Question 24
Consider the polynomial $f(x) = x^3 + x^2 - 4x + 1$.
a. Prove that if $r$ is a root of $f(x)$ then $r^2 + r - 3$ is also a root of $f(x)$.
b. Let $\alpha$, $\beta$, $\gamma$ be the three roots of $f(x)$, in some order. Determine all possible values of
$$
\frac{\alpha}{\beta} + \frac{\beta}{\gamma} + \frac{\gamma}{\alpha}
$$
a. Prove that if $r$ is a root of $f(x)$ then $r^2 + r - 3$ is also a root of $f(x)$.
b. Let $\alpha$, $\beta$, $\gamma$ be the three roots of $f(x)$, in some order. Determine all possible values of
$$
\frac{\alpha}{\beta} + \frac{\beta}{\gamma} + \frac{\gamma}{\alpha}
$$