Find roots of polynomial $$ p(x) = 18x^5+x^4+7x^3+8x^2+21x $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 0.6553+0.8965i\\[1 em]x_3 &= 0.6553-0.8965i\\[1 em]x_4 &= -0.6831+0.6926i\\[1 em]x_5 &= -0.6831-0.6926i \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ x }$ from $ 18x^5+x^4+7x^3+8x^2+21x $ and solve two separate equations:

$$ \begin{aligned} 18x^5+x^4+7x^3+8x^2+21x & = 0\\[1 em] \color{blue}{ x }\cdot ( 18x^4+x^3+7x^2+8x+21 ) & = 0 \\[1 em] \color{blue}{ x = 0} ~~ \text{or} ~~ 18x^4+x^3+7x^2+8x+21 & = 0 \end{aligned} $$

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

Step 2:

Polynomial $ 18x^4+x^3+7x^2+8x+21 $ has no rational roots that can be found using Rational Root Test, so the roots were found using quartic formulas.

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