Question

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

Answer 1

We have,
$x=-5+2\sqrt{-4}$
$⟹x+5=4i$
$⟹(x+5{)}^2=16i^2⟹x^2+10x+25=-16⟹x^2+10x+41=0$
Now, $x^4+9x^3+35x^2-x+4$
$=x^2(x^2+10x+41)-x(x^2+10x+41)+4(x^2+10x+41)-160$
$=x^2\left(0\right)-x\left(0\right)+4\left(0\right)-160=-160$ $[\because x^2+10x+41=0]$
Thus, the value of given polynomial for $x=-5+2\sqrt{-4}$ is $-160$.