Evaluate the following: 2x^3+2x^2-7x+72, when x= 3-5i 2 - Vidyadip

Question

Evaluate the following:
$2\text{x}^3+2\text{x}^2-7\text{x}+72,$ when $\text{x}=\frac{3-5\text{i}}{2}$

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Answer

$\text{x}=\frac{3-5\text{i}}{2}$
$\Rightarrow2\text{x}=3-5\text{i}$
$\Rightarrow2\text{x}-3=-5\text{i}$
$\Rightarrow(2\text{x}-3)^2=(-5\text{i})^2$
$\Rightarrow4\text{x}^2+9-12\text{x}=-25$
$\Rightarrow4\text{x}^2-12\text{x}+34=0$
$\Rightarrow2(2\text{x}^2-6\text{x}+17)=0$
$\Rightarrow2\text{x}^2-6\text{x}+17=0 \ ...(\text{i})$
$\therefore2\text{x}^3+2\text{x}^2-7\text{x}+72$
$=\text{x}(2\text{x}^2-6\text{x}+17)+6\text{x}^2-17\text{x}+2\text{x}^2-7\text{x}+72$ (Adding and subtracting $6x^2$ and 17x)
$\text{x}\times0+8\text{x}^2-24\text{x}+72$ (Using(i))
$=4(2\text{x}^2-6\text{x}+17)+4$
$=4\times0+4$ (Using(i))
$=4$

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