Write down the third root, z3, of the equation.

Question

Given that z1 = 2 and z2 = 1 + i√3 are roots of the cubic equation z^3 + bz^2 + c*z + d = 0 where b, c, d are real numbers:

Write down the third root, z3, of the equation.

The answer: 1 – i√3

Doubt: I don't know how to get to this answer.

Answer 1

If $r$ is a root of your polynomial, then, by definition, $$ r^3+br^2+cr+d=0 $$ Take complex conjugates: $$ 0=\bar{0}=\overline{r^3+br^2+cr+d}=\bar{r}^3+b\bar{r}^2+c\bar{r}+d $$ (because $\bar{b}=b$ and the same for the other coefficients, which are supposed to be real). Hence also $\bar{r}$ is a root.

This yields nothing new for the root $z_1=2$, but it says that $\overline{z_2}=1-i\sqrt{3}$ is a root. Hence $z_3=1-i\sqrt{3}$, because the polynomial has at most three roots.