# Irreducible Palindromic Polynomials, Reducible and Irreducible Polynomials and Degrees of Polynomials

3.- a) Determine the number of polynomials f(x) of degree 6 over for which f(1)=1 and f(0)=1.

b) Determine the number of polynomials of degree 6 over that are reducible but have no linear factors. How this is possible, please can you explain this?

Hint Consider the possible factorizations.

c) Determine the number of irreducible polynomials of degree 6 over . Explain.

In order to find this, is there a formula o we need to figure out by inspection?

4.- Show that there is an irreducible palindromic polynomial of degree 6 over .

Hint use problem #3

Please see the attached file for the fully formatted problems.

keywords: palindrome

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Problem #3

(a) The polynomial with degree 6 over must have form

, where .

From the condition, we know

, ,

Thus we have , so there are odd number of 1's from to . There are possibilities.

Thus there are 16 polynomials with degree 6 over such that .

(b) Since has no linear factor, then . Because ...

#### Solution Summary

Irreducible Palindromic Polynomials, Reducible and Irreducible Polynomials and Degrees of Polynomials are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.