Let a `ne` 0 and P(x) be a polynomial of degree greater then 2.If P(x) leaves remianders a and a- when divided, respectively, by x + a and x - a, then find the remainder when P(x) is divided by `x^(2) - a^(2)`.

Explanation

According to the question P(-a) = and P(a) = - a. Let the remainder, when P(x) is divided by `x^(2)-a^(2)` be Ax + B. Then `P(x) = Q (x) (x^(2)-a^(2))` + Ax + B, where Q (x) is the quotient
Putting x = a, we get
P(a) = 0 + Aa + B
or Aa + B = - a ...(1)
Putting x = - a, we get
`-Aa + B = a` ...(2)
Solving (1) and (2) , we get
B = 0 and A = - 1
Hence, the requried remainder = Ax + B = - x