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  4. The value of x3 + 7x2 - x + 16 when x = 1 + 2i is ,

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Q. The value of $x^{3} + 7x^{2} - x + 16$ when $x = 1 + 2i$ is ,

Complex Numbers and Quadratic Equations

Solution:

We have, $x=1 + 2i$
$\Rightarrow \, x-1=2i$
$\Rightarrow \, x^{2}-2x+5=0$
Now, $x^{3} + 7x^{2}- x + 16$
$=x\left(x^{2}-2x+5\right)+9\left(x^{2}-2x+5\right)+\left(12x-29\right)$
$=x\left(0\right)+9\left(0\right)+12x-29$
$=12\left(1+2i\right)-29$
$=-17+24\, i$