Find roots of polynomial $$ p(x) = 6x^5+15x^4+10x $$

The roots of polynomial $ p(x) $ are:

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= -1.0467\\[1 em]x_3 &= -2.3757\\[1 em]x_4 &= 0.4612+0.6764i\\[1 em]x_5 &= 0.4612-0.6764i \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ x }$ from $ 6x^5+15x^4+10x $ and solve two separate equations:

$$ \begin{aligned} 6x^5+15x^4+10x & = 0\\[1 em] \color{blue}{ x }\cdot ( 6x^4+15x^3+10 ) & = 0 \\[1 em] \color{blue}{ x = 0} ~~ \text{or} ~~ 6x^4+15x^3+10 & = 0 \end{aligned} $$

One solution is $ \color{blue}{ x = 0 } $. Use second equation to find the remaining roots.

Step 2:

Polynomial $ 6x^4+15x^3+10 $ has no rational roots that can be found using Rational Root Test, so the roots were found using quartic formulas.

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