Find roots of polynomial $$ p(x) = 6x^5+21x^2 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= -1.5183\\[1 em]x_3 &= 0.7592+1.3149i\\[1 em]x_4 &= 0.7592-1.3149i \end{aligned} $$Explanation
Step 1:
Factor out $ \color{blue}{ 3x^2 }$ from $ 6x^5+21x^2 $ and solve two separate equations:
$$ \begin{aligned} 6x^5+21x^2 & = 0\\[1 em] \color{blue}{ 3x^2 }\cdot ( 2x^3+7 ) & = 0 \\[1 em] \color{blue}{ 3x^2 = 0} ~~ \text{or} ~~ 2x^3+7 & = 0 \end{aligned} $$One solution is $ \color{blue}{ x = 0 } $. Use second equation to find the remaining roots.
Step 2:
Polynomial $ 2x^3+7 $ has no rational roots that can be found using Rational Root Test, so the roots were found using qubic formulas.
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