Find roots of polynomial $$ p(x) = x^4+\frac{1}{81} $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= 0.2357+0.2357i\\[1 em]x_2 &= 0.2357-0.2357i\\[1 em]x_3 &= -0.2357+0.2357i\\[1 em]x_4 &= -0.2357-0.2357i \end{aligned} $$Explanation
Step 1:
Get rid of fractions by multipling equation by $ \color{blue}{ 81 } $.
$$ \begin{aligned} x^4+\frac{1}{81} & = 0 ~~~ / \cdot \color{blue}{ 81 } \\[1 em] 81x^4+1 & = 0 \end{aligned} $$Step 2:
Polynomial $ 81x^4+1 $ has no rational roots that can be found using Rational Root Test, so the roots were found using quartic formulas.
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