Find roots of polynomial $$ p(x) = 120x^3-72x $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= \frac{\sqrt{ 15 }}{ 5 }\\[1 em]x_3 &= - \frac{\sqrt{ 15 }}{ 5 } \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ 24x }$ from $ 120x^3-72x $ and solve two separate equations:

$$ \begin{aligned} 120x^3-72x & = 0\\[1 em] \color{blue}{ 24x }\cdot ( 5x^2-3 ) & = 0 \\[1 em] \color{blue}{ 24x = 0} ~~ \text{or} ~~ 5x^2-3 & = 0 \end{aligned} $$

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

Step 2:

The solutions of $ 5x^2-3 = 0 $ are: $ x = - \dfrac{\sqrt{ 15 }}{ 5 } ~ \text{and} ~ x = \dfrac{\sqrt{ 15 }}{ 5 }$.

You can use step-by-step quadratic equation solver to see a detailed explanation on how to solve this quadratic equation.

Polynomial Roots Calculator