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Complex Numbers

Find the valu...

Question

Find the value of $x^{3} + 2 x^{2} - 3 x + 21$ if $x = 1 + 2 i$

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Solution

$x^{3} + 2 x^{2} - 3 x + 21$ if $x = 1 + 2 i$
$(x - 1)^{2} = (2 i)^{2}$
$x^{2} - 2 x + 1 = - 4$
$x^{2} - 2 x + 5 = 0$
$∴ x^{3} + 2 x^{2} - 3 x + 21$
$= x (x^{2} - 2 x + 5) + 4 x^{2} - 8 x + 21$
$= 0 + 4 x^{2} - 8 x + 21$
$= 4 (x^{2} - 2 x + 5) + 1$
$= 0 + 1$
$= 1$
Value is $1$ .

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