The cubic equation $x^3 - 4 x^2 + x + 1 =0$

Question

• The cubic polynomial $P(x) = x^{3} - 4x^{2} + x + 1$ has discriminant $\Delta = 169 = 13^{2}$ which tells us that the extension $\mathbb{Q}(a)/\mathbb{Q}$ is normal, where $a$ is any root of the equation $P(x) = 0$.
• Therefore, given one root $a$, one can find the other as polynomial expressions in $a$.
• For instance, in this case it is not hard to check that the other roots are $a^{2} - 4a + 2$ and $-a^{2} + 3 a + 2$. But what if we didn't know these expressions? Is there a way to get them?