Question

If $x=-5+2\sqrt{-4}$ find the value of $x^4+9x^3+35x^2-x+4$

Answer 1

We have $x=-5+2\sqrt{-4}$

$\Rightarrow x+5=4i$

Squaring,$x^2+10x+25=-16$

$\Rightarrow x^2+10x+41=0$

Now $x^4+9x^3+35x^2-x+4$

$=(x^2+10x+41)(x^2-10x+41)-160$

$=0-160=0$ since $x^2+10x+41=0$