<s>[INST] The complex number $(1+i)$ is root of polynomial $x^3-x^2+2$. Find the other two roots. The complex number $(1+i)$ is root of polynomial $x^3-x^2+2$.
Find the other two roots.
$(1+i)^3 -(1+i)^2+2= (1-i-3+3i)-(1-1+2i) +2= (-2+2i)-(2i) +2= 0$.
The other two roots are found by division.
$$
\require{enclose}
\begin{array}{rll}
    x^2 && \hbox{} \\[-3pt]
   x-1-i \enclose{longdiv}{x^3 -x^2 + 2}\kern-.2ex \\[-3pt]
      \underline{x^3-x^2- i.x^2} && \hbox{} \\[-3pt]
      2 +i.x^2
  \end{array}
$$
$x^3-x^2+2= (x-1-i)(x^2) +2+i.x^2$
How to pursue by this or some other approach?
 [/INST] One of the other roots is $1-i,$ since complex roots of real polynomials come in complex conjugate pairs.
The sum of the roots is $-1$ times the coefficient of $x^2.$ So, if $r$ is the third root, $(1+i)+(1-i)+r=1,$ or $r=-1.$
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