The Rational Root Theorem is a tool that helps us identify potential rational roots of a polynomial equation. For the polynomial equation you provided, x3 – 2x + 9 = 0, we will apply the theorem to find all possible rational roots.
First, let’s identify the components needed to apply the Rational Root Theorem:
- Leading Coefficient: The leading coefficient is the coefficient of the highest degree term. In this case, the leading coefficient is 1 (from the term x3).
- Constant Term: The constant term is the term without any variable. Here, the constant term is 9.
According to the Rational Root Theorem, any potential rational root can be expressed as a fraction p/q, where:
- p is a factor of the constant term (9).
- q is a factor of the leading coefficient (1).
Let’s start by finding the factors of the constant term (9):
- Factors of 9: ±1, ±3, ±9
Next, we find the factors of the leading coefficient (1):
- Factors of 1: ±1
Now we can list all the potential rational roots:
- By substituting the factors of 9 for p and the factors of 1 for q, the possible rational roots are:
- ±1, ±3, ±9
Thus, the complete list of all possible rational roots for the equation x3 – 2x + 9 = 0 is:
- 1
- -1
- 3
- -3
- 9
- -9
These candidates can be tested to determine if they are indeed roots of the polynomial equation. You can substitute each value back into the original equation to see which, if any, satisfy the equation x3 – 2x + 9 = 0.