<s>[INST] $P\left(x\right)=x^{100}+x^{50}-2x^4-x^3+x+1$, $\frac{P(x)}{x^3+x}$ 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.
 [/INST] $x^3+x=x(x+i)(x-i)$
Remainder of $P(x)/(x^3+x)$ is a (at most) second degree polynomial 
$R(x)=a x^2+bx+c$
We have $P(i)=-1+2i$ and $R(i)=-a + i b + c$ so $-a + i b + c=-1+2i$
then $P(-i)=-1-2i$ and $R(-i)=-a - i b + c$ so $-a - i b + c=-1-2i$
and $P(0)=1$ and $R(0)=c$
So $c=1$. Substitute in the first two equations
$-a+ib+1=-1+2i;\;-a-ib+1=-1-2i$
add the two equations $-2a+2=-2$ then $a=2$
substitute and get $b=2$
Therefore the remainder is $R(x)=2x^2+2x+1$
Hope this helps
</s>