To determine the factors of the polynomial f(x) = 4x³ + 11x² – 75x + 18, we can use several methods, including the Rational Root Theorem or synthetic division. First, we can look for possible rational roots using the factors of the constant term (18) and the leading coefficient (4).
The factors of 18 are ±1, ±2, ±3, ±6, ±9, and ±18. The factors of 4 are ±1, ±2, and ±4. By applying the Rational Root Theorem, we can create a list of potential rational roots, which are the factors of 18 divided by the factors of 4. This gives us candidates such as ±1, ±1/2, ±1/4, ±2, ±3, ±3/2, ±6, ±9, ±18, etc.
Next, we can substitute these values into f(x) to see which one, if any, yields 0. Once we find a root, say r, we can then use synthetic division to divide f(x) by (x – r) to find the quotient polynomial, which will also help us identify other factors. For example, if through substitution we find that f(3) = 0, then (x – 3) is a factor.
After checking a few values, you may discover that 3 is indeed a root. Thus, (x – 3) is a factor. Continuing this process will allow us to factor the polynomial completely, revealing its structure and other factors.
In conclusion, the factors of the polynomial can vary, but knowing how to find them systematically through evaluation of potential rational roots is a solid strategy.