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If p(x) is a polynomial of degree greater than 2 such that p(x) leaves remainder a and a when divided by x+a and xa respectively. If p(x) is divided by x2a2 then remainder is

A
2x
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B
2x
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C
x
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D
x
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Solution

The correct option is D x
As degree of remainder is always lesser than degree of divisor,we have taken remainder with degree less than 2 here.

Let ux+v be the remainder when p(x) is divided by (x2a2)

Using division algorithm,
p(x)=(x2a2)q(x)+(ux+v)(i)
where q(x) is the quotient.

We know that, by remainder theorem,
p(a)=a, p(a)=a
Putting x=a and x=a in the equation (i),
a=ua+va=ua+v
Solving them simultaneously,
v=0u=1

Hence, the required remainder is
ux+v=x+0=x

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