Final answer:
The set of possible rational zeros for the function f(x) = 2x³+6x²+7x - 8 can be found using the Rational Root Theorem. By listing the factors of the constant term (8) and the leading coefficient (2), and taking all possible combinations of these factors, the possible rational zeros are ±[1, 0.5, 2, 4, 8].
Step-by-step explanation:
To find the set of possible rational zeros for a polynomial function, we use the Rational Root Theorem. This theorem states that if the polynomial function f(x) = 2x³+6x²+7x - 8 has any rational zeros, then they must be of the form ±p/q, where p is a factor of the constant term (in this case, 8), and q is a factor of the leading coefficient (in this case, 2).
Let's list the factors of 8 (the constant term): ±1, ±2, ±4, ±8. Now let's list the factors of 2 (the leading coefficient): ±1, ±2. We can now create a list of all possible rational zeros by taking all possible ratios of p to q: ±1/1, ±1/2, ±2/1, ±2/2, ±4/1, ±4/2, ±8/1, ±8/2. This simplifies to the set of possible rational zeros: ±1, ±0.5, ±2, ±4, ±8.
Remember to include both positive and negative versions of these numbers to get the complete list of possible rational zeros: ±[1, 0.5, 2, 4, 8].
Steps to solve the mathematical problem completely:
List factors of the constant term (8).
List factors of the leading coefficient (2).
Create ratios p/q using the factors from steps 1 and 2.
Include both positive and negative versions to get the full set of possible zeros.
After you have the list of possible rational zeros, you can use synthetic division or another method to test which, if any, are actual zeros of the function.