Cambridge Additional Mathematics

174 Polynomials (Chapter 6)

2 Carry out the following divisions:

a

x^3

x+2

b

x^3

(x+ 2)(x+3)

3 Findallcubic polynomials with zeros^14 , 1 §

p

5.

4 If f(x)=x^3 ¡ 3 x^2 ¡ 9 x+b has (x¡k)^2 as a factor, show that there are two possible

values ofk. For each of these two values ofk, find the corresponding value forb, and hence solve

f(x)=0.

5 Find the remainder when:

a x^3 ¡ 5 x^2 +9 is divided by x¡ 2 b 4 x^3 +7x¡ 11 is divided by 2 x¡ 1.

6 When f(x)=2x^3 ¡x^2 +ax¡ 4 is divided by x¡ 3 , the remainder is 56.

a Finda. b Find the remainder when f(x) is divided by x+1.

7aUse the Factor theorem to show that x¡ 2 is a factor of x^3 ¡ 13 x+18.

b Write x^3 ¡ 13 x+18 in the form (x¡2)(ax^2 +bx+c), where a,b,c 2 Z.

c Find the real roots of x^3 +18=13x.

8 x¡ 2 and x+3are factors of ax^3 ¡ 3 x^2 ¡ 11 x+b. Findaandb.

9 x+1is a factor of f(x)=x^3 +5x^2 +kx+4. Findk, and the zeros of f(x).

10 2 x¡ 1 is a factor of f(x)=2x^3 ¡ 9 x^2 +ax+b, and when f(x) is divided by x¡ 1 the

remainder is¡ 15.

a Findaandb. b Write f(x) as a product of linear factors.

11 Solve forx: 2 x^3 ¡ 2 x^2 ¡ 28 x+48=0

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_06\174CamAdd_06.cdr Thursday, 3 April 2014 5:19:24 PM BRIAN