×
Get Full Access to Discrete Mathematics And Its Applications - 7 Edition - Chapter 4.se - Problem 23e
Get Full Access to Discrete Mathematics And Its Applications - 7 Edition - Chapter 4.se - Problem 23e

×

# Prove that if f(x) is a nonconstant polynomial with ISBN: 9780073383095 37

## Solution for problem 23E Chapter 4.SE

Discrete Mathematics and Its Applications | 7th Edition

• Textbook Solutions
• 2901 Step-by-step solutions solved by professors and subject experts
• Get 24/7 help from StudySoup virtual teaching assistants Discrete Mathematics and Its Applications | 7th Edition

4 5 1 249 Reviews
16
5
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.]

Step-by-Step Solution:
Step 1 of 3

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.

Step 2 of 3

Step 3 of 3

##### 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.

Unlock Textbook Solution