Find the solutions of the equation

Question

$x^3- 3x + 2 = 0$

Using the Horner scheme, I can find easily the roots of the equation:

$x_1 = x_2 = 1 $ and $x_3 = 2.$

How can I find them using other method ?

Answer 1

Notice that $x=1$ is a root from rational root test.

$$x^3-3x+2=(x-1)(x^2+x-2)=(x-1)(x-1)(x+2)=(x-1)^2(x+2)$$

Answer 2

There are formulas to solve the ternary and quaternary equations. This is classical mathematics, done hundreds of years ago. Look up Cardano's formula and Ferrari's formula.