Question
Find roots of polynomial $$ p(x) = x^3+\frac{7}{1000}x $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= \dfrac{\sqrt{ 70 }}{ 100 }i\\[1 em]x_3 &= - \dfrac{\sqrt{ 70 }}{ 100 }i \end{aligned} $$Explanation
Step 1:
Get rid of fractions by multipling equation by $ \color{blue}{ 1000 } $.
$$ \begin{aligned} x^3+\frac{7}{1000}x & = 0 ~~~ / \cdot \color{blue}{ 1000 } \\[1 em] 1000x^3+7x & = 0 \end{aligned} $$Step 2:
Factor out $ \color{blue}{ x }$ from $ 1000x^3+7x $ and solve two separate equations:
$$ \begin{aligned} 1000x^3+7x & = 0\\[1 em] \color{blue}{ x }\cdot ( 1000x^2+7 ) & = 0 \\[1 em] \color{blue}{ x = 0} ~~ \text{or} ~~ 1000x^2+7 & = 0 \end{aligned} $$One solution is $ \color{blue}{ x = 0 } $. Use second equation to find the remaining roots.
Step 3:
The solutions of $ 1000x^2+7 = 0 $ are: $ x = \dfrac{\sqrt{ 70 }}{ 100 } i ~ \text{and} ~ x = - \dfrac{\sqrt{ 70 }}{ 100 } 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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