Find roots of polynomial $$ p(x) = -2x^4+6x^3+51x $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 4.3485\\[1 em]x_3 &= -0.6743+2.3258i\\[1 em]x_4 &= -0.6743-2.3258i \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ -x }$ from $ -2x^4+6x^3+51x $ and solve two separate equations:

$$ \begin{aligned} -2x^4+6x^3+51x & = 0\\[1 em] \color{blue}{ -x }\cdot ( 2x^3-6x^2-51 ) & = 0 \\[1 em] \color{blue}{ -x = 0} ~~ \text{or} ~~ 2x^3-6x^2-51 & = 0 \end{aligned} $$

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

Step 2:

Polynomial $ 2x^3-6x^2-51 $ has no rational roots that can be found using Rational Root Test, so the roots were found using qubic formulas.

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