Find roots of polynomial $$ p(x) = r^4+32r^2+256r $$

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

$$ \begin{aligned}r_1 &= 0\\[1 em]r_2 &= -4.718\\[1 em]r_3 &= 2.359+6.9782i\\[1 em]r_4 &= 2.359-6.9782i \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ r }$ from $ r^4+32r^2+256r $ and solve two separate equations:

$$ \begin{aligned} r^4+32r^2+256r & = 0\\[1 em] \color{blue}{ r }\cdot ( r^3+32r+256 ) & = 0 \\[1 em] \color{blue}{ r = 0} ~~ \text{or} ~~ r^3+32r+256 & = 0 \end{aligned} $$

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

Step 2:

Polynomial $ r^3+32r+256 $ has no rational roots that can be found using Rational Root Test, so the roots were found using qubic formulas.

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