Find roots of polynomial $$ p(x) = \frac{1}{2}x^4-x^3-x^2-x+1 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= 0.56\\[1 em]x_2 &= 2.8587\\[1 em]x_3 &= -0.7094+0.8638i\\[1 em]x_4 &= -0.7094-0.8638i \end{aligned} $$Explanation
Step 1:
Get rid of fractions by multipling equation by $ \color{blue}{ 2 } $.
$$ \begin{aligned} \frac{1}{2}x^4-x^3-x^2-x+1 & = 0 ~~~ / \cdot \color{blue}{ 2 } \\[1 em] x^4-2x^3-2x^2-2x+2 & = 0 \end{aligned} $$Step 2:
Polynomial $ x^4-2x^3-2x^2-2x+2 $ has no rational roots that can be found using Rational Root Test, so the roots were found using quartic formulas.
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