[Bezout's Theorem. Factorization]

For which $a$ does the polynomial $P(x)=a^{3} x^{5}+(1-a) x^{4}+\left(1+a^{3}\right) x^{2}+(1-3 a) x-a^{3}$ divide by $x-1$?

#
$P(1)=a^{3}+(1-a)+\left(1+a^{3}\right)+(1-3 a)-a^{3}=a^{3}-4 a+3=(a-1)\left(a^{2}+a-3\right)$.

## Answer

$a_{1}=1, a_{2,3}=\frac{-1 \pm \sqrt{13}}{2}$