HMMT-Nov 2015 guts, Problem 22

Find all the roots of the polynomial $x^{5}-5 x^{4}+11 x^{3}-13 x^{2}+9 x-3$.

Solution: The $x^{5}-5 x^{4}$ at the beginning of the polynomial motivates us to write it as $(x-1)^{5}+x^{3}-3 x^{2}+4 x-2$ and again the presence of the $x^{3}-3 x^{2}$ motivates writing the polynomial in the form $(x-1)^{5}+(x-1)^{3}+(x-1)$. Let $a$ and $b$ be the roots of the polynomial $x^{2}+x+1$. It's clear that the roots of our polynomial are given by 1 and the roots of the polynomials $(x-1)^{2}=a$ and $(x-1)^{2}=b$. The quadratic formula shows that WLOG $a=\frac{-1+\sqrt{3} i}{2}$ and $b=\frac{-1-\sqrt{3} i}{2}$ so we find that either $x-1= \pm \frac{1+\sqrt{3} i}{2}$ or $x-1= \pm \frac{1-\sqrt{3} i}{2}$. Hence our roots are $1, \frac{3+\sqrt{3} i}{2}, \frac{1-\sqrt{3} i}{2}, \frac{3-\sqrt{3} i}{2}, \frac{1+\sqrt{3} i}{2}$. $\fbox{1, \frac{3+\sqrt{3} i}{2}, \frac{1-\sqrt{3} i}{2}, \frac{3-\sqrt{3} i}{2}, \frac{1+\sqrt{3} i}{2}}$.

Answer: 1, \frac{3+\sqrt{3} i}{2}, \frac{1-\sqrt{3} i}{2}, \frac{3-\sqrt{3} i}{2}, \frac{1+\sqrt{3} i}{2}