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Question

If f(x)=x42x3+3x2ax+b is a polynomial such that when it is divided by (x1) and (x+1), the remainders are 5 and 19 respectively, the remainder when f(x) is divisible by (x2) 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 10
When x42x3+3x2ax+b is divide by x1, remainder is 5.
So, substituting for x is 1, in the above, we get
5=12+3a+b
a+b=3----(1)
When x42x3+3x2ax+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+b
a+b=13----(2)
Solving (1) and (2), we get a=5,b=8
So polynomial becomes x42x3+3x25x+8
The remainder when x42x3+3x25x+8 is divided by x2 is by plugging in x as 2 in the given polynomial, we get
1616+1210+8=10
so remainder is 10
So, option D.

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