Find roots of polynomial $$ p(x) = x^7-5x^4-91x^3 $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 3.2169\\[1 em]x_3 &= -2.9548\\[1 em]x_4 &= -0.131+3.0914i\\[1 em]x_5 &= -0.131-3.0914i \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ x^3 }$ from $ x^7-5x^4-91x^3 $ and solve two separate equations:

$$ \begin{aligned} x^7-5x^4-91x^3 & = 0\\[1 em] \color{blue}{ x^3 }\cdot ( x^4-5x-91 ) & = 0 \\[1 em] \color{blue}{ x^3 = 0} ~~ \text{or} ~~ x^4-5x-91 & = 0 \end{aligned} $$

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

Step 2:

Polynomial $ x^4-5x-91 $ has no rational roots that can be found using Rational Root Test, so the roots were found using quartic formulas.

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