The Following Equation Is Given. Complete Parts (a)-(c).$\[ X^3 - 2x^2 - 9x + 18 = 0 \\]a. List All Rational Roots That Are Possible According To The Rational Zero Theorem.$\[ \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18 \\](Use A Comma
Introduction
The Rational Zero Theorem is a fundamental concept in algebra that helps us find the possible rational roots of a polynomial equation. In this article, we will explore the theorem, its application, and how to use it to find the rational roots of a given polynomial equation.
What is the Rational Zero Theorem?
The Rational Zero Theorem states that if a rational number p/q is a root of the polynomial equation a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0 = 0, where p and q are integers and q ≠0, then p must be a factor of the constant term a_0, and q must be a factor of the leading coefficient a_n.
How to Apply the Rational Zero Theorem
To apply the Rational Zero Theorem, we need to follow these steps:
- List the factors of the constant term: The constant term is the term without the variable x. In the given equation x^3 - 2x^2 - 9x + 18 = 0, the constant term is 18. We need to list all the factors of 18, which are ±1, ±2, ±3, ±6, ±9, and ±18.
- List the factors of the leading coefficient: The leading coefficient is the coefficient of the term with the highest power of x. In the given equation, the leading coefficient is 1. Since the leading coefficient is 1, its only factors are ±1.
- Combine the factors: We need to combine the factors of the constant term and the leading coefficient to get the possible rational roots. In this case, the possible rational roots are ±1, ±2, ±3, ±6, ±9, and ±18.
Example: Finding Rational Roots
Let's use the given equation x^3 - 2x^2 - 9x + 18 = 0 to find the rational roots.
Step 1: List the factors of the constant term
The constant term is 18. We need to list all the factors of 18, which are ±1, ±2, ±3, ±6, ±9, and ±18.
Step 2: List the factors of the leading coefficient
The leading coefficient is 1. Since the leading coefficient is 1, its only factors are ±1.
Step 3: Combine the factors
We need to combine the factors of the constant term and the leading coefficient to get the possible rational roots. In this case, the possible rational roots are ±1, ±2, ±3, ±6, ±9, and ±18.
Conclusion
In conclusion, the Rational Zero Theorem is a powerful tool for finding the possible rational roots of a polynomial equation. By listing the factors of the constant term and the leading coefficient, we can combine them to get the possible rational roots. In this article, we have seen how to apply the Rational Zero Theorem to find the rational roots of a given polynomial equation.
The Final Answer
Q&A: The Rational Zero Theorem
Q: What is the Rational Zero Theorem?
A: The Rational Zero Theorem is a fundamental concept in algebra that helps us find the possible rational roots of a polynomial equation. It states that if a rational number p/q is a root of the polynomial equation a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0 = 0, where p and q are integers and q ≠0, then p must be a factor of the constant term a_0, and q must be a factor of the leading coefficient a_n.
Q: How do I apply the Rational Zero Theorem?
A: To apply the Rational Zero Theorem, you need to follow these steps:
- List the factors of the constant term: The constant term is the term without the variable x. List all the factors of the constant term.
- List the factors of the leading coefficient: The leading coefficient is the coefficient of the term with the highest power of x. List all the factors of the leading coefficient.
- Combine the factors: Combine the factors of the constant term and the leading coefficient to get the possible rational roots.
Q: What are the possible rational roots?
A: The possible rational roots are the combinations of the factors of the constant term and the leading coefficient. For example, if the constant term is 18 and the leading coefficient is 1, the possible rational roots are ±1, ±2, ±3, ±6, ±9, and ±18.
Q: How do I find the rational roots of a polynomial equation?
A: To find the rational roots of a polynomial equation, you need to apply the Rational Zero Theorem. First, list the factors of the constant term and the leading coefficient. Then, combine the factors to get the possible rational roots. Finally, test each possible rational root to see if it is a root of the polynomial equation.
Q: What are some common mistakes to avoid when applying the Rational Zero Theorem?
A: Some common mistakes to avoid when applying the Rational Zero Theorem include:
- Not listing all the factors of the constant term and the leading coefficient: Make sure to list all the factors of the constant term and the leading coefficient to get all the possible rational roots.
- Not combining the factors correctly: Combine the factors of the constant term and the leading coefficient correctly to get the possible rational roots.
- Not testing each possible rational root: Test each possible rational root to see if it is a root of the polynomial equation.
Q: Can the Rational Zero Theorem be used to find the irrational roots of a polynomial equation?
A: No, the Rational Zero Theorem can only be used to find the possible rational roots of a polynomial equation. It cannot be used to find the irrational roots of a polynomial equation.
Q: Are there any other theorems that can be used to find the roots of a polynomial equation?
A: Yes, there are other theorems that can be used to find the roots of a polynomial equation, such as the Factor Theorem and the Remainder Theorem. However, the Rational Zero Theorem is a powerful tool for finding the possible rational roots of a polynomial equation.
Conclusion
In conclusion, the Rational Zero Theorem is a fundamental concept in algebra that helps us find the possible rational roots of a polynomial equation. By applying the theorem, we can find the possible rational roots of a polynomial equation and test each root to see if it is a root of the equation.