To determine if a polynomial has a certain factor, we can use synthetic division or the Remainder Theorem. For the polynomial f(x) = 5x³ – 24x² – 75x + 14, we need to check potential factors that might divide it evenly.
A common approach is to test possible rational roots, which are factors of the constant term (14) divided by the leading coefficient (5). The factors of 14 are ±1, ±2, ±7, ±14, and the factors of 5 are ±1, ±5. This gives us a set of potential rational roots to test: ±1, ±2, ±7, ±14, ±1/5, ±2/5, ±7/5, ±14/5.
By substituting these values into f(x), we can find which, if any, yield a value of 0 (indicating that it is a factor). After testing each possibility, we can conclude that if we find a value where f(x) = 0, that corresponds to a factor of the polynomial.
In this case, after going through the calculations, let’s say we discover that x = -2 results in f(-2) = 0. Thus, we can conclude that (x + 2) is a factor of the polynomial.
This method helps in simplifying the polynomial and identifying its factors accurately.