Question

# If p(x) is a polynomial of degree greater than 2 such that p(x) leaves remainder a and âˆ’a when divided by x+a and xâˆ’a respectively. If p(x) is divided by x2âˆ’a2 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 −xAs 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 (x2−a2) Using division algorithm, p(x)=(x2−a2)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+v−a=ua+v Solving them simultaneously, ⇒v=0⇒u=−1 Hence, the required remainder is ux+v=−x+0=−x

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