Question

Solve the following equation:
$x^3$ - 5x${}^2$ - 2x + 24 = 0, given that the product of two roots is 12.

Answer 1

Let the roots be $\alpha$,$\beta$ ,$\gamma$.
$\alpha \beta \gamma$= -24, $\alpha \beta$ = 12
$\therefore$ $\gamma$ = - 2 is a root. Dividing the equation by x + 2, we get
x${}^2$ - 7x +12 = 0 giving x = 4, 3.
Thus the roots are $4,3,-2.$