<s>[INST] Polynomial with real roots Consider the polynomial: $$f=X^4+4X^3+6X^2+aX+b$$
We know that $f$ has four real roots. Let $x_1,x_2,x_3,x_4$ be the roots of this polynomial. How can one compute $$x_1^{2015}+x_2^{2015}+x_3^{2015}+x_4^{2015}?$$
If $a=4$ and $b=1$, we obtain a self-reciprocal (palindromic) polynomial. We can write $f=(X+1)^4$, thus $x_1=x_2=x_3=x_4=1$. Hence the sum computes to $-4$.
Are there any other cases to consider ($a,b$)? I thought using the formula for the quartic equation and paying attention to the cases where we have only real roots.
Any ideas? Thank you!
 [/INST] We know the following to be true:
$$
\tag1x_1+x_2+x_3+x_4 = -4
$$
$$
\tag2x_1x_2+x_2x_3+x_3x_4+x_4x_1+x_1x_3+x_2x_4 = 6
$$
$$
\tag3x_1 x_2x_3+x_2x_3x_4+x_3x_4x_1+x_4x_1x_2 = -a
$$
$$
\tag4x_1x_2x_3x_4 = b
$$
Now we can square $(1)$ to get
$$
\begin{align}
(x_1+x_2+x_3+x_4)^2 = &x^2_1+x^2_2+x^2_3+x^2_4\\
&+2[x_1x_2+x_2x_3+x_3x_4+x_4x_1+x_1+x_3+x_2x_4]
\end{align}
$$
Substituting in values from $(1)$ gives
$$
\begin{align}
(-4)^2 &= x^2_{1}+x^2_{2}+x^2_{3}+x^2_{4}+2(6)\\
16&=x^2_{1}+x^2_{2}+x^2_{3}+x^2_{4}+12\\
4&=x^2_{1}+x^2_{2}+x^2_{3}+x^2_{4}
\end{align}
$$
Using the Cauchy-Schwartz Inequality, we get
$$
\left(x^2_{1}+x^2_{2}+x^2_{3}+x^2_{4}\right)(1^2+1^2+1^2+1^2)\geq (x_{1}+x_{2}+x_{3}+x_{4})^2
$$ 
where equality holds when
$$
\tag5x_{1} = x_{2} = x_{3} = x_{4}
$$
Thus, the equality condition holds, because $4\cdot 4= (-4)^2$. So from $(1)$ and $(5)$, we have $$x_{1} = x_{2} = x_{3} = x_{4}=-1$$
Setting the values of $x_{1} = x_{2} = x_{3} = x_{4}=-1$ in $(3)$ and $(4)$, we get $(a,b)=(4,1)$, which is thus the unique solution. So we can evaluate the sum as
$$
\begin{align}
x^{2015}_{1}+x^{2015}_{2}+x^{2015}_{3}+x^{2015}_{4} &= (-1)+(-1)+(-1)+(-1)\\
&=-4
\end{align}
$$
</s>