$P\left(x\right)=x^{100}+x^{50}-2x^4-x^3+x+1$, $\frac{P(x)}{x^3+x}$ remainder Given the polynomial:
$P\left(x\right)=x^{100}+x^{50}-2x^4-x^3+x+1$
What is the remainder of $\frac{P(X)}{x^3+x}$?
I don't think the long division is efficient the way to go, and the remainder theorem doesn't seem to be applicable here as $x^3+x$ is not linear. Could I have some hints on how to approach this? Thank you.

$P(x)=x^{100}+x^{50}-2x^4-x^3+x+1$
Let's divide $x^{100}+x^{50}$ by $x^3+x$
-> $Q=x^{97},r=-x^{98}+x^{50}$
Divide $-x^{98}+x^{50}$ by $x^3+x$
-> $Q=-x^{95}, r=x^{96}+x^{50}$
Divide $x^{96}+x^{50}$ by $x^3+x$
-> $Q=x^{93}, r=-x^{94}+x^{50}$
...
Can you see a pattern?
...
The pattern ends here:
Divide $x^{52}+x^{50}$ by $x^3+x$
-> $Q=x^{49}, r=0$

Now return to the original polynomial (without the first two term):
$-2x^4-x^3+x+1$
And do polynomial division as you do normally!
