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Prove that if f(x) is a nonconstant polynomial with
Chapter 5, Problem 23E(choose chapter or problem)
Prove that if f(x) is a nonconstant polynomial with integer coefficients, then there is an integer y such that f(y) is composite. [Hint: Assume thai f(x0) = p is prime. Show that p divides f(x0 + kp) for all integers k. Obtain a contradiction of the fact that a polynomial of degree n, where n > 1, takes on each value at most n times.]
Questions & Answers
QUESTION:
Prove that if f(x) is a nonconstant polynomial with integer coefficients, then there is an integer y such that f(y) is composite. [Hint: Assume thai f(x0) = p is prime. Show that p divides f(x0 + kp) for all integers k. Obtain a contradiction of the fact that a polynomial of degree n, where n > 1, takes on each value at most n times.]
ANSWER:Solution:Step-1:In this problem we need to prove that if f(x) is a nonconstant polynomial with integer coefficients , then there is an integer y such that f(y) is composite.[Given hint: Assume that is a prime. .We need to show that p divides for all integers k.Obtain a contradiction of the fact that a polynomial of degree n, where