Question

Evaluate the mentioned below
(ii)$x^4-4x^3+4x^2+8x+44,whenx=3+2i$

Answer 1

Given that-
$x=3+2i$
$\Rightarrow x-3=2i$
$\Rightarrow (x-3{)}^2=(2i{)}^{2}squaringonbothsides$
$\Rightarrow x^2-6x+9=4i^2$
$\Rightarrow x^2-6x+13=0\dots \left(1\right)$
$\Rightarrow x^4-4x^3+4x^2+8x+44$
$=x^2(x^2-6x+13)+2x^3-9x^2+8x+44$
$=x^2(x^2-6x+13)+2x(x^2-6x+13)+3x^2-18x+44$
$=x^2(x^2-6x+13)+2x(x^2-6x+13)+3(x^2-6x+13)+5$
$=x^2\times 0+2x\times 0+3\times 0+5fromeqn\left(1\right)$
$=5$
Therefore , $x^4-4x^3+4x^2+8x+44=5$