Question

use the rational root theorem to solve for x.
$x^3 + 2x^2 - 5x - 6 = 0$
$x = ? , , $
enter the solutions in ascending order.

Explanation:

Step1: List possible rational roots

Possible roots: $\pm1, \pm2, \pm3, \pm6$

Step2: Test $x=2$ in polynomial

$2^3 + 2(2)^2 -5(2)-6=8+8-10-6=0$

Step3: Factor out $(x-2)$ via division

$\frac{x^3+2x^2-5x-6}{x-2}=x^2+4x+3$

Step4: Factor quadratic

$x^2+4x+3=(x+1)(x+3)$

Step5: Find all roots

Set each factor to 0: $x-2=0$, $x+1=0$, $x+3=0$

Answer:

$x = -3, -1, 2$