Find roots of polynomial $$ p(x) = 2x^4-10x^3+16x^2-10x $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 2.7549\\[1 em]x_3 &= 1.1226+0.7449i\\[1 em]x_4 &= 1.1226-0.7449i \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ 2x }$ from $ 2x^4-10x^3+16x^2-10x $ and solve two separate equations:

$$ \begin{aligned} 2x^4-10x^3+16x^2-10x & = 0\\[1 em] \color{blue}{ 2x }\cdot ( x^3-5x^2+8x-5 ) & = 0 \\[1 em] \color{blue}{ 2x = 0} ~~ \text{or} ~~ x^3-5x^2+8x-5 & = 0 \end{aligned} $$

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

Step 2:

Polynomial $ x^3-5x^2+8x-5 $ has no rational roots that can be found using Rational Root Test, so the roots were found using qubic formulas.

Polynomial Roots Calculator