Evaluate the following:
(i) $2 x^{3}+2 x^{2}-7 x+72$, when $x=\frac{3-5 i}{2}$
(ii) $x^{4}-4 x^{3}+4 x^{2}+8 x+44$, when $x=3+2 i$
(iii) $x^{4}+4 x^{3}+6 x^{2}+4 x+9$, when $x=-1+i \sqrt{2}$
(iv) $x^{6}+x^{4}+x^{2}+1$, when $x=\frac{1+i}{\sqrt{2}}$
(v) $2 x^{4}+5 x^{3}+7 x^{2}-x+41$, when $x=-2-\sqrt{3} i$
(i) $x=\frac{3-5 i}{2}$
$\Rightarrow x^{2}=\left(\frac{3-5 i}{2}\right)^{2}$
$=\frac{9+25 i^{2}-30 i}{4}$
$=\frac{-16-30 i}{4}$
$\Rightarrow x^{3}=\frac{-16-30 i}{4} \times \frac{3-5 i}{2}$
$=\frac{-48+80 i-90 i+150 i^{2}}{8}$
$=\frac{-198-10 i}{8}$
$\therefore 2 x^{3}+2 x^{2}-7 x+72=2\left(\frac{-198-10 i}{8}\right)+2\left(\frac{-16-30 i}{4}\right)-7\left(\frac{3-5 i}{2}\right)+72$
$=\frac{-198-10 i-32-60 i-42+70 i+288}{4}$
$=\frac{16}{4}$
= 4
(ii) $x=3+2 i$
$\Rightarrow x^{2}=(3+2 i)^{2}$
$=9+4 i^{2}+12 i$
$=5+12 i$
$\Rightarrow x^{3}=x^{2} \times x$
$=(5+12 i) \times(3+2 i)$
$=15+10 i+36 i-24$
$=-9+46 i$
$\Rightarrow x^{4}=\left(x^{2}\right)^{2}$
$=(5+12 i)^{2}$
$=25+144 i^{2}+120 i$
$=-119+120 i$
$\Rightarrow x^{4}-4 x^{3}+4 x^{2}+8 x+44=-119+120 i-4(-9+46 i)+4(5+12 i)+8(3+2 i)+44$
$=-119+120 i+36-184 i+20+48 i+24+16 i+44$
= 5
(iii) $x=-1+\sqrt{2} i$
$\Rightarrow x^{2}=(-1+\sqrt{2} i)^{2}$
$=1+2 i^{2}-2 \sqrt{2} i$
$=-1-2 \sqrt{2} i$
$\Rightarrow x^{3}=(-1-2 \sqrt{2} i) \times(-1+\sqrt{2} i)$
$=1-\sqrt{2} i+2 \sqrt{2} i-4 i^{2}$
$=5+\sqrt{2} i$
$\Rightarrow x^{4}=(-1-2 \sqrt{2} i)^{2}$
$=1+8 i^{2}+4 \sqrt{2} i$
$=-7+4 \sqrt{2} i$
$\Rightarrow x^{4}+4 x^{3}+6 x^{2}+4 x+9=-7+4 \sqrt{2} i+4(5+\sqrt{2} i)+6(-1-2 \sqrt{2} i)+4(-1+\sqrt{2} i)+9$
$=-7+4 \sqrt{2} i+20+4 \sqrt{2} i-6-12 \sqrt{2} i-4+4 \sqrt{2} i+9$
= 12
(iv) $x=\frac{1+i}{\sqrt{2}}$
$\Rightarrow x^{2}=\left(\frac{1+i}{\sqrt{2}}\right)^{2}$
$=\left(\frac{1+i^{2}+2 i}{2}\right)$
$=\frac{2 i}{2}$
$=i$
$\Rightarrow x^{6}=\left(x^{2}\right)^{3}$
$=i^{3}$
$=-i$
$\Rightarrow x^{2}=i$
$\Rightarrow x^{4}=\left(x^{2}\right)^{2}$
$=i^{2}$
$=-1$
Now, $x^{6}+x^{4}+x^{2}+1=-i-1+i+1$
= 0
$(\mathrm{v}) x=-2-\sqrt{3} i$
$\Rightarrow x^{2}=(-2-\sqrt{3} i)^{2}$
$=(-2)^{2}+(-\sqrt{3} i)^{2}+2(-2)(-\sqrt{3} i)$
$=4+3 i^{2}+4 \sqrt{3} i$
$=4-3+4 \sqrt{3} i$ $\left[\because i^{2}=-1\right]$
$=1+4 \sqrt{3} i$
$\Rightarrow x^{3}=(1+4 \sqrt{3} i) \times(-2-\sqrt{3} i)$
$=-2-\sqrt{3} i-8 \sqrt{3} i-12 i^{2}$
$=10-9 \sqrt{3} i$ $\left[\because i^{2}=-1\right]$
$\Rightarrow x^{4}=(1+4 \sqrt{3} i)^{2}$
$=1+48 i^{2}+8 \sqrt{3} i$
$=-47+8 \sqrt{3} i$ $\left[\because i^{2}=-1\right]$
$\Rightarrow 2 x^{4}+5 x^{3}+7 x^{2}-x+41=2(-47+8 \sqrt{3} i)+5(10-9 \sqrt{3} i)+7(1+4 \sqrt{3} i)-(-2-\sqrt{3} i)+41$
$=-94+16 \sqrt{3} i+50-45 \sqrt{3} i+7+28 \sqrt{3} i+2+\sqrt{3} i+41$
= 6
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