<s>[INST] Sum of real roots of two different cubic polynomials Let $\alpha$ be the only real root of $p(x)=x^3-3x^2+5x-17$ and $\beta$ the only real root of $q(x)=x^3-3x^2+5x+11$. Compute $\alpha + \beta$.
I've noticed that the two graphs are just the same shifted vertically (the first coefficients are equal except for the last), but I don't know if it is useful. 
(This is supposed to be doable without any calculator)
 [/INST] Re-arrange $p(x)$ and $q(x)$:
\begin{align*}
  p(x) &= x^3-3x^2+5x-17 \\
       &= (x-1)^3+2(x-1)-14 \\
  q(x) &= x^3-3x^2+5x+11 \\
       &= (x-1)^3+2(x-1)+14
\end{align*}

Note that $$p(1+x) = -q(1-x)$$

If $p(1+x)=-q(1-x)=0$, then $$
\left \{
  \begin{align*}
    1+x &= \alpha \\
    1-x &= \beta
  \end{align*}
\right.$$
$$\fbox{$\alpha+\beta=2$}$$
</s>