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Question

The polynomial x42x3+3x2ax+b when divided by x+1 and x1 gives remainders 19 and 5 respectively. Find the remainder when the polynomial is divided by x3. [3 MARKS]

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Solution

Let p(x)=x42x3+3x2ax+b

p(1)=19;p(1)=5 [1 Mark]

p(1)=(1)42(1)3+3(1)2a(1)+b=19

a+b=13...............(1)

p(1)=(1)42(1)3+3(1)2a(1)+b=5

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

On solving (1) and (2), we get a=5,b=8 [1 Mark]

p(x)=x42x3+3x25x+8

When p(x) is divided by(x-3),
Remainder = p(3)=(3)42(3)3+3(3)25(3)+8 [1 Mark]

p(3)=47


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