Final Answer:
The correct factorization of the polynomial f(x) = x³ - 4x² - 19x + 14 is c) (x + 2)(x - 1)(x + 7).
Step-by-step explanation:
To find the factors of the given polynomial, we can use the factor theorem or synthetic division. By trying different values for x and checking for roots, we find that x = -2, 1, and -7 are the roots that make the polynomial equal to zero. Therefore, the factors are (x + 2)(x - 1)(x + 7).
Performing synthetic division or using the factor theorem with these values confirms that when x = -2, 1, and -7, the polynomial evaluates to zero. This implies that (x + 2), (x - 1), and (x + 7) are indeed factors of the polynomial. Comparing this result with the provided options, we find that option c) (x + 2)(x - 1)(x + 7) matches the correct factorization.
In summary, the correct answer is c) because it accurately represents the factors obtained by finding the roots of the polynomial, and further confirmation through synthetic division or the factor theorem validates this factorization.