We say that the polynomials f(x) and g(x) are congruent modulo n as polynomials if for each power of x the coefficients of that power in f(x) and g(x) are congruent modulo n. For example, 11x3 + x2 + 2 and x3 − 4x2 + 5x + 22 are congruent as polynomials modulo 5. The notation f(x) ≡ g(x)(mod n) is often used to denote that f(x) and g(x) are congruent as polynomials modulo n. In Exercises, assume that n is a positive integer with n > 1 and that all polynomials have integer coefficients.
Suppose that p is prime, f(x) is a polynomial with integer coefficients, a1 a2,… , ak are incongruent integers modulo p, and f(aj) ≡ 0 (mod p) for j = 1, 2,…, k. Show that there exists a polynomial g(x) with integer coefficients such that f(x) and (x − a1)(x − a2) ⋯ (x − ak)g(x) are congruent as polynomials modulo p.
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