Find roots of polynomial $$ p(x) = -10x^4+20x^3-21x^2-10x $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= -0.3441\\[1 em]x_3 &= 1.172+1.2381i\\[1 em]x_4 &= 1.172-1.2381i \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ -x }$ from $ -10x^4+20x^3-21x^2-10x $ and solve two separate equations:

$$ \begin{aligned} -10x^4+20x^3-21x^2-10x & = 0\\[1 em] \color{blue}{ -x }\cdot ( 10x^3-20x^2+21x+10 ) & = 0 \\[1 em] \color{blue}{ -x = 0} ~~ \text{or} ~~ 10x^3-20x^2+21x+10 & = 0 \end{aligned} $$

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

Step 2:

Polynomial $ 10x^3-20x^2+21x+10 $ has no rational roots that can be found using Rational Root Test, so the roots were found using qubic formulas.

Polynomial Roots Calculator