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Question

Let f(x)=x5+ax3+bx. The remainder when f(x) is divided by x+1 is 3. Then the remainder when f(x) is divided by x21, is

A
3
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B
3x
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C
3x
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D
3
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Solution

The correct option is C 3x
f(x)=x5+ax3+bx
When f(x) is divided by x+1, the remainder is 3, so
f(1)=3
1ab=3a+b=2 (1)

Assuming
f(x)=p(x)×(x21)+qx+r
Putting x=1, we get
3=q+r (2)
Putting x=1, we get
f(1)=q+r1+a+b=q+r
Using equation (1), we get
3=q+r (3)
Solving equations (2) and (3), we get
q=3, r=0

Therefore, f(x)=p(x)×(x21)+3x
Hence, the remainder is 3x.

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