Cambridge Additional Mathematics

170 Polynomials (Chapter 6)

Example 18 Self Tutor

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

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

Since x¡ 2 is a factor, P(2) = 0 fFactor theoremg

) (2)^3 +k(2)^2 ¡3(2) + 6 = 0

) 8+4k=0

) k=¡ 2

The coefficient of

x^3 is 1 £1=1

The constant term

is ¡ 2 £¡3=6

So, P(x)=x^3 ¡ 2 x^2 ¡ 3 x+6=(x¡2)(x^2 +bx¡3)

=x^3 +(b¡2)x^2 +(¡ 2 b¡3)x+6

Equatingx^2 s: b¡2=¡ 2

) b=0

So, P(x)=(x¡2)(x^2 ¡3)

=(x¡2)(x+

p

3)(x¡

p

3)

Example 19 Self Tutor

2 x¡ 1 is a factor of f(x)=4x^3 ¡ 4 x^2 +ax+b, and the remainder

when f(x) is divided by x¡ 1 is¡ 1. Find the values ofaandb.

2 x¡ 1 is a factor of f(x),sof(^12 )=0

) 4(^12 )^3 ¡4(^12 )^2 +a(^12 )+b=0

)^12 a+b=^12 .... (1)

Also, f(1) =¡ 1 fRemainder theoremg

) 4(1)^3 ¡4(1)^2 +a(1) +b=¡ 1

) a+b=¡ 1 .... (2)

Solving simultaneously: ¡a¡ 2 b=¡ 1 f¡ 2 £(1)g

a+b=¡ 1 f(2)g

Adding, ¡b=¡ 2

) b=2 and a=¡ 3

EXERCISE 6D

1 Use the Factor theorem to determine whether:

a x¡ 1 is a factor of 4 x^3 ¡ 7 x^2 +5x¡ 2 b x¡ 3 is a factor of x^4 ¡x^3 ¡ 4 x^2 ¡ 15

c x+2 is a factor of 3 x^3 +5x^2 ¡ 6 x¡ 8 d x+4 is a factor of 2 x^3 +6x^2 +4x+16.

If is a factor of ,

then.

21 ()

( )=0

xfx

f

¡

__Qw

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_06\170CamAdd_06.cdr Friday, 20 December 2013 1:00:36 PM BRIAN