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