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Here is the question I'm working on:

List and completely factor all the polynomials of the form $x^5 + ax^2 +bx +c$ over $\mathbb{Z_2}$.

I'm trying to relate this problem to the previous problem I already completed. I know how to list and factor all the polynomials of degree $\leq 4$ over $\mathbb{Z_2}$. So in $\mathbb{Z_2}$, we have

$1, x, x+1, x^2, x^2 + 1 = (x+1)^2, x^2 + x = x(x+1), x^2 + x + 1, x^3, x^3 +1$, .. and the list continues. But in the original problem, I'm working with degree $5$ and and have the constants $a,b,c$How would one approach this problem?

John W. Smith
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    Surely you just continue what you did for degree 4, and extend into degree 5? – unseen_rider Mar 10 '17 at 20:34
  • Yes, but the problem is that I don't have an $x^4$ or $x^3$ term in my polynomial. That's why I didn't want to write down all of the polynomials since I wasn't sure. – John W. Smith Mar 10 '17 at 20:37
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    The lack of $x^4$ or $x^3$ makes it easier. Since you have three unknowns over $\mathbb{Z}_2$, you have only $8$ possibilities for your polynomial. In fact, if $c = 0$, you can factor out an $x$ and reduce half the cases to the work you already did for degree $4$ polynomials. – NoName Mar 10 '17 at 20:39
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    Ahh so would I have $x^5$, $x^5 + c$, $x^5 + bx$, $x^5 + ax^2$ $x^5 + bx + c$...? – John W. Smith Mar 10 '17 at 20:41
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    @i8Σπ_821, over $\mathbb{Z}_2$, each of the constants $a$, $b$, and $c$ is either $0$ or $1$. E.g., for $(a,b,c)=(1,0,1)$, the polynomial to list and factor is $x^5+x^2+1$. Since $2\cdot2\cdot2=8$, that's why NoName said you have only $8$ possibilities to consider. – Barry Cipra Mar 10 '17 at 21:17

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