1976 USAMO 第 5 题

If $P(x)$, $Q(x)$, $R(x)$, and $S(x)$ are all polynomials such that

$$P(x^5) + xQ(x^5) + x^2 R(x^5) = (x^4 + x^3 + x^2 + x +1) S(x),$$

prove that $x-1$ is a factor of $P(x)$.

如果 $P(x)$、$Q(x)$、$R(x)$ 和 $S(x)$ 都是多项式，使得

$$P(x^5) + xQ(x^5) + x^2 R(x^5) = (x^4 + x^3 + x^2 + x +1) S(x),$$

证明$x-1$是$P(x)$的因数。