Find the value of expression $x^{4} - 4 x^{3} + 3 x^{2} - 2 x + 1$ when $x = 1 + i$ is substituted in the expression.

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Solution

$x = 1 + i$
$\Rightarrow$ $x - 1 = i$
$\Rightarrow$ ${(x - 1)}^{2} = - 1$
$\Rightarrow$ $x^{2} - 2 x + 2 = 0$
Now $x^{4} - 4 x^{3} + 3 x^{2} - 2 x + 1 = (x^{2} - 2 x + 2) (x^{2} - 3 x - 3) - 4 x + 7$
$∴$ when $x = 1 + i$ i.e. $x^{2} - 2 x + 2 = 0$
$x^{4} - 4 x^{3} + 3 x^{2} - 2 x + 1 = 0 - 4 (1 + i) + 7 = - 4 + 7 - 4 i = 3 - 4 i$

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