Find roots of polynomial $$ p(x) = t^3-15t^2+72t $$

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

$$ \begin{aligned}t_1 &= 0\\[1 em]t_2 &= \frac{ 15 }{ 2 }+\frac{ 3 \sqrt{ 7}}{ 2 }i\\[1 em]t_3 &= \frac{ 15 }{ 2 }-3 \frac{\sqrt{ 7 }}{ 2 }i \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ t }$ from $ t^3-15t^2+72t $ and solve two separate equations:

$$ \begin{aligned} t^3-15t^2+72t & = 0\\[1 em] \color{blue}{ t }\cdot ( t^2-15t+72 ) & = 0 \\[1 em] \color{blue}{ t = 0} ~~ \text{or} ~~ t^2-15t+72 & = 0 \end{aligned} $$

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

Step 2:

The solutions of $ t^2-15t+72 = 0 $ are: $ t = \dfrac{ 15 }{ 2 }+\dfrac{ 3 \sqrt{ 7}}{ 2 }i ~ \text{and} ~ t = \dfrac{ 15 }{ 2 }-\dfrac{ 3 \sqrt{ 7}}{ 2 }i$.

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

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