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Question

If x3+ax2+bx+6 is divisible by (x2) and leaves remainder 3 when divided by (x3) then find the value of a and b.

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Solution

Let the given polynomial be p(x)=x3+ax2+bx+6

Given p(x) is divisible by (x2), hence p(2)=0

Put x=2 in p(x)

p(2)=23+a(2)2+b(2)+6

0=8+4a+2b+6

4a+2b=14

2a+b=7 .........(1)

It is also given that when p(x) is divided by (x3) the remainder is 3

That p(3)=3

Put x=3 in p(x)

p(3)=33+a(3)2+b(3)+6

3=27+9a+3b+6

9a+3b=30

3a+b=10 .....(2)

Subtract (2) from (1)

3a+b2ab=10+7

a=3

Substitute a=3 in (1), we get

2(3)+b=7

b=1

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