Applying the Rational Zero Theorem to the Given Equation

Oct 13, 2025 by pynkocean.com 19 views

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 and apply it to the given equation to find the possible rational roots.

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_nx^n + a_(n-1)x^(n-1) + ... + a_1x + 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.

Applying the Rational Zero Theorem to the Given Equation

The given equation is 12x^3 + 77x^2 - 48x + 7 = 0. To find the possible rational roots, we need to find the factors of the constant term 7 and the leading coefficient 12.

Factors of the Constant Term 7

The factors of 7 are ±1 and ±7.

Factors of the Leading Coefficient 12

The factors of 12 are ±1, ±2, ±3, ±4, ±6, and ±12.

Possible Rational Roots

Using the Rational Zero Theorem, we can list all possible rational roots by combining the factors of the constant term 7 and the leading coefficient 12. The possible rational roots are:

±1
±7
±1/2
±1/3
±1/4
±1/6
±1/12
±7/2
±7/3
±7/4
±7/6
±7/12

Choosing the Correct Answer

Based on the Rational Zero Theorem, the possible rational roots of the given equation are:

±1
±7
±1/2
±1/3
±1/4
±1/6
±1/12
±7/2
±7/3
±7/4
±7/6
±7/12

The correct answer is:

A. ±1, ±7, ±1/2, ±1/3, ±1/4, ±1/6, ±1/12, ±7/2, ±7/3, ±7/4, ±7/6, ±7/12

Conclusion

In the previous article, we explored the Rational Zero Theorem and applied it to the given equation to find the possible rational roots. In this article, we will answer some frequently asked questions about 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_nx^n + a_(n-1)x^(n-1) + ... + a_1x + 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 to a polynomial equation?

A: To apply the Rational Zero Theorem, you need to find the factors of the constant term and the leading coefficient of the polynomial equation. Then, you can list all possible rational roots by combining the factors of the constant term and the leading coefficient.

Q: What are the possible rational roots of a polynomial equation?

A: The possible rational roots of a polynomial equation are all combinations of the factors of the constant term and the leading coefficient. For example, if the constant term is 6 and the leading coefficient is 12, the possible rational roots are ±1, ±2, ±3, ±6, ±1/2, ±1/3, ±1/4, ±1/6, ±2/3, ±2/4, ±2/6, ±3/2, ±3/4, ±3/6, ±6/2, ±6/3, ±6/4, ±6/6.

Q: How do I choose the correct answer when applying the Rational Zero Theorem?

A: When applying the Rational Zero Theorem, you need to list all possible rational roots and then choose the correct answer based on the given options. Make sure to check if the options are correct and complete.

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 possible rational roots
Not checking if the options are correct and complete
Not considering the factors of the constant term and the leading coefficient
Not applying the theorem correctly to the given equation

Q: Can the Rational Zero Theorem be used to find the roots of a polynomial equation with complex coefficients?

A: No, the Rational Zero Theorem can only be used to find the possible rational roots of a polynomial equation with rational coefficients. If the polynomial equation has complex coefficients, you need to use other methods to find the roots.

Q: Can the Rational Zero Theorem be used to find the roots of a polynomial equation with irrational coefficients?

A: No, the Rational Zero Theorem can only be used to find the possible rational roots of a polynomial equation with rational coefficients. If the polynomial equation has irrational coefficients, you need to use other methods to find the roots.

Conclusion

The Rational Zero Theorem is a powerful tool for finding the possible rational roots of a polynomial equation. By understanding the theorem and applying it correctly, you can find the possible rational roots of a polynomial equation. Remember to avoid common mistakes and use the theorem only for polynomial equations with rational coefficients.