Find roots of polynomial $$ p(x) = x^8-8x^6+2x^4-16x^2 $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= -2.8284\\[1 em]x_3 &= 2.8284\\[1 em]x_4 &= 0.8409+0.8409i\\[1 em]x_5 &= 0.8409-0.8409i\\[1 em]x_6 &= -0.8409+0.8409i\\[1 em]x_7 &= -0.8409-0.8409i \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ x^2 }$ from $ x^8-8x^6+2x^4-16x^2 $ and solve two separate equations:

$$ \begin{aligned} x^8-8x^6+2x^4-16x^2 & = 0\\[1 em] \color{blue}{ x^2 }\cdot ( x^6-8x^4+2x^2-16 ) & = 0 \\[1 em] \color{blue}{ x^2 = 0} ~~ \text{or} ~~ x^6-8x^4+2x^2-16 & = 0 \end{aligned} $$

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

Step 2:

Polynomial $ x^6-8x^4+2x^2-16 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.

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