Find the zeroes of the polynomial: $4 x^{4}+0 x^{3}+0 x^{2}+500+7$.

Given: Polynomial: $4 x^{4}+0 x^{3}+0 x^{2}+500+7$.

To do: To find the zeroes of the given polynomial.

Solution:

Given polynomial: $4 x^{4}+0 x^{3}+0 x^{2}+500+7$

Let $p(x)=4 x^{4}+0 x^{3}+0 x^{2}+500+7=0$

$\Rightarrow p( x)=4 x^{4}+0 x^{3}+0 x^{2}+500+7=0$

$\Rightarrow 4x^4+0+0+507=0$

$\Rightarrow 4x^4+507=0$

$\Rightarrow4x^4=-507$

$\Rightarrow x^4=-\frac{507}{4}$

Let $u=x^2$ and $u^2=x^4$

$\Rightarrow u^2=-\frac{507}{4}$

$\Rightarrow u=i\frac{13\sqrt{3}}{2},\ -i\frac{13\sqrt{3}}{2}$

$\because u=x^2$, on solving for $x$,

$\Rightarrow x^2=i\frac{13\sqrt{3}}{2},\ -i\frac{13\sqrt{3}}{2}$

$\Rightarrow x=\frac{\sqrt{13}\sqrt[4]{3}}{2}+\frac{\sqrt[4]{507}}{2}i$, $x=-\frac{\sqrt{13}\sqrt[4]{3}}{2}-\frac{\sqrt[4]{507}}{2}i$, $x=-\frac{\sqrt{13}\sqrt[4]{3}}{2}+\frac{\sqrt[4]{507}}{2}i$, $x=\frac{\sqrt{13}\sqrt[4]{3}}{2}-\frac{\sqrt[4]{507}}{2}i$.

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Updated on: 10-Oct-2022

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