Find roots of polynomial $$ p(x) = -x^5+2x^4-3x^3-2x^2+8x $$

The roots of polynomial $ p(x) $ are:

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= -1.2259\\[1 em]x_3 &= 1.4943\\[1 em]x_4 &= 0.8658+1.902i\\[1 em]x_5 &= 0.8658-1.902i \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ -x }$ from $ -x^5+2x^4-3x^3-2x^2+8x $ and solve two separate equations:

$$ \begin{aligned} -x^5+2x^4-3x^3-2x^2+8x & = 0\\[1 em] \color{blue}{ -x }\cdot ( x^4-2x^3+3x^2+2x-8 ) & = 0 \\[1 em] \color{blue}{ -x = 0} ~~ \text{or} ~~ x^4-2x^3+3x^2+2x-8 & = 0 \end{aligned} $$

One solution is $ \color{blue}{ x = 0 } $. Use second equation to find the remaining roots.

Step 2:

Polynomial $ x^4-2x^3+3x^2+2x-8 $ has no rational roots that can be found using Rational Root Test, so the roots were found using quartic formulas.

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