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 (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