remainder theorem??

Write $x+x^9+x^{25}+x^{49}+x^{81}=p(x)(x^3-x)+a+bx+cx^2$ . Plugging in $x=0$ gives $a=0$ , and plugging in $x=1,-1$ gives $b+c=5$ and $-b+c=-5$ so $b=5,c=0$ . The remainder is $\boxed{5x}$ .

Since you're dividing by x^3 - x, a trick to find the remainder is to introduce the relation x^3 = x and use it to simplify x^81 + x^49 + x^25 + x^9 + x. The result of that is the remainder.

With x^3 =x, we also have x^4 = x^2. Inductively, x^(3^n) = x and x^(2^n) = x^2 for all positive integers n. Thus

x^81 + x^49 + x^25 + x^9 + x

= x^(3^4) + (x^3)^(2^4) * x + (x^3)^(2^3) * x + x^(3^2) + x

= x + x^(2^4) * x + x^(2^3) * x + x + x

= x + x^2 * x + x^2 * x + x + x

= x + x + x + x + x

= 5x.

Thus, 5x is the remainder of x^81 + x^49 + x^25 + x^9 + x upon division by x^3 - x.