Exercise 10

In each polynomial below, assume $x$ is some variable. Use the binomial theorem to expand and reduce modulo the appropriate number:

1. $(x+1)^2 \pmod 2$
2. $(x+1)^5 \pmod 5$
3. $(x+1)^7 \pmod 7$

Prove that $(x+1)^p \equiv x^p + 1 \pmod p$ for any prime $p$.