Factorising $t ^6 − 10t ^4 + 31t ^2 − 30$

Question

Does anyone know how I can factorise $t ^6 − 10t ^4 + 31t ^2 − 30$? I can see the answer using WolframAlpha but I want to know how to do it by hand without guessing roots.

Answer 1

Hint:

$$t^6 - 10t^4 + 31t^2 - 30 = t^2(t^4 - 10t^2 + 25) + (6t^2 - 30)$$

Can you take it from here?

Answer 2

Write the polynomial in terms of $X:=t^2$ as $X^3-10X^2+31X-30$. Its roots $\alpha$, $\beta$ and $\gamma$ can be found by the rational root test. The roots of $t^6-10t^4+31t^2-30$ are then $\pm\sqrt{\alpha}$, $\pm\sqrt{\beta}$ and $\pm\sqrt{\gamma}$.