I want to find the roots of the polynomial $2T^7 + 3T^6 + 6T^5 + 11T^3 + 9T + 8$ in the field $\mathbb{Z}_{13}$.
I did a polynomial division by $3T^4 + 11T^2 + 2T + 9$ which resulted in:
$(3T^4 + 11T^2 + 2T + 9)(5T^3 + T^2 + T + 6) + (5T^3 + T^2+ T + 6)$
This can be rewritten to
$(5T^3 + 3T^2 + T + 6)(3T^4 + 11T^2 + 2T +10)$
I tried to find a root for the term $(5T^3 + 3T^2 + T + 6)$ by just inserting 1 to 12 for $T$, but it was not valid in any case. For the term $(3T^4 + 11T^2 + 2T +10)$ I do not have any clue at all.
How can I find a root here?