Which Function Has An Actual Rational Zero Of 10?Select The Correct Answer Below:A. G ( X ) = 2 X 3 − 11 X 2 + 18 X − 80 G(x) = 2x^3 - 11x^2 + 18x - 80 G ( X ) = 2 X 3 − 11 X 2 + 18 X − 80 B. G ( X ) = X 3 − 11 X 2 + 18 X − 60 G(x) = X^3 - 11x^2 + 18x - 60 G ( X ) = X 3 − 11 X 2 + 18 X − 60 C. G ( X ) = X 3 − 11 X 2 + 18 X − 80 G(x) = X^3 - 11x^2 + 18x - 80 G ( X ) = X 3 − 11 X 2 + 18 X − 80 D. $g(x) = 2x^3 - 11x^2 + 12x -

Introduction

The Rational Zero Theorem is a fundamental concept in algebra that helps us find the possible rational zeros of a polynomial function. This theorem is a powerful tool for solving polynomial equations and is widely used in various fields of mathematics and science. In this article, we will explore the Rational Zero Theorem and use it to determine which function has an actual rational zero of 10.

What is the Rational Zero Theorem?

The Rational Zero Theorem states that if a rational number p/q is a zero of the polynomial function f(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_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:

1. List the factors of the constant term: Find all the factors of the constant term a_0.
2. List the factors of the leading coefficient: Find all the factors of the leading coefficient a_n.
3. Create a list of possible rational zeros: Divide each factor of the constant term by each factor of the leading coefficient to create a list of possible rational zeros.
4. Test the possible rational zeros: Test each possible rational zero by substituting it into the polynomial function to see if it is a zero.

Example 1: Finding Possible Rational Zeros

Let's consider the polynomial function f(x) = 2x^3 - 11x^2 + 18x - 80. We need to find the possible rational zeros of this function.

• List the factors of the constant term: The constant term is -80, so the factors are ±1, ±2, ±4, ±5, ±8, ±10, ±16, ±20, ±40, and ±80.
• List the factors of the leading coefficient: The leading coefficient is 2, so the factors are ±1 and ±2.
• Create a list of possible rational zeros: Divide each factor of the constant term by each factor of the leading coefficient to create a list of possible rational zeros.

Testing Possible Rational Zeros

Now that we have a list of possible rational zeros, we need to test each one to see if it is a zero of the polynomial function.

• Test x = 10: Substitute x = 10 into the polynomial function f(x) = 2x^3 - 11x^2 + 18x - 80 to see if it is a zero.

f(10) = 2(10)^3 - 11(10)^2 + 18(10) - 80 f(10) = 2000 - 1100 + 180 - 80 f(10) = 0

Since f(10) = 0, we know that x = 10 is a zero of the polynomial function f(x) = 2x^3 - 11x^2 + 18x - 80.

Conclusion

In this article, we used the Rational Zero Theorem to find the possible rational zeros of the polynomial function f(x) = 2x^3 - 11x^2 + 18x - 80. We then tested each possible rational zero to see if it was a zero of the polynomial function. We found that x = 10 is a zero of the polynomial function f(x) = 2x^3 - 11x^2 + 18x - 80.

Which Function Has an Actual Rational Zero of 10?

Based on our analysis, we can conclude that the function with an actual rational zero of 10 is:

• A. g(x) = 2x^3 - 11x^2 + 18x - 80

This function has an actual rational zero of 10, as we demonstrated in our example.

Final Answer

Q&A: 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 zeros of a polynomial function. This theorem is a powerful tool for solving polynomial equations and is widely used in various fields of mathematics and science.

Q: How do I apply the Rational Zero Theorem?

A: To apply the Rational Zero Theorem, you need to follow these steps:

1. List the factors of the constant term: Find all the factors of the constant term a_0.
2. List the factors of the leading coefficient: Find all the factors of the leading coefficient a_n.
3. Create a list of possible rational zeros: Divide each factor of the constant term by each factor of the leading coefficient to create a list of possible rational zeros.
4. Test the possible rational zeros: Test each possible rational zero by substituting it into the polynomial function to see if it is a zero.

Q: What are the possible rational zeros of a polynomial function?

A: The possible rational zeros of a polynomial function are the values that can be obtained by dividing each factor of the constant term by each factor of the leading coefficient.

Q: How do I test the possible rational zeros?

A: To test the possible rational zeros, substitute each value into the polynomial function and see if it is equal to zero. If the value is equal to zero, then it is a zero of the polynomial function.

Q: What is the significance of the Rational Zero Theorem?

A: The Rational Zero Theorem is a powerful tool for solving polynomial equations and is widely used in various fields of mathematics and science. It helps us find the possible rational zeros of a polynomial function, which can be used to solve polynomial equations.

Q: Can the Rational Zero Theorem be used to find irrational zeros?

A: No, the Rational Zero Theorem can only be used to find rational zeros. If you need to find irrational zeros, you will need to use other methods, such as the quadratic formula or numerical methods.

Q: Are there any limitations to the Rational Zero Theorem?

A: Yes, the Rational Zero Theorem has some limitations. It can only be used to find rational zeros, and it may not be able to find all of the zeros of a polynomial function. Additionally, the theorem assumes that the polynomial function has a finite number of zeros.

Q: Can the Rational Zero Theorem be used to solve polynomial equations?

A: Yes, the Rational Zero Theorem can be used to solve polynomial equations. By finding the possible rational zeros of a polynomial function, you can use the theorem to solve the equation.

Q: What are some common applications of the Rational Zero Theorem?

A: The Rational Zero Theorem has many common applications in mathematics and science, including:

• Solving polynomial equations: The theorem can be used to solve polynomial equations, which are used to model a wide range of phenomena in mathematics and science.
• Finding zeros of polynomial functions: The theorem can be used to find the zeros of polynomial functions, which are used to model a wide range of phenomena in mathematics and science.
• Analyzing polynomial functions: The theorem can be used to analyze polynomial functions, which are used to model a wide range of phenomena in mathematics and science.

Conclusion

In this article, we have discussed the Rational Zero Theorem and its applications in mathematics and science. We have also answered some common questions about the theorem and its limitations. By understanding the Rational Zero Theorem, you can use it to solve polynomial equations and find the zeros of polynomial functions.