The Rational Root Theorem states that any rational solution, or root, of a polynomial equation 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 given polynomial equation x^3 – 2x + 9 = 0, let’s identify the constant term and the leading coefficient:
- Constant term: 9
- Leading coefficient: 1 (the coefficient of x^3)
Next, we need to find the factors of 9 and 1:
- Factors of 9: ±1, ±3, ±9
- Factors of 1: ±1
Now, we can form the possible rational roots using the factors of the constant term (9) divided by the factors of the leading coefficient (1):
- Possible rational roots: ±1, ±3, ±9
In summary, the possible rational roots for the equation x^3 – 2x + 9 = 0 are ±1, ±3, ±9.