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Question

# Let a≠0 and p(x) be a polynomial of degree greater than 2. If p(x) leaves remainders a and −a when divided respectively, by x+a and x−a, the remainder when p(x) is divided by x2−a2 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 −xWe are given that p(−a)=a and p(a)=−a[When a polynomial f(x) is divided by x−a , remainder is f(a)].Let the remainder, when p(x) is divided by x2−a2, be Ax+B. Then,p=(x)=Q(x)(x2−a2)+Ax+B (1)where Q(x) is the quotient. Putting x=a and −a in (1), we get p(a)=0+Aa+B⇒−a=Aa+B (2)and p(−a)=0−aA+B⇒a=−aA+B (3)Solving (2) and (3), we getB=0 and A=−1Hence, the required remainder is −x.

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