Dear Uncle Colin,

I had a question in an exam that gave a cubic, $f(x) = x^3 – 8x^2 + cx + d$, with roots $\alpha$, $\beta$ and $\gamma$. When plotted on an Argand diagram, the triangle formed by the three roots has area 8. Given that $\alpha=2$, find $c$ and $d$. I couldn’t get near it – any ideas?

– Failed At Complex Task, Only Real Solution

Hi, FACTORS, and thanks for your message!

There are several steps to getting to the bottom of this, but the key one is that the roots of the polynomial $ax^n + bx^{n-1} + …=0$ sum to $-\frac{b}{a}$. You’ll definitely have seen that with…

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