How can I show that the all of the roots of the polynomial $f = 2x^4-12x^2+2$ are real. I am given the polynomial
$$f = 2x^4-12x^2+2$$
and I have to show that all of the roots of this polynomial are real. I have no idea how to approach this. I tried plugging in the rational roots given by the rational root theorem, but none of them ($\pm1, \pm 2$) turned out to actually be roots, so I'm kind of lost.

Just find them all.
By quadratic equation
$x^2 = \frac {12\pm \sqrt{144- 16}}{4}$.
So the four solutions are 
$\sqrt{\frac {12+ \sqrt{144- 16}}{4}}, -\sqrt{\frac {12+ \sqrt{144- 16}}{4}},\sqrt{\frac {12- \sqrt{144- 16}}{4}}, -\sqrt{\frac {12- \sqrt{144- 16}}{4}} $
Now as $144 > 164$ then $\sqrt{144-16}$ is real.  And as $\sqrt{144-16} < \sqrt {144} = 12$ then $12 - \sqrt{144-16} > 0$ so $\sqrt{\frac {12- \sqrt{144- 16}}{4}}$ is real and all the roots are real.
