> 
> If the polynomial $x^{19}+x^{17}+x^{13}+x^{11}+x^{7}+x^{5}+x^{3}$
> is divided by $(x^ 
> 2
>  +1)$, then the remainder is:
> 
> 
> 

How Do I solve this question without the tedious long division?

Using remainder theorem , we can take $x^3$ common and put $x^2 =-1$ although $x$ is not a real number. By this method, I got the right answer as $-x$. Is it the right way? Because $x$ comes out to be $i$ which is not real.

Also , can I apply remainder theorem to quadratic divisor polynomials in this way?