Find roots of polynomial $$ p(x) = 10x^4+56x^3-36x^2-288x $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 2.1707\\[1 em]x_3 &= -2.5332\\[1 em]x_4 &= -5.2374 \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ 2x }$ from $ 10x^4+56x^3-36x^2-288x $ and solve two separate equations:

$$ \begin{aligned} 10x^4+56x^3-36x^2-288x & = 0\\[1 em] \color{blue}{ 2x }\cdot ( 5x^3+28x^2-18x-144 ) & = 0 \\[1 em] \color{blue}{ 2x = 0} ~~ \text{or} ~~ 5x^3+28x^2-18x-144 & = 0 \end{aligned} $$

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

Step 2:

Polynomial $ 5x^3+28x^2-18x-144 $ has no rational roots that can be found using Rational Root Test, so the roots were found using qubic formulas.

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