Cambridge Additional Mathematics

Polynomials (Chapter 6) 159

If P(x) is divided by ax+b until a constant remainderRis obtained, then

P(x)

ax+b

=Q(x)+

R

ax+b

where ax+b is thedivisor, D(x),

Q(x) is thequotient,

and R is theremainder.

Notice that P(x)=Q(x)£(ax+b)+R.

Division algorithm

We can divide a polynomial by another polynomial using an algorithm similar to that used for division of

whole numbers:

Step 1: What do we multiplyxby to get 2 x^3?

The answer is 2 x^2 ,

and 2 x^2 (x+2)=2x^3 +4x^2.

Step 2: Subtract 2 x^3 +4x^2 from 2 x^3 +7x^2.

The answer is 3 x^2.

Step 3: Bring down the 10 xto obtain 3 x^2 +10x.

Return toStep 1with the question:

“What must we multiplyxby to get 3 x^2 ?”

The answer is 3 x, and 3 x(x+2)=3x^2 +6x ....

We continue the process until we are left with a constant.

2 x^2 +3x+4

x+2 2 x^3 +7x^2 +10x+15

¡( 2 x^3 +4x^2 )

3 x^2 +10x

¡( 3 x^2 +6x)

4 x+15

¡( 4 x+8)

7

So,

2 x^3 +7x^2 +10x+15

x+2

=2x^2 +3x+4+

7

x+2

Example 4 Self Tutor

Find the quotient and remainder for

x^3 ¡x^2 ¡ 3 x¡ 5

x¡ 3

.

Hence write x^3 ¡x^2 ¡ 3 x¡ 5 in the form Q(x)£(x¡3) +R.

x^2 +2x+3

x¡ 3 x^3 ¡ x^2 ¡ 3 x¡ 5

¡(x^3 ¡ 3 x^2 )

2 x^2 ¡ 3 x

¡( 2 x^2 ¡ 6 x)

3 x¡ 5

¡( 3 x¡ 9 )

4

The quotient is x^2 +2x+3

and the remainder is 4.

)

x^3 ¡x^2 ¡ 3 x¡ 5

x¡ 3

=x^2 +2x+3+

4

x¡ 3

) x^3 ¡x^2 ¡ 3 x¡5=(x^2 +2x+ 3)(x¡3) + 4.

Check your answer by

expanding the RHS.

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_06\159CamAdd_06.cdr Thursday, 3 April 2014 5:18:23 PM BRIAN