Final answer:
The cubic equation f(x) = x³ - 6x² + 11x - 6 can be factored using synthetic division, revealing that (x - 1), (x - 2), and (x - 3) are its factors.
Step-by-step explanation:
To solve the mathematical problem completely represented by the cubic equation f(x) = x³ - 6x² + 11x - 6, we can attempt to find a factor using synthetic division or by inspection.
First, we look for rational roots that are factors of the constant term (in this case, ±6, ±3, ±2, and ±1). Synthetic division will quickly tell us if our guess is a root since the remainder will be zero.
Let's test if x=1 is a root:
1 | 1 -6 11 -6
| 1 -5 6
-----------------
1 -5 6 0
The remainder is 0, which means x=1 is a root and (x - 1) is a factor. After dividing, the quotient we obtain is x² - 5x + 6.
This quadratic can be factored further into (x - 2)(x - 3).
Therefore, the factors of the original cubic equation are (x - 1)(x - 2)(x - 3).
To check if these factors are correct, we can multiply them back together to see if we get the original polynomial:
(x - 1)(x - 2)(x - 3)
= (x - 1)((x - 2)(x - 3))
= (x - 1)(x² - 5x + 6)
= x³ - 6x² + 11x - 6
This confirms our factors are correct.