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Q7) Factorise the polynomial f(x) = x3 − 2x2 + 2x − 1 into irreducible polynomials...

Q7) Factorise the polynomial f(x) = x3 − 2x2 + 2x − 1 into irreducible polynomials in Z5[x], i.e. represent f(x) as a product of irreducible polynomials in Z5[x]. Demonstrate that the polynomials you obtained are irreducible.

I think i manged to factorise this polynomial.

I found a factor to be 1 so i divided the polynomial by (x-1) as its a linear factor.

So i get the form

(x3 − 2x2 + 2x − 1) = (x2-x+1)*(x-1) which is congruent to (x2+4x+1)*(x+4) (Mod 5)

Is this further reducible? Or have i factorised this polynomial wrong? Because i still get f(1) giving me 0 meaning its a linear factor

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