Factorization of Polynomials over a field

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The discussion focuses on factoring the polynomial x^3 - 23x^2 - 97x + 291 over the fields Z3[x], Z7[x], and Z11[x]. In Z3, the polynomial factors to X(X^2 + X + 2), but X^2 + X + 2 has no roots, indicating irreducibility. In Z7, the polynomial is shown to be irreducible as it has no roots. For Z11, after checking potential roots, it is concluded that the polynomial is also irreducible. The original poster ultimately resolves their confusion regarding the irreducibility of the polynomial across these fields.
lilcoley23@ho
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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!
 
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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.
 
I think I might have phrased this question wrong, but I figured it out.

THANKS
 

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