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.
4.- Show that there is an irreducible palindromic polynomial of degree 6 over .
Hint use problem #3
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(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 ...
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.