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Question

# If f(x)=x4âˆ’2x3+3x2âˆ’ax+b is a polynomial such that when it is divided by (xâˆ’1) and (x+1), the remainders are 5 and 19 respectively, the remainder when f(x) is divisible by (xâˆ’2) is

A
7
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B
8
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C
9
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D
10
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Solution

## The correct option is D 10When x4−2x3+3x2−ax+b is divide by x−1, remainder is 5.So, substituting for x is 1, in the above, we get5=1−2+3−a+b ∴−a+b=3----(1)When x4−2x3+3x2−ax+b is divide by x+1, remainder is 19.So, substituting for x is −1, in the above, we get 19=1+2+3+a+ba+b=13----(2)Solving (1) and (2), we get a=5,b=8So polynomial becomes x4−2x3+3x2−5x+8The remainder when x4−2x3+3x2−5x+8 is divided by x−2 is by plugging in x as 2 in the given polynomial, we get 16−16+12−10+8=10so remainder is 10So, option D.

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