Q:

Let $f(x)=7x^{32}+5x^{22}+3x^{12}+x^2$. Find the remainder when $x^2+1$ divides $f(x)$ and $xf(x)$.

Let $f(x)=7x^{32}+5x^{22}+3x^{12}+x^2$. Find the remainder when $x^2+1$ divides $f(x)$ and $xf(x)$.
I tried this problem two ways, substituting $x=1,-1$ in $f(x)$ to find the remainder, and by long division, but, that's not getting me to the answer.  I think there is a shorter and elegant technique for solving this question.  Please help.  Thank you.r

A:

Let $t=x^2$. We want the remainder when $Q(t)=7t^{16}+5t^{11}+3t^6+t$ is divided by $t+1$. By the Remainder Theorem, this is $Q(-1)$.
For the second question, multiply the answer to the first by $x$.

