A cubic by factoring

#cubic#factoring#rational root

Statement

Solve in $\mathbb{R}$ the equation
$x^3-3x^2-4x+12=0.$

Hint

Look for an obvious integer root, then factor.

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Rational root test. Test $x=2$ : $8-12-8+12=0$ . So $x=2$ is a root.

Factor. Divide by $x-2$ :
$x^3-3x^2-4x+12=(x-2)(x^2-x-6)=(x-2)(x-3)(x+2).$

Roots. $x=2$ , $x=3$ , $x=-2$ .

Conclusion. $x\in\{-2,2,3\}$ .