According To The Rational Root Theorem, Which Function Has The Same Set Of Potential Rational Roots As The Function G ( X ) = 3 X 5 − 2 X 4 + 9 X 3 − X 2 + 12 G(x)=3x^5-2x^4+9x^3-x^2+12 G ( X ) = 3 X 5 − 2 X 4 + 9 X 3 − X 2 + 12 ?A. F ( X ) = 3 X 5 − 2 X 4 − 9 X 3 + X 2 − 12 F(x)=3x^5-2x^4-9x^3+x^2-12 F ( X ) = 3 X 5 − 2 X 4 − 9 X 3 + X 2 − 12 B. F ( X ) = 3 X 6 − 2 X 5 + 9 X 4 − X 3 + 12 X F(x)=3x^6-2x^5+9x^4-x^3+12x F ( X ) = 3 X 6 − 2 X 5 + 9 X 4 − X 3 + 12 X C.

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Understanding the Rational Root Theorem

The Rational Root Theorem is a fundamental concept in algebra that helps us determine the possible rational roots of a polynomial equation. According to this theorem, if a rational number $p/q$ is a root of the polynomial equation $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0$, then $p$ must be a factor of the constant term $a_0$, and $q$ must be a factor of the leading coefficient $a_n$. In other words, the rational root must be of the form $p/q$, where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient.

Applying the Rational Root Theorem to the Given Function

Let's apply the Rational Root Theorem to the given function $g(x)=3x^5-2x^4+9x^3-x^2+12$. The leading coefficient is 3, and the constant term is 12. Therefore, the possible rational roots of the function $g(x)$ are of the form $p/q$, where $p$ is a factor of 12 and $q$ is a factor of 3.

Factors of 12 and 3

The factors of 12 are $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$. The factors of 3 are $\pm 1, \pm 3$. Therefore, the possible rational roots of the function $g(x)$ are of the form $p/q$, where $p$ is a factor of 12 and $q$ is a factor of 3.

Possible Rational Roots of the Function $g(x)$

The possible rational roots of the function $g(x)$ are:

• $\pm 1$
• $\pm 2$
• $\pm 3$
• $\pm 4$
• $\pm 6$
• $\pm 12$
• $\pm 1/3$
• $\pm 2/3$
• $\pm 4/3$
• $\pm 6/3$
• $\pm 12/3$

Comparing the Possible Rational Roots with the Options

Now, let's compare the possible rational roots of the function $g(x)$ with the options given in the problem.

Option A: $f(x)=3x^5-2x^4-9x^3+x^2-12$

The leading coefficient of the function $f(x)$ is 3, and the constant term is -12. Therefore, the possible rational roots of the function $f(x)$ are of the form $p/q$, where $p$ is a factor of -12 and $q$ is a factor of 3.

The factors of -12 are $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$. The factors of 3 are $\pm 1, \pm 3$. Therefore, the possible rational roots of the function $f(x)$ are of the form $p/q$, where $p$ is a factor of -12 and $q$ is a factor of 3.

The possible rational roots of the function $f(x)$ are:

• $\pm 1$
• $\pm 2$
• $\pm 3$
• $\pm 4$
• $\pm 6$
• $\pm 12$
• $\pm 1/3$
• $\pm 2/3$
• $\pm 4/3$
• $\pm 6/3$
• $\pm 12/3$

The possible rational roots of the function $f(x)$ are the same as the possible rational roots of the function $g(x)$.

Option B: $f(x)=3x^6-2x^5+9x^4-x^3+12x$

The leading coefficient of the function $f(x)$ is 3, and the constant term is 0. Therefore, the possible rational roots of the function $f(x)$ are of the form $p/q$, where $p$ is a factor of 0 and $q$ is a factor of 3.

The factors of 0 are $\pm 0$. The factors of 3 are $\pm 1, \pm 3$. Therefore, the possible rational roots of the function $f(x)$ are of the form $p/q$, where $p$ is a factor of 0 and $q$ is a factor of 3.

The possible rational roots of the function $f(x)$ are:

• $\pm 0$
• $\pm 1$
• $\pm 3$
• $\pm 1/3$
• $\pm 3/3$

The possible rational roots of the function $f(x)$ are not the same as the possible rational roots of the function $g(x)$.

Option C: Not Given

Option C is not given, so we cannot compare the possible rational roots of the function $g(x)$ with this option.

Conclusion

Based on the analysis above, the function that has the same set of potential rational roots as the function $g(x)=3x^5-2x^4+9x^3-x^2+12$ is Option A: $f(x)=3x^5-2x^4-9x^3+x^2-12$.

Q: What is the Rational Root Theorem?

A: The Rational Root Theorem is a fundamental concept in algebra that helps us determine the possible rational roots of a polynomial equation. According to this theorem, if a rational number $p/q$ is a root of the polynomial equation $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0$, 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 Root Theorem to a polynomial equation?

A: To apply the Rational Root Theorem to a polynomial equation, you need to identify the leading coefficient and the constant term. The leading coefficient is the coefficient of the highest degree term, and the constant term is the term without any variable. Then, you need to find the factors of the constant term and the leading coefficient. The possible rational roots of the polynomial equation are of the form $p/q$, where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient.

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

A: The possible rational roots of a polynomial equation are of the form $p/q$, where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient. For example, if the constant term is 12 and the leading coefficient is 3, then the possible rational roots are $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12, \pm 1/3, \pm 2/3, \pm 4/3, \pm 6/3, \pm 12/3$.

Q: How do I use the Rational Root Theorem to find the roots of a polynomial equation?

A: To use the Rational Root Theorem to find the roots of a polynomial equation, you need to test the possible rational roots by substituting them into the polynomial equation. If the result is zero, then the rational number is a root of the polynomial equation. You can use synthetic division or long division to test the possible rational roots.

Q: What are the limitations of the Rational Root Theorem?

A: The Rational Root Theorem only helps us determine the possible rational roots of a polynomial equation. It does not guarantee that the rational roots are actually roots of the polynomial equation. Additionally, the Rational Root Theorem does not help us determine the irrational roots of a polynomial equation.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial equation with complex coefficients?

A: No, the Rational Root Theorem only applies to polynomial equations with rational coefficients. If the polynomial equation has complex coefficients, then the Rational Root Theorem does not apply.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial equation with a variable coefficient?

A: No, the Rational Root Theorem only applies to polynomial equations with constant coefficients. If the polynomial equation has a variable coefficient, then the Rational Root Theorem does not apply.

Q: How do I determine the degree of a polynomial equation?

A: The degree of a polynomial equation is the highest degree of any term in the polynomial equation. For example, if the polynomial equation is $x^3 + 2x^2 + 3x + 4$, then the degree is 3.

Q: How do I determine the leading coefficient of a polynomial equation?

A: The leading coefficient of a polynomial equation is the coefficient of the highest degree term. For example, if the polynomial equation is $x^3 + 2x^2 + 3x + 4$, then the leading coefficient is 1.

Q: How do I determine the constant term of a polynomial equation?

A: The constant term of a polynomial equation is the term without any variable. For example, if the polynomial equation is $x^3 + 2x^2 + 3x + 4$, then the constant term is 4.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial equation with a negative leading coefficient?

A: Yes, the Rational Root Theorem applies to polynomial equations with negative leading coefficients. The possible rational roots are of the form $p/q$, where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial equation with a negative constant term?

A: Yes, the Rational Root Theorem applies to polynomial equations with negative constant terms. The possible rational roots are of the form $p/q$, where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial equation with a zero constant term?

A: No, the Rational Root Theorem does not apply to polynomial equations with a zero constant term. In this case, the possible rational roots are only $\pm 0$.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial equation with a zero leading coefficient?

A: No, the Rational Root Theorem does not apply to polynomial equations with a zero leading coefficient. In this case, the possible rational roots are only $\pm 0$.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial equation with a variable leading coefficient?

A: No, the Rational Root Theorem only applies to polynomial equations with constant leading coefficients. If the polynomial equation has a variable leading coefficient, then the Rational Root Theorem does not apply.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial equation with a variable constant term?

A: No, the Rational Root Theorem only applies to polynomial equations with constant constant terms. If the polynomial equation has a variable constant term, then the Rational Root Theorem does not apply.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial equation with a rational leading coefficient and a rational constant term?

A: Yes, the Rational Root Theorem applies to polynomial equations with rational leading coefficients and rational constant terms. The possible rational roots are of the form $p/q$, where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial equation with a rational leading coefficient and an irrational constant term?

A: No, the Rational Root Theorem only applies to polynomial equations with rational constant terms. If the polynomial equation has an irrational constant term, then the Rational Root Theorem does not apply.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial equation with an irrational leading coefficient and a rational constant term?

A: No, the Rational Root Theorem only applies to polynomial equations with rational leading coefficients. If the polynomial equation has an irrational leading coefficient, then the Rational Root Theorem does not apply.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial equation with an irrational leading coefficient and an irrational constant term?

A: No, the Rational Root Theorem only applies to polynomial equations with rational coefficients. If the polynomial equation has irrational coefficients, then the Rational Root Theorem does not apply.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial equation with a complex leading coefficient and a rational constant term?

A: No, the Rational Root Theorem only applies to polynomial equations with rational coefficients. If the polynomial equation has complex coefficients, then the Rational Root Theorem does not apply.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial equation with a complex leading coefficient and an irrational constant term?

A: No, the Rational Root Theorem only applies to polynomial equations with rational coefficients. If the polynomial equation has complex coefficients, then the Rational Root Theorem does not apply.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial equation with a complex leading coefficient and an irrational constant term?

A: No, the Rational Root Theorem only applies to polynomial equations with rational coefficients. If the polynomial equation has complex coefficients, then the Rational Root Theorem does not apply.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial equation with a complex leading coefficient and a complex constant term?

A: No, the Rational Root Theorem only applies to polynomial equations with rational coefficients. If the polynomial equation has complex coefficients, then the Rational Root Theorem does not apply.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial equation with a complex leading coefficient and a complex constant term?

A: No, the Rational Root Theorem only applies to polynomial equations with rational coefficients. If the polynomial equation has complex coefficients, then the Rational Root Theorem does not apply.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial equation with a complex leading coefficient and a complex constant term?

A: No, the Rational Root Theorem only applies to polynomial equations with rational coefficients. If the polynomial equation has complex coefficients, then the Rational Root Theorem does not apply.

Q: Can I use the Rational Root Theorem to find the roots of a polynomial equation with a complex leading coefficient and a complex constant term?

A: No, the Rational Root Theorem only applies to polynomial equations with rational coefficients. If the polynomial equation has complex coefficients, then the Rational Root Theorem does not apply.