Question

Which of the following is a polynomial with roots 2,3i,-3i

Answer 1

`\bf \begin{cases}x=2\implies &x-2=0\\x=3i\implies &x-3i=0\\x=-3i\implies &x+3i=0\end{cases}~\hspace{8em}(x-2)\stackrel{\textit{difference of squares}}{(x-3i)(x+3i)}=\stackrel{y}{0}\\\\\\(x-2)~~[x^2-(3i)^2]=0\implies (x-2)~~[x^2-(3^2i^2)]=0\\\\\\(x-2)~~[x^2-(9(-1))]=0\implies (x-2)~~(x^2+9)=0\\\\\\\stackrel{x(x^2+9)}{x^3+9x}~~+~~\stackrel{-2(x^2+9)}{(-2x^2-18)}\implies x^3-2x^2+9x-18=y`