1. Factoring polynomials

This is a question from a previous exam:

Express the following polynomial in Z3[x] as a product of irreducibles:

p(x) = 2x4 + x2 + x + 2.

I can divide this by x + 2, but x + 2 is zero when x = 1 (in Z3[x]), so it's not irreducible, right?

2. Originally Posted by arfa brane
This is a question from a previous exam:

Express the following polynomial in Z3[x] as a product of irreducibles:

p(x) = 2x4 + x2 + x + 2.

I can divide this by x + 2, but x + 2 is zero when x = 1 (in Z3[x]), so it's not irreducible, right?
In this context what do you mean by irreducible? Ordinarily, a linear factor is irreducible.

3. $2x^4 + x^2 + x + 2 \equiv (x+2)(2x^3 + 2 x^2 + 1 ) \equiv 2(x+2)(x^3 + x^2 + 2 )$
Proof that the last term is not factorisable.
If it was factorisable, it would be into terms of order 1 and 2. So it suffices to check all irreducible monomials polynomials with order 1.
$x^3 + x^2 + 2 \equiv (x)(x^2 + x) +2 \equiv (x + 1)(x^2) + 2 \equiv (x + 2) (x^2 + 2x + 2) + 1$

4. Originally Posted by mathman
Ordinarily, a linear factor is irreducible.
Yeah, trick question.
Irreducible means it can't be written as a product of lower degree polynomials.
Originally Posted by rpenner
Proof that the last term is not factorisable.
The other way to prove $(x^3 + x^2 + 2)$ is not factorisable is to show it has no zeros in Z3[x], which clearly it doesn't. This only applies to degree 3 or 2 polynomials (why?).

5. A degree four polynomial could be a product of two irreducible polynomials of degree 2.

Examples (in Z_3[x]):
x^4 + 1
x^4 + 2x^2 + 1
x^4 + x^3 + x + 2
x^4 + x^3 + 2 x^2 + 2 x + 1
x^4 + 2 x^3 + 2x + 2
x^4 + 2 x^3 + 2 x^2 + x + 1

So factoring in Z_3[x] an polynomial like x^8 + 2 can be tricky.

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