Question

If $f\left(x\right)=x^4+9x^3+35x^2-x+4$, find $f(-5+4i)$

Answer 1

Given:$f\left(x\right)=x^4+9x^3+35x^2-x+4$

if $x=-5+4i$

$\Rightarrow (x+5)=4i\Rightarrow {(x+5)}^2={\left(4i\right)}^2\Rightarrow x^2+25+10x=-16\Rightarrow x^2+10x+41=0-\left(i\right)$

All we have to do is divide the expression $x^4+9x^3+35x^2-x+4$ by $x^2+10x+41$ via the long division method.

$\therefore x^4+9x^3+35x^2-x+4=(x^2+10x+41)(x^2-x+4)-160\therefore f\left(x\right)=x^4+9x^3+35x^2-x+4=-160$