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Question

# Let f(x) be a polynomial with degree greater than 2 for which remainders when divided by (x−1),(x−2),(x−3) are 3,7,13 respectively, then the remainder of f(x) when divided by (x−1)(x−2)(x−3), is:

A
2x+1
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B
x2+x+1
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C
x2+1
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D
x+2
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Solution

## The correct option is B x2+x+1Let f(x)=(x−1)(x−2)(x−3)g(x)+rx2+sx+tf(1)=(1−1)(1−2)(1−3)g(1)+r×12+s×1+t⇒r+s+t=3 ...(1)f(2)=(2−1)(2−2)(2−3)g(2)+r×22+s×2+t⇒4r+2s+t=7 ...(2)f(3)=(3−1)(3−2)(3−3)g(3)+r×32+s×3+t⇒9r+3s+t=13 ...(3)Solving (1), (2) and (3), we getr=1,s=1,t=1Hence remainder is x2+x+1, when f(x) is divided by (x−1)(x−2)(x−3)

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