Which Of The Numbers Below Are Potential Roots Of P ( X ) = X 3 + 6 X 2 − 7 X − 60 P(x)=x^3+6x^2-7x-60 P ( X ) = X 3 + 6 X 2 − 7 X − 60 According To The Rational Root Theorem?A. − 10 -10 − 10 B. − 7 -7 − 7 C. − 5 -5 − 5 D. 3 3 3 E. 15 15 15 F. 24 24 24

The Rational Root Theorem is a fundamental concept in algebra that helps us identify potential roots of a polynomial equation. This theorem states that if a rational number $p/q$ is a root of the polynomial $f(x)$, then $p$ must be a factor of the constant term, and $q$ must be a factor of the leading coefficient. In this article, we will apply the Rational Root Theorem to find potential roots of the polynomial $p(x)=x3+6x2-7x-60$.

Understanding the Rational Root Theorem

The Rational Root Theorem is based on the following statement:

• If $p/q$ is a root of the polynomial $f(x)$, then $p$ must be a factor of the constant term, and $q$ must be a factor of the leading coefficient.

In other words, if we want to find potential roots of a polynomial, we need to find all the factors of the constant term and all the factors of the leading coefficient.

Factors of the Constant Term

The constant term of the polynomial $p(x)=x^3+6x^2-7x-60$ is $-60$. To find the factors of $-60$, we can list all the numbers that divide $-60$ without leaving a remainder.

• Factors of $-60$: $\pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 10, \pm 12, \pm 15, \pm 20, \pm 30, \pm 60$

Factors of the Leading Coefficient

The leading coefficient of the polynomial $p(x)=x^3+6x^2-7x-60$ is $1$. Since the leading coefficient is $1$, the only factor of the leading coefficient is $1$.

Applying the Rational Root Theorem

Now that we have found the factors of the constant term and the leading coefficient, we can apply the Rational Root Theorem to find potential roots of the polynomial.

• Potential roots: $\pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 10, \pm 12, \pm 15, \pm 20, \pm 30, \pm 60$

However, we are given a list of specific numbers as potential roots: A. $-10$ B. $-7$ C. $-5$ D. $3$ E. $15$ F. $24$. We need to determine which of these numbers are potential roots of the polynomial according to the Rational Root Theorem.

Checking the Potential Roots

To check if a number is a potential root of the polynomial, we can substitute the number into the polynomial and see if the result is equal to zero.

• A. $-10$: $(-10)^3+6(-10)^2-7(-10)-60 = -1000 + 600 + 70 - 60 = -390 \neq 0$
• B. $-7$: $(-7)^3+6(-7)^2-7(-7)-60 = -343 + 294 + 49 - 60 = -60 \neq 0$
• C. $-5$: $(-5)^3+6(-5)^2-7(-5)-60 = -125 + 150 + 35 - 60 = 0$
• D. $3$: $(3)^3+6(3)^2-7(3)-60 = 27 + 54 - 21 - 60 = 0$
• E. $15$: $(15)^3+6(15)^2-7(15)-60 = 3375 + 1350 - 105 - 60 = 3560 \neq 0$
• F. $24$: $(24)^3+6(24)^2-7(24)-60 = 13824 + 3456 - 168 - 60 = 15552 \neq 0$

Based on the Rational Root Theorem, the potential roots of the polynomial $p(x)=x^3+6x^2-7x-60$ are $\pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 10, \pm 12, \pm 15, \pm 20, \pm 30, \pm 60$. However, only two of the given numbers are potential roots of the polynomial: C. $-5$ and D. $3$.

Conclusion

In our previous article, we applied the Rational Root Theorem to find potential roots of the polynomial $p(x)=x^3+6x^2-7x-60$. We found that the potential roots of the polynomial are $\pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 10, \pm 12, \pm 15, \pm 20, \pm 30, \pm 60$. However, only two of the given numbers are potential roots of the polynomial: C. $-5$ and D. $3$. In this article, we will answer some frequently asked questions about the Rational Root Theorem.

Q: What is the Rational Root Theorem?

A: The Rational Root Theorem is a fundamental concept in algebra that helps us identify potential roots of a polynomial equation. This theorem states that if a rational number $p/q$ is a root of the polynomial $f(x)$, then $p$ must be a factor of the constant term, and $q$ must be a factor of the leading coefficient.

Q: How do I apply the Rational Root Theorem?

A: To apply the Rational Root Theorem, you need to find all the factors of the constant term and all the factors of the leading coefficient. Then, you can use these factors to find potential roots of the polynomial.

Q: What are the factors of the constant term and the leading coefficient?

A: The factors of the constant term are all the numbers that divide the constant term without leaving a remainder. The factors of the leading coefficient are all the numbers that divide the leading coefficient without leaving a remainder.

Q: How do I check if a number is a potential root of the polynomial?

A: To check if a number is a potential root of the polynomial, you can substitute the number into the polynomial and see if the result is equal to zero.

Q: What are some common mistakes to avoid when applying the Rational Root Theorem?

A: Some common mistakes to avoid when applying the Rational Root Theorem include:

• Not finding all the factors of the constant term and the leading coefficient.
• Not checking if a number is a potential root of the polynomial.
• Not using the correct formula to find potential roots.

Q: Can the Rational Root Theorem be used to find all the roots of a polynomial?

A: No, the Rational Root Theorem can only be used to find potential roots of a polynomial. To find all the roots of a polynomial, you need to use other methods such as factoring, synthetic division, or numerical methods.

Q: Is the Rational Root Theorem only used for polynomials with integer coefficients?

A: No, the Rational Root Theorem can be used for polynomials with rational coefficients. However, if the coefficients are not rational, then the theorem may not be applicable.

Q: Can the Rational Root Theorem be used to find roots of polynomials with complex coefficients?

A: No, the Rational Root Theorem is only applicable to polynomials with rational coefficients. If the coefficients are complex, then the theorem may not be applicable.

Conclusion

In this article, we answered some frequently asked questions about the Rational Root Theorem. We hope that this article has helped you understand the Rational Root Theorem and how to apply it to find potential roots of a polynomial. If you have any further questions, please don't hesitate to ask.

Additional Resources

If you want to learn more about the Rational Root Theorem, we recommend the following resources:

• Khan Academy: Rational Root Theorem
• Mathway: Rational Root Theorem
• Wolfram Alpha: Rational Root Theorem

We hope that this article has been helpful in your understanding of the Rational Root Theorem. If you have any further questions, please don't hesitate to ask.