Standard Form of a Polynomial Function and Its Degree

There is a polynomial $P$ such that for every real number $x$, $$x^{512} + x^{256} + 1 = (x^2 + x + 1) P(x).$$ When $P$ is written in standard polynomial form, how many of its coefficients are nonzero?

(Hint: First multiply both sides by $x-1$, and then think about using polynomial long division.)

Source: Math Prize for Girls