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.
 A: Let $t=x^2$, then the equation becomes: $2t^2-12t+2$. From here you have:
$$x=\pm \sqrt{\frac{12\pm\sqrt{128}}{4}}$$ Because $12>\sqrt{128}$ all the roots are real.
A: HINT: Consider the polynomial $2y^2-12y+2$
A: We have
$$\begin{align*}
f(-3) &= +56, \\
f(-1) &= -8, \\
f(0) &= +2, \\
f(1) &= -8, \\
f(3) &= +56. \\
\end{align*}$$
Thus, $f$ changes sign four times, so all four roots of $f$ are real.
A: Factor it into a product of two quadrtatics, and check that the discriminants of those quadratics are positive.
So, $2(x^4-6x^2+1)=2(x^2-2x  -1 )(x^2+2x    -1 )$.  The discriminants are $8$.
A: You can prove it without solving the biquadratic: this function has a local maximum and two local minima, and it tends to $+\infty$ at $\pm\infty$. Therefore, it suffices, by the intermediate value theorem, to show that the local maximum is nonnegative and the local minima are negative.
Indeed, $f'(x)=8x^3-24x)=8x(x^2-3)$ , so the critical values are $0, \sqrt 3,-\sqrt 3$. Also $f''(x)=24(x^2-1)$, which is negative on the interval $(-1,1)$, positive outside this interval. By the second derivative test, $0$ is the local maximum  (and it  equals $2$); the local minima are attained at $\pm\sqrt 3$ and $f(\pm\sqrt 3)=18-36+2=-16$.
A: 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.
