g(x)= x^4 x^3+x^2+3x 5 Given, x=2+3i ⇒ x^2= 5+12i ⇒ x^3= 46+9i g(2+3i)=2+3i^(4 (2+3i)^3)+(2+3i)^2+3(2+3i) 5 =(2+3i)^(4 ( 46+9i))+( ...

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Complex Numbers

gx= x4-x3+x2+...

Question

$g (x) = x^{4 - x^{3}} + x^{2} + 3 x - 5$

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Solution

$g (x) = x^{4 - x^{3}} + x^{2} + 3 x - 5$

Given, $x = 2 + 3 i$

$\Rightarrow x^{2} = - 5 + 12 i$

$\Rightarrow x^{3} = - 46 + 9 i$

$g (2 + 3 i) = {2 + 3 i}^{(4 - {(2 + 3 i)}^{3})} + {(2 + 3 i)}^{2} + 3 (2 + 3 i) - 5$

$= {(2 + 3 i)}^{(4 - (- 46 + 9 i))} + (- 5 + 12 i) + 6 + 9 i - 5$

$= {(2 + 3 i)}^{(50 - 9 i)} - 5 + 12 i + 6 + 9 i - 5$

$= {(2 + 3 i)}^{(50 - 9 i)} - 4 + 21 i$

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