To find a factor of the polynomial f(x) = 5x³ + 24x² + 75x + 14, we can use the Rational Root Theorem, synthetic division, or polynomial long division.
We can start by testing some possible rational roots based on the constant term (14) and the leading coefficient (5). The candidates for potential rational roots could be ±1, ±2, ±7, ±14 for the factors of the constant 14, divided by ±1, ±5 for the factors of 5.
After testing these values, we find that when we substitute x = -2 into f(x), it equals 0. Therefore, x + 2 is a factor of the polynomial. We can verify this by performing synthetic division with -2:
- 5 | 24 | 75 | 14
- | -10 | -28 | -14
- ———————-
- 5 | 14 | 47 | 0
This leaves us with the quotient 5x² + 14x + 47 as the other factor. Thus, one of the factors of f(x) is (x + 2), and the polynomial can be expressed as:
f(x) = (x + 2)(5x² + 14x + 47).
In conclusion, a factor of f(x) = 5x³ + 24x² + 75x + 14 is (x + 2).