

24.

1 1 2 − + x +1 x − 2 x+3

(b) a = 4, b = -12, c = 9

1 x+4 + 2 4( x + 2) 4( x − 2 x + 4)

3 2 (a) Q( x) = x − 6 x + 3 x + 10

2 (b) a = 1, b = −3, P ( x) = ( x + 2)( x − 3)

4 1 5 − + 1 − x 1 + x (1 + x) 2 m = 3, n = −5, P ( x) = ( x − 3)(3 x − 2)( x + 1)

25. 26.

(a)

a = −3

(b)

P ( x ) = ( x + 2)( x − 5)( 2 x + 3)

(c)

3 x ∈ (−∞,−2) ∪ (− ,5) 2

27.

a = −2 , b = −15

28.

x+

29.

(a) a = 0, b = 0

b = −29

5   x − 2  ln  +c 4   x + 2  a = −2, b = 4

(b) x = 2, x = − 2 , x = 2

30.

p = 4 and q = −3 , Q(x ) = 3 x + 1

MATHEMATICS UNIT

Page 59

2 x = 3, x = −1, x = , x = 2 3