Find the irreducible factors of the polynomial $x^3 + 5x^2 + 2x + 6 \in \mathbb{Z}_7 [ x ]$ .

Question

Find the irreducible factors of the polynomial $$x^3 + 5x^2 + 2x + 6 \in \mathbb{Z}_7 [ x ]$$

I am having trouble understanding irreducible factors for my algebra module and have an exam next week. If any one could help me understand this sample question, I would be so grateful

Answer 1

Hint: Over a field, a cubic polynomial either is irreducible or has a factor of degree one, that is, a root.

Or:

$$ x^3 + 5x^2 + 2x + 6 = x^3 - 2x^2 + 2x -1 = x^3 - 2x^2 + x + x-1 = x(x^2 - 2x + 1) + x-1 = x(x-1)^2 + x-1 =(x-1)(x(x-1)+1) =(x-1)(x^2-x+1) =(x-1)(x^2+6x+8) =(x-1)(x+2)(x+4) =(x-1)(x-3)(x-5) $$

Answer 2

As suggested in the comments, there are only seven values to test in $\mathbb{Z}_7$. Testing the value 1, we get:

$$(1^3 + 5 \cdot 1^2 + 2 \cdot 1 + 6)\,mod\,7 = 14\,mod\,7 = 0$$

If we do the same for the values 3 and 5, we get 0 as well. Indeed, in $\mathbb{Z}_7$:

$$(x-1)(x-3)(x-5) = x^3 - 9 x^2 + 23 x - 15 = x^3 + 5 x^2 + 2 x + 6$$