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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

Swadhin
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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 $x$.

André Nicolas
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  • That was simply brilliant. Thank you very much! – Swadhin May 15 '15 at 07:13
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    You are welcome. It was really very natural, and not at all brilliant. – André Nicolas May 15 '15 at 07:14
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    This might be a little off-topic, but would you recommend some books where I can find some brilliant natural techniques? – Swadhin May 15 '15 at 07:17
  • Unfortunately not for this class of problem. The Remainder Theorem is a standard tool. Your substitution process will not work as stated, though using $\pm i$ (the square roots of $-1$, the roots of $x^2+1=0$, will work even for polynomials $g(x)$ which involve both even and odd powers of $x$. – André Nicolas May 15 '15 at 07:23
  • Ok, that is fine. But, if you have any books in mind, according to my level, then please do recommend. And, thank you for your valuable comment. – Swadhin May 15 '15 at 07:31
  • +1 for including the natural suggestion to look at $f(\pm i)$. – Jyrki Lahtonen May 15 '15 at 07:34