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Question

# Suppose p(x) is a polynomial with integer coefficients. The remainder when p(x) is divided by xâˆ’1 is 1 and the remainder when p(x) is divided by xâˆ’4 is 10. If r(x) is the remainder when p(x) is divided by (xâˆ’1)(xâˆ’4), 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 6016Let Q1,Q2 be the quotients when p(x) is divided by (x−1),(x−4) respectively.p(x)=(x−1)Q1(x)+1 ....(1)p(x)=(x−4)Q2(x)+10 .....(2)⇒(x−4)p(x)=(x−4)(x−1)Q1(x)+(x−4) ....(3)and (x−1)p(x)=(x−1)(x−4)Q2(x)+10x−10 ....(4)Subtracting (3) from (4), we get⇒p(x)=(x−1)(x−4)[Q2(x)−Q1(x)]3+9x−63Here, p(x) when divided by (x−1)(x−4), remainder =9x−63⇒r(x)=9x−63=3x−2⇒r(2006)=3×2006−2=6016

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