Find roots of polynomial $$ p(x) = -x+\frac{1}{2}+1-x^3 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= 0.8612\\[1 em]x_2 &= -0.4306+1.2475i\\[1 em]x_3 &= -0.4306-1.2475i \end{aligned} $$Explanation
Step 1:
Get rid of fractions by multipling equation by $ \color{blue}{ 2 } $.
$$ \begin{aligned} -x+\frac{1}{2}+1-x^3 & = 0 ~~~ / \cdot \color{blue}{ 2 } \\[1 em] -2x+1+2-2x^3 & = 0 \end{aligned} $$Step 2:
Combine like terms:
$$ -2x+ \color{blue}{1} + \color{blue}{2} -2x^3 = -2x^3-2x+ \color{blue}{3} $$Step 3:
Polynomial $ -2x^3-2x+3 $ has no rational roots that can be found using Rational Root Test, so the roots were found using qubic formulas.
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