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

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Solution

Consider the given expression.
$x^{2} - 4 x^{3} + 3 x^{2} - 2 x + 1$

Put $x = 1 + i$ in given polynomial. Now, we get

$x^{2} - 4 x^{3} + 3 x^{2} - 2 x + 1$
$= 4 x^{2} - 4 x^{3} - 2 x + 1$

$= 4 {(1 + i)}^{2} - 4 {(1 + i)}^{3} - 2 (1 + i) + 1$

$= 4 (1 + i^{2} + 2 i) - 4 (1^{3} + i^{3} + 3.1. i^{2} + {3.1}^{2} . i) - 2 - 2 i + 1$

$= - 4 i^{3} - 8 i^{2} - 6 i + 1$

$∵ i^{2} = - 1$

$∴ = - 4 i^{3} + 8 i^{2} - 6 i + 1 = - 4 i^{2} . i + 8 i^{2} - 6 i + 1$

$- 4 (- 1) . i + (- 1) 8 - 6 i + 1 = 4 i - 8 - 6 i + 1$

$= - 2 i - 7$

Hence, this is the answer .

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