Factorization of Polynomials over a field

[lilcoley23@ho]

I don't understand how to factor a polynomial over Z₃ [x], Z₇ [x], and Z₁₁ [x]

I need to factor the polynomail x³ - 23x² - 97x + 291

PLEASE HELP!

 

[LorenzoMath]

Note that for a polynomial is of degree 2 and 3, reducibility is equivalent to the existence of roots.

mod 3:

X^3-23X^2-97X+291=X^3+X^2+2X=X(X^2+X+2). A calculation shows X^2+X+2 doesn't have a root in Z/3Z. Done.

mod 7:
X^3+5X^2+2X+4. A calculation shows it has no root in Z/7Z. The polynomial is irreducible.

mod 11:
291=5=1*5=10*6. So if the polynomial has a root, it should be 1, 5, 6, or 10. A calculation shows X^3-X^2+2X+5 has no root. The given poly is irreducible.

 

[lilcoley23@ho]

I think I might have phrased this question wrong, but I figured it out.

THANKS