Let p be prime and f (x) ≡ f₀ + f₁x + ··· + f_tx^t (mod p) be a polynomial of degree t, with coefficients f_i drawn from ℤ_p. We say that a ∈ ℤ_p is a zero of f if f (a) ≡ 0 (mod p). Prove that if a is a zero of f, then F (x) ≡ (x ≡ a) g (x) (mod p) for some polynomial g (x) of degree t − 1. Prove by induction on t that if p is prime, then a polynomial f (x) of degree t can have at most t distinct zeros modulo p.