Cambridge Additional Mathematics

Polynomials (Chapter 6) 161

DIVISION BY QUADRATICS

As with division by linears, we can use thedivision algorithmto divide polynomials by quadratics. The

division process stops when the remainder has degree less than that of the divisor, so

If P(x) is divided by ax^2 +bx+c then

P(x)

ax^2 +bx+c

=Q(x)+

ex+f

ax^2 +bx+c

where ax^2 +bx+c is thedivisor,

Q(x) is thequotient,

and ex+f is theremainder.

The remainder will be linear if e 6 =0, and constant if e=0.

Example 6 Self Tutor

Find the quotient and remainder for

x^4 +4x^3 ¡x+1

x^2 ¡x+1

.

Hence write x^4 +4x^3 ¡x+1 in the form Q(x)£(x^2 ¡x+1)+R(x).

x^2 +5x+4

x^2 ¡x+1 x^4 +4x^3 +0x^2 ¡ x+1

¡(x^4 ¡ x^3 + x^2 )

5 x^3 ¡ x^2 ¡ x

¡( 5 x^3 ¡ 5 x^2 +5x)

4 x^2 ¡ 6 x+1

¡( 4 x^2 ¡ 4 x+4)

¡ 2 x¡ 3

The quotient is x^2 +5x+4

and the remainder is ¡ 2 x¡ 3.

) x^4 +4x^3 ¡x+1

=(x^2 +5x+ 4)(x^2 ¡x+1)¡ 2 x¡ 3

EXERCISE 6A.3

1 Find the quotient and remainder for:

a

x^3 +2x^2 +x¡ 3

x^2 +x+1

b

3 x^2 ¡x

x^2 ¡ 1

c

3 x^3 +x¡ 1

x^2 +1

d

x¡ 4

x^2 +2x¡ 1

2 Carry out the following divisions and also write each in the form P(x)=Q(x)D(x)+R(x):

a

x^2 ¡x+1

x^2 +x+1

b

x^3

x^2 +2

c

x^4 +3x^2 +x¡ 1

x^2 ¡x+1

d

2 x^3 ¡x+6

(x¡1)^2

e

x^4

(x+1)^2

f

x^4 ¡ 2 x^3 +x+5

(x¡1)(x+2)

3 Suppose P(x)=(x¡2)(x^2 +2x+3)+7. Find the quotient and remainder when P(x)is divided

by x¡ 2.

4 Suppose f(x)=(x¡1)(x+ 2)(x^2 ¡ 3 x+5)+15¡ 10 x. Find the quotient and remainder when

f(x) is divided by x^2 +x¡ 2.

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_06\161CamAdd_06.cdr Thursday, 3 April 2014 5:18:29 PM BRIAN