The Rational Zeros Theorem states that any rational solution, or zero, of a polynomial function can be expressed as a fraction p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
For the function f(x) = 2x³ + 8x² – 7x – 8, we first identify the constant term and the leading coefficient:
- Constant term (the term without x): -8
- Leading coefficient (the coefficient of x³): 2
Next, we find the factors of these terms:
Factors of -8:
- 1, -1, 2, -2, 4, -4, 8, -8
Factors of 2:
- 1, -1, 2, -2
Now, we can use these factors to list all possible rational zeros. By forming fractions p/q for all combinations of p (factors of -8) and q (factors of 2), we get:
- 1/1, 1/2, -1/1, -1/2
- 2/1, 2/2, -2/1, -2/2
- 4/1, 4/2, -4/1, -4/2
- 8/1, 8/2, -8/1, -8/2
Now, we simplify these fractions and combine like terms:
- ±1, ±2, ±4, ±8
- ±1/2
Combining all these gives us the complete list of the potential rational zeros:
Possible Rational Zeros:
- 1, -1, 2, -2, 4, -4, 8, -8, 1/2, -1/2
Thus, using the Rational Zeros Theorem, we have determined that the possible rational zeros of the given function are ±1, ±2, ±4, ±8, ±1/2.