Remainder of the polynomial: What is wrong with this approach? 
Find the remainder when $P(x)=5x^6 + x^5 - 2x^3 - x^2 + 1$ is divided
  by $x^2+1$.

$$
P(x)=(x^2+1)\cdot Q(x)+R(x)
$$
When $x^2=-1$,
$$
P(x)=-5+x+2x+1+1 = 3x-3
$$
Exactly, why the method above, does not work for the following question?

Find the remainder when $P(x)=x^{81}+x^{49}+x^{25}+x^9 + x$ is divided
  by $x^3-x$.

$$
P(x)=(x^3-x)\cdot Q(x)+R(x)=x(x+1)(x-1)\cdot Q(x) + R(x)
$$
$P(1)=5$, but $P(-1)=-5$, and $P(0)=0$
However, if I factor $P(x)$ by $x$,
$$
P(x)=x(x^{80}+x^{48}+x^{24}+x^8 + 1)
$$
and consider $P(1)$ and $P(-1)$ without the factored $x$, then I get $5x$ in both cases. (The correct answer.)
Question: I can solve the question by slightly different methods where there is no confusion or ambiguity involved. I just like to know the above method does not work with this problem. (And by “exactly why” I mean to ask, how am I supposed to foresee that the above method should not be applied directly.)

The first method does work:
When $x^3=x$, we have $P(x)=x^{81}+x^{49}+x^{25}+x^9 + x= x+x+x+x+x = 5x$.
When $x^3=x$, we have $P(x)=x(x^{80}+x^{48}+x^{24}+x^8 + 1)= x(x^2+x^2+x^2+x^2+1) = x(4x^2+1)=4x^3+x=5x$.
