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

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= -1.436\\[1 em]x_3 &= 1.9523\\[1 em]x_4 &= -0.2582+1.0017i\\[1 em]x_5 &= -0.2582-1.0017i \end{aligned} $$

Step 1:

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

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

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

Step 2:

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

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