# Determine if this polynomial has a repeated factor

1. Feb 7, 2012

### Firepanda

f(t) = t4 - 23 + 3t2 - 2t + 1 in Q[t]

Am i right in thinking I just show by the rational root theorem that the only possible roots are +-1

f(+-1) =/= 0 so there are no repeated factors?

Seems too easy..

2. Feb 7, 2012

### Staff: Mentor

No. Your work shows there are no rational roots, but there could be real or complex roots.

Are you limiting your search only to rational numbers? I'm not sure what your Q[t] notation means, although it suggests to me polynomial functions of t with rational coefficients. Correct me if I have misinterpreted this.

3. Feb 7, 2012

### HallsofIvy

Mark44, I would interpret Q(t) to mean the set of polynomials in t with rational coefficients. A polynomial is "factorable in Q(t)" if and only it is the product of factors with rational coefficients.
Since a cubic, if factorable, must have at least one linear factor, Firepanda is quite correct in noting that, since the polynomial has no rational roots, it has no linear factor with rational coefficients and so is irreducible in Q(t).