<s>[INST] Solve the cubic polynomial $a^3 - 10a + 5 = 0$ I am not able to understand how to solve a cubic polynomial. I have tried hit and trial for finding the first root but in vain. How do I solve for $a$ in
$$
a^3 - 10a + 5 = 0\ ?
$$
 [/INST] Let $a=2\sqrt{\frac{10}{3}}\cos\phi.$
Thus, $$\frac{80\sqrt{10}\cos^3\phi}{3\sqrt3}-\frac{20\sqrt{10}\cos\phi}{\sqrt{3}}+5=0$$ or
$$4\cos^3\phi-3\cos\phi=-\frac{3\sqrt3}{4\sqrt{10}}$$ or
$$\cos3\phi=-\frac{3\sqrt3}{4\sqrt{10}}$$ and from here we can get three real roots.
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