Cambridge Additional Mathematics

Polynomials (Chapter 6) 167

9 2 x+1and x¡ 2 are factors of P(x)=2x^4 +ax^3 +bx^2 +18x+8.

a Findaandb. b Hence, solve P(x)=0.

10 x^3 +3x^2 ¡ 9 x+c, c 2 R, has two identical linear factors. Prove thatcis either 5 or¡ 27 , and

factorise the cubic into linear factors in each case.

Consider the cubic polynomial P(x)=x^3 +5x^2 ¡ 11 x+3.

If we divide P(x) by x¡ 2 , we find that

x^3 +5x^2 ¡ 11 x+3

x¡ 2

=x^2 +7x+3+

9

x¡ 2

remainder

So, when P(x) is divided by x¡ 2 , the remainder is 9.

Notice also that P(2) = 8 + 20¡22 + 3

=9, which is the remainder.

By considering other examples like the one above, we formulate theRemainder theorem.

The Remainder Theorem

When a polynomial P(x) is divided by x¡k until a constant remainderR

is obtained, then R=P(k).

Proof: By the division algorithm, P(x)=Q(x)(x¡k)+R

Letting x=k, P(k)=Q(k)£0+R

) P(k)=R

When using the Remainder theorem, it is important to realise that the following statements are equivalent:

² P(x)=(x¡k)Q(x)+R

² P(k)=R

² P(x) divided by x¡k leaves a remainder ofR.

Example 15 Self Tutor

Use the Remainder theorem to find the remainder when x^4 ¡ 3 x^3 +x¡ 4 is divided by x+2.

If P(x)=x^4 ¡ 3 x^3 +x¡ 4 , then

P(¡2) = (¡2)^4 ¡3(¡2)^3 +(¡2)¡ 4

=16+24¡ 2 ¡ 4

=34

) when x^4 ¡ 3 x^3 +x¡ 4 is divided by x+2,

the remainder is 34. fRemainder theoremg

C The Remainder theorem

The Remainder theorem

allows us to find a

remainder without having

to perform the division.

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Y:\HAESE\CAM4037\CamAdd_06\167CamAdd_06.cdr Monday, 14 April 2014 5:56:37 PM BRIAN