Question
Find roots of polynomial $$ p(x) = m^4-4m^3+7m^2 $$
Answer
The roots of polynomial $ p(m) $ are:
$$ \begin{aligned}m_1 &= 0\\[1 em]m_2 &= 2+\sqrt{ 3 }i\\[1 em]m_3 &= 2-\sqrt{ 3 }i \end{aligned} $$Explanation
Step 1:
Factor out $ \color{blue}{ m^2 }$ from $ m^4-4m^3+7m^2 $ and solve two separate equations:
$$ \begin{aligned} m^4-4m^3+7m^2 & = 0\\[1 em] \color{blue}{ m^2 }\cdot ( m^2-4m+7 ) & = 0 \\[1 em] \color{blue}{ m^2 = 0} ~~ \text{or} ~~ m^2-4m+7 & = 0 \end{aligned} $$One solution is $ \color{blue}{ m = 0 } $. Use second equation to find the remaining roots.
Step 2:
The solutions of $ m^2-4m+7 = 0 $ are: $ m = 2+\sqrt{ 3 }i ~ \text{and} ~ m = 2-\sqrt{ 3 }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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