If the polynomial f(x) = x^3 - 6x^2 + 11x - 6 is factored, which of the following is one of its factors?

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Q: If the polynomial f(x) = x^3 - 6x^2 + 11x - 6 is factored, which of the following is one of its factors?

Solution: The polynomial can be factored as (x - 1)(x - 2)(x - 3), so x - 1 is one of its factors.

Steps: 9

Step 1: Identify the polynomial given in the question, which is f(x) = x^3 - 6x^2 + 11x - 6.
Step 2: Understand that we need to factor this polynomial to find its factors.
Step 3: Look for possible rational roots using the Rational Root Theorem, which suggests testing factors of the constant term (-6).
Step 4: Test x = 1 by substituting it into the polynomial: f(1) = 1^3 - 6(1^2) + 11(1) - 6 = 0. Since f(1) = 0, x - 1 is a factor.
Step 5: To find the complete factorization, divide the polynomial by (x - 1) using synthetic division or polynomial long division.
Step 6: After dividing, you will find that f(x) can be expressed as (x - 1)(x^2 - 5x + 6).
Step 7: Factor the quadratic x^2 - 5x + 6 further to get (x - 2)(x - 3).
Step 8: Combine all factors to get the complete factorization: f(x) = (x - 1)(x - 2)(x - 3).
Step 9: Conclude that one of the factors of the polynomial is x - 1.