Cambridge Additional Mathematics

Polynomials (Chapter 6) 171

2aFindcgiven that x+1is a factor of 5 x^3 ¡ 3 x^2 +cx+10.

b Findcgiven that x¡ 3 is a factor of x^4 ¡ 2 x^3 +cx^2 ¡ 4 x+3.

c Findbgiven that x+2is a factor of x^6 +bx^5 ¡ 2 x^3 ¡ 5 x+6.

3 x+2is a factor of P(x)=2x^3 +x^2 +kx¡ 4.

Findk, and hence write P(x) as a product of linear factors.

4 x¡ 3 is a factor of P(x)=3x^3 +kx^2 ¡ 5 x+6.

a Findk. b Write P(x) in the form P(x)=(x¡3)(ax^2 +bx+c).

c Find all solutions to P(x)=0.

5 2 x^3 +ax^2 +bx+5 has factors x¡ 1 and x+5. Findaandb.

6 x¡ 2 is a factor of f(x)=x^3 +ax^2 ¡ 11 x+b. The remainder when f(x) is divided by x+1is

15. Findaandb.

7 x+3is a factor of P(x)=2x^3 +9x^2 +ax+b. When P(x) is divided by x+4, the remainder

is¡ 18.

a Findaandb.

b Find the remainder when P(x) is divided by x¡ 2.

c Write P(x) in the form P(x)=(x+ 3)(px^2 +qx+r).

d Find the zeros of P(x).

8 2 x¡ 1 is a factor of P(x)=2x^3 +ax^2 ¡ 8 x+b. When P(x) is divided by x¡ 1 , the remainder

is 3.

a Findaandb.

b Find the irrational roots of P(x)=0, giving your answer in the form x=p§

p

q where

p,q 2 Z.

9aConsider P(x)=x^3 ¡a^3 whereais real.

i Find P(a). What is the significance of this result?

ii Factorise x^3 ¡a^3 as the product of a real linear and a quadratic factor.

b Now consider P(x)=x^3 +a^3 , whereais real.

i Find P(¡a). What is the significance of this result?

ii Factorise x^3 +a^3 as the product of a real linear and a quadratic factor.

10 Find the real numberasuch that (x¡ 1 ¡a) is a factor of P(x)=x^3 ¡ 3 ax¡ 9.

E CUBIC EQUATIONS

InDiscovery 4inChapter 3on page 97 , we considered the sum and products of roots of a quadratic.

In particular, we saw that

(x¡®)(x¡ ̄)=x^2 ¡(®+ ̄)x+® ̄.

If we perform a similar expansion for a cubic, we find that

(x¡®)(x¡ ̄)(x¡°)=x^3 ¡(®+ ̄+°)x^2 +(® ̄+ ̄°+°®)x¡® ̄°.

In both cases, theproduct of the rootshas the same size as the constant term in the expanded polynomial.

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_06\171CamAdd_06.cdr Monday, 23 December 2013 1:37:02 PM BRIAN