Find roots of polynomial $$ p(x) = 1-5x+\frac{1}{2}x^2-\frac{1}{2}x^3 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= 0.2033\\[1 em]x_2 &= 0.3984+3.1112i\\[1 em]x_3 &= 0.3984-3.1112i \end{aligned} $$Explanation
Step 1:
Get rid of fractions by multipling equation by $ \color{blue}{ 2 } $.
$$ \begin{aligned} 1-5x+\frac{1}{2}x^2-\frac{1}{2}x^3 & = 0 ~~~ / \cdot \color{blue}{ 2 } \\[1 em] 2-10x+x^2-x^3 & = 0 \end{aligned} $$Step 2:
Write polynomial in descending order
$$ \begin{aligned} 2-10x+x^2-x^3 & = 0\\[1 em] -x^3+x^2-10x+2 & = 0 \end{aligned} $$Step 3:
Polynomial $ -x^3+x^2-10x+2 $ has no rational roots that can be found using Rational Root Test, so the roots were found using qubic formulas.
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