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 $x=-1$

$(-1)^3 + 2(-1)^2 -5(-1)-6 = -1+2+5-6=0$

Step3: Factor out $(x+1)$

Use polynomial division: $\frac{x^3+2x^2-5x-6}{x+1}=x^2+x-6$

Step4: Factor quadratic

$x^2+x-6=(x+3)(x-2)$

Step5: Find all roots

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

Answer:

$x = -3, -1, 2$