Polynomial And Quadratic equation

Question

Let $p(x)$ be a polynomial of degree grater than 2. If $p(x)$ leaves a remainder $a$ and $-a$ when divided by $x+a$ and $x-a$ respectively, then what is the remainder when $p(x)$ is divided by $x^2-a^2$?

Answer 1

Hint: the two conditions give $p(-a)=a$ and $p(a)=-a$.

Writing the Euclidian division $p(x)=(x^2-a^2)\,q(x) + r(x)$ the remainder $r(x)$ must be a $1^{st}$ degree polynomial $r(x)=r_1x + r_2$. Evaluate the division equation at $x = \pm a$ to determine $r_1,r_2$.