Cambridge Additional Mathematics

Polynomials (Chapter 6) 169

7 Consider f(x)=2x^3 +ax^2 ¡ 3 x+b. When f(x) is divided byx+1, the remainder is 7. When

f(x) is divided by x¡ 2 , the remainder is 28. Find the remainder when f(x) is divided by x+3.

8aSuppose a polynomial P(x) is divided by 2 x¡ 1 until a constant remainderRis obtained.

Show that R=P(^12 ).

Hint: P(x)=Q(x)(2x¡1) +R.

b Find the remainder when:

i 4 x^2 ¡ 10 x+1 is divided by 2 x¡ 1 ii 2 x^3 ¡ 5 x^2 +8 is divided by 2 x¡ 1

iii 4 x^3 +7x¡ 3 is divided by 2 x+1.

9 When 2 x^3 +ax^2 +bx+4 is divided by x+1the remainder is¡ 5 , and when divided by 2 x¡ 1

the remainder is 10. Findaandb.

10 When P(z) is divided by z^3 ¡ 3 z+2 the remainder is 4 z¡ 7.

Find the remainder when P(z) is divided by: a z¡ 1 b z¡ 2.

For any polynomial P(x), kis a zero of P(x) , x¡k is a factor of P(x).

Proof: kis a zero of P(x),P(k)=0 fdefinition of a zerog

,R=0 fRemainder theoremg

,P(x)=Q(x)(x¡k) fdivision algorithmg

,x¡k is a factor of P(x) fdefinition of a factorg

TheFactor theoremsays that if 2 is a zero of P(x) then x¡ 2 is a factor of P(x), and vice versa.

We can use the Factor theorem to determine whether x¡k is a factor of a polynomial, without having to

perform the long division.

Example 17 Self Tutor

Determine whether:

a x¡ 2 is a factor of x^3 +3x^2 ¡ 13 x+6 b x+3is a factor of x^3 ¡ 8 x+7.

a Let P(x)=x^3 +3x^2 ¡ 13 x+6

) P(2) = (2)^3 + 3(2)^2 ¡13(2) + 6

=8+12¡26 + 6

=0

Since P(2) = 0, x¡ 2 is a factor of x^3 +3x^2 ¡ 13 x+6. fFactor theoremg

b Let P(x)=x^3 ¡ 8 x+7

) P(¡3) = (¡3)^3 ¡8(¡3) + 7

=¡27 + 24 + 7

=4

Since P(¡3) 6 =0, x+3isnota

factor of x^3 ¡ 8 x+7. fFactor theoremg

D The Factor theorem

When

is divided by ,

a remainder of

is left over.

xx

x

3 ¡8+7

+3

4

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Y:\HAESE\CAM4037\CamAdd_06\169CamAdd_06.cdr Monday, 14 April 2014 5:55:38 PM BRIAN