Tutorial polynomials .Doc

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POLYNOMIALS

1. Find the quotient and remainder by using long division process when

a) 2 6 5 2 3− − + x x x _{is divided by}x−3 b) 4x3+3x2 +2x−4 is divided by x2−x+1 c) 3x3+2x2 −5 is divided by x2+2

2. Find the remainder when 2x3−5x2 −28x+15 is divided by x−2

3. Determine if (x−1)is a factor of ( ) 12 22 2 8 2 3 − + + = x x x x P

4. Find the value of r if P(x)=2x −9x +3x+r 2

3

leaves a remainder -54 when divided by ) 2 (x− 5. Factorise ( ) 3 3 2 3₋ ₋ ₊ =x x x x

P _{and hence solve the equation}P(x)=0

(SCIENCE ONLY)

6. Factorise the polynomial completely x4 −15x2 −10x+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. When x4 −ax3+bx2+15x is divided by x2 −4x+3, the remainder is 5x+3.Find the values of a and b.

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(SCIENCE ONLY)

9. Given that P(x)=2x4 +3x3−10x2−12x+8=(2x−a)(x−b)(x+c)2,where a,bandcare positive integers

(a) Determine the values of a,bandc (b) State the zeroes of P(x)

10. Express the following in partial fractions

(a) ( 1)( 2) 12 7 − − − x x x (c) ( 1) ( 1) 1 2 2 + − + x x x (b) ( 5)(2 1) 4 2 + − + x x x (d) 1 1 3− + x x

11. Express the following in partial fractions (a) ) 2 )( 1 ( 4 2 2 − − + + x x x x (c) ( 1)( 2) 12 3 2 3 + − + x x x (b) 2 1 1 2 2 + + + + x x x x (d) 2 3 2 3 + − x x x 12. Polynomial P(x)is defined by ( ) 4 7 10. 2 3₋ ₋ ₊ =x x x x P

(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 ) (x R _{is the remainder} 13. Given that ( )

(

3

)

2 2 2 ₋ ₊ ₊ ₊ = x k x kx x

P n _{where k and n are positive integers}

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(c) Hence, factorise P(x) completely

14. Given that P(x)=2x3 −x2 −5x−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) Express ( 1) ( ) 10 20 2 x P x x + + in partial fractions

15. Given that (x+3) and (2x−1)are factors of the polynomial 12 11 2 ) (_x = _x4 +_ax3+_bx2 + _x− P

(a) By using the factor theorem, find the values of the constants aandb.

(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)is one factor of P(x)=9−12x−11x2 −2x3. Factorise P(x) completely,

and express ( ) 18 13 x P x+

as a sum of partial fractions [8 marks]

17. A polynomial has the form P x = x − x − px+q 2

3 3 2 )

( _{, with x real and p, q constants.} (a) When P(x) is divided by (x−1)the remainder is (2−4x). Find the values of p

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2005/2006

18. (a) Polynomial P(x)=2x3+ax2 −x+bhas (x+1)as a factor and leaves a remainder 12 when divided by (x−3). Determine the values of a and b . [6 marks]

19. Two factors of the polynomial P(x)=x3 +ax2+bx−6are (x+1)and (x−2). Determine the values of a and b and find the third factor of the polynomial. Hence, express

) ( 13 5 2 2 x P x x − −

as a sum of partial fractions [13 marks]

2006/2007

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

where A, B, C and D are constants [5 marks] 21. (a) Show that (x−3)is a factor of the polynomial ( ) 2 5 6

2 3₋ ₋ ₊

=x x x

x

P _{. Hence,}

factorise P(x) completely [4 marks] (b) If f x =ax +bx+c

2 )

( _{leaves remainder 1, 25 and 1 on division by}(x−1)_, )

1

(x+ _and(x−2)_{respectively, find the values of a, b and c. Hence show that} )

(x

f _{has two equal real roots.} _{[9 marks]}

2007/2008 22. Express ( 2)( 2 4) 1 2 2 − + + + x x x x

in partial fractions [6 marks]

23. (a) Find a cubic polynomial Q(x)=(x+a)(x+b)(x+c)satisfying the following conditions : the coefficient of x is 1, Q(-1) = 0 , Q(2) = 0 and Q(3) = -83

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) 18 2

( x+ _{when divided by}(x+1)_{. Find the value of a and b. Hence, factorise} ) (x P _completely _{[8 marks]} 2008/2009 24. Express (1 )(1 ) 8 3 5 2 2 x x x x + − + +

in partial fractions [5 marks]

25. Polynomial P(x)=mx3 −8x2 +nx+6can be divided exactly by (x2 −2x−3). Find the values of m and n. Using these values of m and n, factorise the polynomial completely. Hence, solve the equation 3x4 −14x3 +11x2 ++16x−12=0using the polynomial P(x)

[13 marks]

2009/2010

26. Given a polynomial P(x)=2x3 +ax2 +bx−30 has factors (x+2)and (x−5). (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. Dividing M(x)=x2 +ax+b by (x+1) and (x−1) give a remainder of -12 and -16 respectively. Determine the values of aand .b [6 marks]

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

∫

− + dx x x 4 1 2 2 [6 marks] QS015 2011/2012 29. The polynomial ( ) 2 , 2 3 b ax x x x

p = − + + _wherea_andb_{are constants, has a factor of}

) 2

(x− _{and leaves a remainder of} 3

a _{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).

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

[6 marks]

QA016 2011/2012

30. When3x3+ px2 −qx−3is divided by x2 +x−2, the remainder is 8x−1.

By using the Remainder Theorem, find the values of p and q . Hence, obtain the quotient. [7 marks] SUGGESTED ANSWERS POLYNOMIALS 1. _(a)Q(x)=x2+x−3, R(x)=−4 (b) Q(x)=4x+7, R(x)=5x−11 (c) Q(x)=3x+2, R(x)=−6x−9 2. -45 3. (x−1)is a factor of P(x) 4. r = -40

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6. f(x)=(x−1)(x+2)(x+3)(x−4) 7. ( ) 3 27 93 747 810 2 3 4₊ ₋ ₋ ₊ = x x x x x h 8. a = 2, b = -6 9. (a) 2 , 2 , 1 = = = b c a (b) 2 , 2 , 2 , 2 1 − − 10. (a) 2 2 1 5 − + − x x _(c) 4( 1) 1 ) 1 ( 2 3 ) 1 ( 4 1 2 − + − + − x x x (b) 17(2 1) 13 6 ) 5 ( 17 3 2 + + − − x x x _(d) 3

(

1

)

1 2 ) 1 ( 3 2 2 + + + − − x x x x 11. (a) 2 12 1 7 1 − + − − x x _(c) ( 2) 8 2 ) 1 ( 5 3 ₂ + + − − + x x x (b)

( )

2 1 1 1 1 1 + + + − x x _(d) 2 8 1 1 3 − + − − + x x x 12. (a) (x+2)is a factor of P(x) (b) -2, 1, 5 (c) R(x)=−20,P(x)=(x−3)(x2 −x−10)−20 13. (b) 2 (c)P(x)=(x−1)(x−2)(x2+3x+1) 14. (a) 2 (b) 1 , 2 1 , 2− − (c) 2 1 24 2 2 ) 1 ( 10 1 10 2 + − − + + + + x x x x 15. (a) a=11,b=20 (b) ( ) ( 3)(2 1)( 3 4) 2+ + − + = x x x x x P (c) ) , 2 1 ( ) 3 , (−∞− ∪ ∞ 16. ) 2 1 ( ) 3 ( ) (x x 2 x P = + − 2 ) 3 ( 3 3 1 2 1 2 + − + + − x x x 17. (a) p=3,q=2,P(x)=(x−2)(2x−1)(x+1)

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18. (a) a = -5, b = 6 19. a = 2, b = -5, ) 3 (x+ 3 2 2 1 1 1 + + − − + x x x 20. 0 , 3 10 , 3 , 3 2 ₌₋ ₌ ₌ = B C D A 21. (a) (x−1)(x+2)(x−3) (b) a = 4, b = -12, c = 9 22. 4( 2 4) 4 ) 2 ( 4 1 2 − + + + + − x x x x 23. (a) Q(x)=x3 −6x2 +3x+10 (b) a=1,b=−3,P(x)=(x+2)(x−3)2 24. 2 ) 1 ( 5 1 1 1 4 x x x− + + + − 25. ) 1 )( 2 3 )( 3 ( ) ( , 5 , 3 =− = − − + = n P x x x x m 2 , 3 2 , 1 , 3 =− = = = x x x x 26. (a) a=−3 b=−29 (b) P(x)=(x+2)(x−5)(2x+3) (c) ,5) 2 3 ( ) 2 , (−∞− ∪ − ∈ x 27. a=−2 , b=−15 28. c x x x _+            + − + 2 2 ln 4 5 29. _(a)a=0,b=0 a=−2,b=4 (b) x=2,x=− 2,x= 2 30. p=4 and q=−3 ,Q

( )

x =3x+1