Ask Uncle Colin: A Complex Roots Problem

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 $a{x}^n+b{x}^{n-1}+\dots =0$ sum to $-\frac{b}{a}$. You’ll definitely have seen that with quadratics, but it’s true of all polynomials.

In this case, the roots sum to 8, and one of them is 2. The other two must therefore be complex conjugates of the form $3\pm k\mathrm{i}$.

The triangle has a (horizontal) height of 1 and a (vertical) base of 16, so $k=8$.

That means the polynomial can be written as $(x-2)(x-(3+8i))(x-(3-8i))$, which expands to $(x-2)(x^2-6x+73)=x^3-8x^2+85x-146$.

Therefore, $c=85$ and $d=-146$.

Hope that helps!

- Uncle Colin