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Prove that if f(x) is a nonconstant polynomial with

Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen ISBN: 9780073383095 37

Solution for problem 23E Chapter 4.SE

Discrete Mathematics and Its Applications | 7th Edition

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Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen

Discrete Mathematics and Its Applications | 7th Edition

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Problem 23E

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.]

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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 n>1, takes on each value at most n times].Step-2:Given: is a prime number and f(x) is a non constant polynomial.Let us consider, with and integer coefficients.At then the value of ,where p is prime number and k is an integer. We know that for any x , y and n for some C.Because highest degree of that function is n.So, can be written as : since Therefore, .Step-3:p(1+C) is not a constant value.Because for all values of C we have infinite number of values.So, p(1+C) is not a prime number………….(1)But the given is function takes only prime values………..(2)So, from (1) and (2) it is a contradiction.Therefore, there exist an integer y such that f(y) is composite.

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Chapter 4.SE, Problem 23E is Solved
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Textbook: Discrete Mathematics and Its Applications
Edition: 7
Author: Kenneth Rosen
ISBN: 9780073383095

The full step-by-step solution to problem: 23E from chapter: 4.SE was answered by , our top Math solution expert on 06/21/17, 07:45AM. Since the solution to 23E from 4.SE chapter was answered, more than 360 students have viewed the full step-by-step answer. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. This full solution covers the following key subjects: Integer, polynomial, nonconstant, contradiction, degree. This expansive textbook survival guide covers 101 chapters, and 4221 solutions. The answer 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. [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.]” is broken down into a number of easy to follow steps, and 65 words. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7.

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Prove that if f(x) is a nonconstant polynomial with