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

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Step-by-Step Solution

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.