Find roots of polynomial $$ p(x) = 5x^4-21x^3+24x-4x $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 1.1441\\[1 em]x_3 &= -0.8868\\[1 em]x_4 &= 3.9427 \end{aligned} $$

Step 1:

Combine like terms:

$$ 5x^4-21x^3+ \color{blue}{24x} \color{blue}{-4x} = 5x^4-21x^3+ \color{blue}{20x} $$

Step 2:

Factor out $ \color{blue}{ x }$ from $ 5x^4-21x^3+20x $ and solve two separate equations:

$$ \begin{aligned} 5x^4-21x^3+20x & = 0\\[1 em] \color{blue}{ x }\cdot ( 5x^3-21x^2+20 ) & = 0 \\[1 em] \color{blue}{ x = 0} ~~ \text{or} ~~ 5x^3-21x^2+20 & = 0 \end{aligned} $$

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

Step 3:

Polynomial $ 5x^3-21x^2+20 $ has no rational roots that can be found using Rational Root Test, so the roots were found using qubic formulas.

Polynomial Roots Calculator