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.

We 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.