Solve: $x^3 - 6x^2 + 11x - 6 = 0$ using the factor theorem.

Step 1: Identify the polynomial and possible factors

Let the given polynomial be P(x). According to the Factor Theorem, if (x-a) is a factor of P(x), then P(a) = 0. The possible integer roots are the divisors of the constant term, which is -6.

Step 2: Test for $x = 1$

We test $x=1$ by substituting it into the polynomial P(x) $. Since P(1) = 0$, by the Factor Theorem, $(x-1)$ is a factor of P(x).

Step 3: Perform polynomial division

Now that we know $(x-1)$ is a factor, we can divide the polynomial P(x) by $(x-1)$ to find the other factors. We perform polynomial long division or synthetic division.

Step 4: Factor the quadratic expression

The quadratic expression $x^2 - 5x + 6$ can be factored into two linear factors. We look for two numbers that multiply to 6 and add up to -5. These numbers are -2 and -3.

Step 5: State all factors and roots

Combining all the factors, we have P(x) = $(x-1)(x-2)(x-3)$. Setting P(x) = 0 gives us the roots of the polynomial.

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