P-adic Norm not Complete on Rational Numbers/Proof 2/Lemma 3

Theorem
Let $x_1, p, k \in Z_{\gt 0}$ be any positive integers.

Let $a = x_1^k + p$

Let $\map f X \in \Z \sqbrk X$ be the polynomial:
 * $X^k - a$

Then:
 * $\map f {x_1} \equiv 0 \pmod p$