Question
Find roots of polynomial $$ p(x) = 12x^5+40x^4-32x^3 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= \dfrac{ 2 }{ 3 }\\[1 em]x_3 &= -4 \end{aligned} $$Explanation
Step 1:
Factor out $ \color{blue}{ 4x^3 }$ from $ 12x^5+40x^4-32x^3 $ and solve two separate equations:
$$ \begin{aligned} 12x^5+40x^4-32x^3 & = 0\\[1 em] \color{blue}{ 4x^3 }\cdot ( 3x^2+10x-8 ) & = 0 \\[1 em] \color{blue}{ 4x^3 = 0} ~~ \text{or} ~~ 3x^2+10x-8 & = 0 \end{aligned} $$One solution is $ \color{blue}{ x = 0 } $. Use second equation to find the remaining roots.
Step 2:
The solutions of $ 3x^2+10x-8 = 0 $ are: $ x = -4 ~ \text{and} ~ x = \dfrac{ 2 }{ 3 }$.
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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