Polynomials – How to Factor x^6 + 5x^3 + 8

Question

I wanted to know, how can I factor $x^6 +5x^3 +8$, I have no idea. Is there any method to know if a polynomial is factored. Just some advice will do.

Help appreciated.

Thanks.

Answer 1

Let $y = x^3$ to obtain $y^2 + 5y + 8 = 0$. This factors as $y = \frac{-5 \pm \sqrt{-7}}{2} = \frac{-5}{2} \pm \frac{\sqrt{7}}{2}i$. These have modulus $r = \sqrt{25/4 + 7/4} = 2\sqrt{2}$. Now solve $5/2 = 2\sqrt{2} \cos \theta$ to find the angle $\theta = \cos^{-1}(5/4\sqrt{2})$. Our two roots correspond to $re^{\pi-\theta}$ and $re^{\pi+\theta}$.

Now we have $x^3 = re^{\pi-\theta}$ and $x^3 = re^{\pi+\theta}$ to contend with. For the former, one root is $x_1 = \sqrt[3]{r}e^{(\pi-\theta)/3}$, so the other two are $x_2 = \sqrt[3]{r}\exp(\frac{\pi-\theta}{3} + 2\pi/3)$ and $x_3 = \sqrt[3]{r}\exp(\frac{\pi-\theta}{3} + 4\pi/3)$.

Similarly, the roots of the other equation are $x_4 = \sqrt[3]{r}e^{(\pi+\theta)/3}$, $x_5 = \sqrt[3]{r}\exp(\frac{\pi+\theta}{3} + 2\pi/3)$, and $x_6 = \sqrt[3]{r}\exp(\frac{\pi-\theta}{3} + 4\pi/3)$.