Find roots of polynomial $$ p(x) = -x^4+3x^2+112x $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 5.0276\\[1 em]x_3 &= -2.5138+3.9947i\\[1 em]x_4 &= -2.5138-3.9947i \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ -x }$ from $ -x^4+3x^2+112x $ and solve two separate equations:

$$ \begin{aligned} -x^4+3x^2+112x & = 0\\[1 em] \color{blue}{ -x }\cdot ( x^3-3x-112 ) & = 0 \\[1 em] \color{blue}{ -x = 0} ~~ \text{or} ~~ x^3-3x-112 & = 0 \end{aligned} $$

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

Step 2:

Polynomial $ x^3-3x-112 $ has no rational roots that can be found using Rational Root Test, so the roots were found using qubic formulas.

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