What are the values of $m$ and $n$?

Question

The equation $x3+mx^2+2x+n=0$, where $m$ and $n$ are real numbers, admits $1 + i$ as a root. What are the values of $m$ and $n$?

Answer 1

According to Wolfy (or by hand),

$(1+i)^3+m*(1+i)^2+2*(1+i)+n = i (2m+4) + n $ so $n=0, m=-2 $.

Answer 2

Since $$\overline{x^3+mx^2+2x+n}=0$$ its $$\overline{x}^3+m\overline{x}^2+2\overline{x}+n=0,$$ we see the $1-i$ is a root of the equation.

Id est, $x^3+mx^2+2x+n$ is divisible by $$(x-1-i)(x-1+i),$$ which is $$x^2-2x+2$$ and we obtain: $$x^3+mx^2+2x+n=x(x^2-2x+2)+(m+2)x^2+n,$$ which gives $m=-2$ and $n=0.$