Find roots of polynomial $$ p(x) = 6x^5-15x^3-24x^2 $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 2.099\\[1 em]x_3 &= -1.0495+0.8968i\\[1 em]x_4 &= -1.0495-0.8968i \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ 3x^2 }$ from $ 6x^5-15x^3-24x^2 $ and solve two separate equations:

$$ \begin{aligned} 6x^5-15x^3-24x^2 & = 0\\[1 em] \color{blue}{ 3x^2 }\cdot ( 2x^3-5x-8 ) & = 0 \\[1 em] \color{blue}{ 3x^2 = 0} ~~ \text{or} ~~ 2x^3-5x-8 & = 0 \end{aligned} $$

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

Step 2:

Polynomial $ 2x^3-5x-8 $ has no rational roots that can be found using Rational Root Test, so the roots were found using qubic formulas.

Polynomial Roots Calculator