Cambridge Additional Mathematics

168 Polynomials (Chapter 6)

Example 16 Self Tutor

When 2 x^3 +2x^2 +ax+b is divided by x+3, the remainder is¡ 11.

When the same polynomial is divided by x¡ 2 , the remainder is 9.

Findaandb.

Let P(x)=2x^3 +2x^2 +ax+b

Now P(¡3) =¡ 11 and P(2) = 9 fRemainder theoremg

So, 2(¡3)^3 +2(¡3)^2 +a(¡3) +b=¡ 11

) ¡54 + 18¡ 3 a+b=¡ 11

) ¡ 3 a+b=25 .... (1)

and 2(2)^3 + 2(2)^2 +a(2) +b=9

) 16+8+2a+b=9

) 2 a+b=¡ 15 .... (2)

Solving simultaneously: 3 a¡b=¡ 25 f¡ 1 £(1)g

2 a+b=¡ 15 f(2)g

Adding, 5 a =¡ 40

) a=¡ 8

Substituting a=¡ 8 in (2) gives 2(¡8) +b=¡ 15

) b=1

EXERCISE 6C

1 For P(x) a real polynomial, write two equivalent statements for each of:

a If P(2) = 7, then ......

b If P(x)=Q(x)(x+3)¡ 8 , then ......

c If P(x) divided by x¡ 5 has a remainder of 11 then ......

2 Without performing division, find the remainder when:

a x^3 +2x^2 ¡ 7 x+8 is divided by x¡ 1

b 2 x^3 +x^2 ¡ 5 x+11 is divided by x+3

c x^4 ¡ 2 x^2 +3x¡ 1 is divided by x+2.

3 Use the Remainder theorem to find the remainder when x^3 ¡x^2 ¡ 3 x¡ 5 is divided by x¡ 3.

Check that your answer is the same as when this long division was performed on page 159.

4 Find a 2 R such that:

a when x^3 ¡ 2 x+a is divided by x¡ 2 , the remainder is 7

b when 2 x^3 +x^2 +ax¡ 5 is divided by x+1, the remainder is¡ 8.

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

remainder is 16. Findaandb.

6 When x^3 +4x^2 +ax+b is divided byx¡ 2 the remainder is 20 , and when divided byx+5the

remainder is 6. Findaandb.

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_06\168CamAdd_06.cdr Monday, 23 December 2013 1:35:11 PM BRIAN