Question

use the rational root theorem to solve for x.
$x^3 - 6x^2 + 11x - 6 = 0$
$x = ?, , $
enter the solutions in ascending order.

Explanation:

Step1: List possible rational roots

By Rational Root Theorem, possible roots are $\pm1, \pm2, \pm3, \pm6$.

Step2: Test $x=1$

Substitute $x=1$: $1^3 - 6(1)^2 + 11(1) - 6 = 1 - 6 + 11 - 6 = 0$. So $x=1$ is a root.

Step3: Factor out $(x-1)$

Use polynomial division or synthetic division:
$$\frac{x^3 - 6x^2 + 11x - 6}{x-1} = x^2 - 5x + 6$$

Step4: Factor quadratic

Factor $x^2 - 5x + 6$: $(x-2)(x-3)$

Step5: Find all roots

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

Answer:

$x=1, 2, 3$