Find roots of polynomial $$ p(x) = -5x^7+9x^6+3x^2 $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 1.8511\\[1 em]x_3 &= 0.5004+0.628i\\[1 em]x_4 &= 0.5004-0.628i\\[1 em]x_5 &= -0.526+0.4755i\\[1 em]x_6 &= -0.526-0.4755i \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ -x^2 }$ from $ -5x^7+9x^6+3x^2 $ and solve two separate equations:

$$ \begin{aligned} -5x^7+9x^6+3x^2 & = 0\\[1 em] \color{blue}{ -x^2 }\cdot ( 5x^5-9x^4-3 ) & = 0 \\[1 em] \color{blue}{ -x^2 = 0} ~~ \text{or} ~~ 5x^5-9x^4-3 & = 0 \end{aligned} $$

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

Step 2:

Polynomial $ 5x^5-9x^4-3 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.

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