Final answer:
To find a factor of the cubic equation f(x) = x^3 - 6x^2 + 11x - 6, we use synthetic division and the Rational Root Theorem to identify potential roots. Upon testing these roots, 1 is found to be a root, leading to the factored form of the polynomial: (x - 1)(x^2 - 5x + 6), and further (x - 1)(x - 2)(x - 3).
Step-by-step explanation:
The question asks how to find a factor of the cubic equation f(x) = x^3 - 6x^2 + 11x - 6 using synthetic division. The first step in the synthetic division process is to identify a potential root of the equation. The Rational Root Theorem can help us with this by listing all possible rational roots as the factors of the constant term divided by the factors of the leading coefficient. In this case, potential roots could be ±1, ±2, ±3, and ±6.
Performing synthetic division with each potential root, we find that 1 is indeed a root because the remainder is 0 when we divide the polynomial by (x - 1). The synthetic division in this case would be set up by writing down the coefficients of f(x): 1, -6, 11, -6, and then bringing down the 1 and continuing the process to find that the quotient is x^2 - 5x + 6. The factored form of the polynomial is then (x - 1)(x^2 - 5x + 6), and we can further factor the quadratic to get the completely factored form (x - 1)(x - 2)(x - 3).