Question

A polynomial $P\left(x\right)$ contains only terms of even degree, when $P\left(x\right)$ is divided by $(x+1)$, the remainder is $-2$. If $P\left(x\right)$ is divided by $(x^2-1)$ then find the remainder.

Answer 1

$f\left(x\right)$ is even function.

$\therefore p(-x)=p\left(x\right)$

$x^2-1=(x+1)(x-1)$

Put $x=-1$;

$\therefore p\left(1\right)=p(-1)=-2$ (Remainder)

$p\left(x\right)=Q\left(x\right)(x^2-1)+ax+b$ Remainder

$p\left(1\right)=Q\left(1\right)(1^2-1)+a\left(1\right)+b\Rightarrow -2=a+b$

$p(-1)=Q(-1)\left(\right(-1{)}^2-1)+a(-1)+b\Rightarrow -2=-a+b$

____________

$-4=2b$

$\Rightarrow b=-2$

$a=0$

$\therefore$ Remainder$=ax+b=-2$.