Equation: x${}^3$ - 2x${}^2$ - 5x + 6 = 0.
First, bring out a${}_n$ which is the coefficient of x${}^3$ = 1.
Then, a${}_0$ which is the coefficient void of x = 6.
The factors of a${}_n$ = 1; The factors of a${}_0$ = 1, 2, 3 and 6.
The numbers to test for the roots are $\pm (\frac{a_0}{a_n})$.
= $\pm (1,2,3,6)$.
Test for +1: 1${}^3$ - 2(1${}^2$) - 5(1) + 6 = 1 - 2 - 5 + 6 = 0.
Therefore x = 1 is a root of the equation.
Using long division method, $\frac{x^3-2x^2-5x+6}{x-1}$ = x${}^2$ - x - 6.
x${}^2$ - x - 6 = (x - 3)(x + 2).
x = -2, 3.
$\therefore$ The roots of the equation = 1, -2 and 3.