Find roots of polynomial $$ p(x) = 8x^5-15x^4+10x^2 $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= -0.6971\\[1 em]x_3 &= 1.2861+0.373i\\[1 em]x_4 &= 1.2861-0.373i \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ x^2 }$ from $ 8x^5-15x^4+10x^2 $ and solve two separate equations:

$$ \begin{aligned} 8x^5-15x^4+10x^2 & = 0\\[1 em] \color{blue}{ x^2 }\cdot ( 8x^3-15x^2+10 ) & = 0 \\[1 em] \color{blue}{ x^2 = 0} ~~ \text{or} ~~ 8x^3-15x^2+10 & = 0 \end{aligned} $$

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

Step 2:

Polynomial $ 8x^3-15x^2+10 $ has no rational roots that can be found using Rational Root Test, so the roots were found using qubic formulas.

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