To find all possible rational zeros for the polynomial \( f(x) = x^3 - x^2 + 9x + 7 \) using the Rational Root Theorem, follow these steps:
1. Identify the leading coefficient (the coefficient of the highest power of \(x\)) and the constant term (the term without any \(x\)). For the polynomial \( f(x) = x^3 - x^2 + 9x + 7 \), the leading coefficient is 1 (from \(x^3\)) and the constant term is 7.
2. List the factors of the constant term and the leading coefficient.
- Factors of the constant term (7): \(\pm 1, \pm 7\)
- Factors of the leading coefficient (1): \(\pm 1\)
3. Form the possible rational zeros by taking each factor of the constant term and dividing it by each factor of the leading coefficient. Since the leading coefficient's factors are just \(\pm 1\), the possible rational zeros are simply the factors of the constant term.
4. List all combinations and remove duplicates:
- Positive combinations: \( \frac{1}{1} = 1\), \( \frac{7}{1} = 7 \)
- Negative combinations: \( \frac{-1}{1} = -1 \), \( \frac{-7}{1} = -7 \)
So, combining all these values, the list of all possible rational zeros for the polynomial \( f(x) = x^3 - x^2 + 9x + 7 \) is:
[tex]\[
\boxed{1, -1, 7, -7}
\][/tex]
This list ensures that no value appears more than once.
