Find roots of polynomial $$ p(x) = 15x^4-12x^2+162x $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= -2.331\\[1 em]x_3 &= 1.1655+1.8097i\\[1 em]x_4 &= 1.1655-1.8097i \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ 3x }$ from $ 15x^4-12x^2+162x $ and solve two separate equations:

$$ \begin{aligned} 15x^4-12x^2+162x & = 0\\[1 em] \color{blue}{ 3x }\cdot ( 5x^3-4x+54 ) & = 0 \\[1 em] \color{blue}{ 3x = 0} ~~ \text{or} ~~ 5x^3-4x+54 & = 0 \end{aligned} $$

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

Step 2:

Polynomial $ 5x^3-4x+54 $ has no rational roots that can be found using Rational Root Test, so the roots were found using qubic formulas.

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