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Question

Suppose p(x) is a polynomial with integer coefficients. The remainder when p(x) is divided by x1 is 1 and the remainder when p(x) is divided by x4 is 10. If r(x) is the remainder when p(x) is divided by (x1)(x4), then the value of r(2006), is

A
6018
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B
6016
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C
6020
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D
None of these
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Solution

The correct option is D 6016
Let Q1,Q2 be the quotients when p(x) is divided by (x1),(x4) respectively.
p(x)=(x1)Q1(x)+1 ....(1)
p(x)=(x4)Q2(x)+10 .....(2)
(x4)p(x)=(x4)(x1)Q1(x)+(x4) ....(3)
and (x1)p(x)=(x1)(x4)Q2(x)+10x10 ....(4)
Subtracting (3) from (4), we get
p(x)=(x1)(x4)[Q2(x)Q1(x)]3+9x63
Here, p(x) when divided by (x1)(x4), remainder =9x63
r(x)=9x63=3x2
r(2006)=3×20062=6016

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