Question
Find roots of polynomial $$ p(x) = 2s^3+7s^2+8s $$
Answer
The roots of polynomial $ p(s) $ are:
$$ \begin{aligned}s_1 &= 0\\[1 em]s_2 &= -\dfrac{ 7 }{ 4 }+\dfrac{\sqrt{ 15 }}{ 4 }i\\[1 em]s_3 &= -\dfrac{ 7 }{ 4 }- \dfrac{\sqrt{ 15 }}{ 4 }i \end{aligned} $$Explanation
Step 1:
Factor out $ \color{blue}{ s }$ from $ 2s^3+7s^2+8s $ and solve two separate equations:
$$ \begin{aligned} 2s^3+7s^2+8s & = 0\\[1 em] \color{blue}{ s }\cdot ( 2s^2+7s+8 ) & = 0 \\[1 em] \color{blue}{ s = 0} ~~ \text{or} ~~ 2s^2+7s+8 & = 0 \end{aligned} $$One solution is $ \color{blue}{ s = 0 } $. Use second equation to find the remaining roots.
Step 2:
The solutions of $ 2s^2+7s+8 = 0 $ are: $ s = -\dfrac{ 7 }{ 4 }+\dfrac{\sqrt{ 15 }}{ 4 }i ~ \text{and} ~ s = -\dfrac{ 7 }{ 4 }-\dfrac{\sqrt{ 15 }}{ 4 }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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