Rational Root Theorem Potential Root Of Function Explained

Let's delve into the Rational Root Theorem and its application in identifying potential rational roots of polynomial functions. The Rational Root Theorem is a crucial tool in algebra that helps us narrow down the possible rational roots of a polynomial equation, making it easier to find the actual roots. To fully understand and solve the given problem, we will first revisit the theorem itself, then apply it to the provided options to determine which function has -7/8 as a potential rational root. Understanding this theorem is essential for students and anyone involved in mathematical problem-solving, as it simplifies the process of finding roots and enhances overall algebraic skills. This article aims to provide a comprehensive explanation and step-by-step solution, ensuring clarity and a strong grasp of the concept.

Understanding the Rational Root Theorem

The Rational Root Theorem states that if a polynomial function with integer coefficients has rational roots, those roots can be expressed in the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. In simpler terms, we are looking for fractions formed by dividing the factors of the last number (constant term) by the factors of the first number (leading coefficient). This theorem drastically reduces the number of potential roots we need to test, saving significant time and effort. Before the advent of computational tools, this theorem was a cornerstone of polynomial root-finding techniques, and it remains a valuable conceptual tool today. By understanding this theorem, students can develop a more intuitive approach to solving polynomial equations, rather than relying solely on rote memorization or computational methods. The theorem bridges the connection between the coefficients of a polynomial and its possible rational roots, making it an elegant and powerful tool in the field of algebra.

Breaking Down the Theorem

To further illustrate, let's consider a general polynomial equation:

$\qquad a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 = 0$

Here, $a_n$ is the leading coefficient, and $a_0$ is the constant term. According to the Rational Root Theorem, any rational root p/q of this polynomial must satisfy these conditions:

1. p is an integer factor of $a_0$ (the constant term).
2. q is an integer factor of $a_n$ (the leading coefficient).

This means we first identify all possible factors (both positive and negative) of the constant term and the leading coefficient. Then, we form all possible fractions by dividing each factor of the constant term by each factor of the leading coefficient. These fractions are the potential rational roots of the polynomial. It is important to note that the theorem provides a list of potential roots, not actual roots. Each potential root must still be tested using methods such as synthetic division or direct substitution to determine if it is indeed a root of the polynomial. Despite this, the Rational Root Theorem greatly streamlines the root-finding process by providing a manageable set of candidates to test.

Applying the Theorem: An Example

Consider the polynomial $f(x) = 2x^2 + 3x - 2$. Here, the leading coefficient is 2, and the constant term is -2. The factors of -2 are ±1 and ±2. The factors of 2 are ±1 and ±2. Therefore, the possible rational roots are:

$\qquad \pm\frac{1}{1}, \pm\frac{1}{2}, \pm\frac{2}{1}, \pm\frac{2}{2}$

Simplifying these, we get: ±1, ±1/2, ±2. These are the potential rational roots we need to test. By substituting these values into the polynomial or using synthetic division, we can find the actual roots. For instance, testing x = 1/2, we find that it is indeed a root. This example underscores how the Rational Root Theorem provides a systematic approach to finding roots, reducing the guesswork involved.

Solving the Problem

Now, let's apply the Rational Root Theorem to the problem at hand. We are given that $-\frac{7}{8}$ is a potential rational root of a function. This means that -7 must be a factor of the constant term, and 8 must be a factor of the leading coefficient. We need to examine the given options and identify the function that satisfies this condition.

Analyzing the Options

We are given four options:

A. $f(x)=24 x^7+3 x^6+4 x^3-x-28$ B. $f(x)=28 x^7+3 x^6+4 x^3-x-24$ C. $f(x)=30 x^7+3 x^6+4 x^3-x-56$ D. $f(x)=56 x^7+3 x^6+4 x^3-x-30$

For $-\frac{7}{8}$ to be a potential rational root:

• -7 must be a factor of the constant term.
• 8 must be a factor of the leading coefficient.

Let's analyze each option:

Option A

$f(x)=24 x^7+3 x^6+4 x^3-x-28$

• Leading coefficient: 24
• Constant term: -28

Factors of -28 include ±1, ±2, ±4, ±7, ±14, ±28. -7 is indeed a factor of -28.

Factors of 24 include ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24. 8 is a factor of 24.

Therefore, $-\frac{7}{8}$ is a potential rational root for this function.

Option B

$f(x)=28 x^7+3 x^6+4 x^3-x-24$

• Leading coefficient: 28
• Constant term: -24

Factors of -24 include ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24. -7 is not a factor of -24.

Factors of 28 include ±1, ±2, ±4, ±7, ±14, ±28. 8 is not a factor of 28.

Therefore, $-\frac{7}{8}$ is not a potential rational root for this function.

Option C

$f(x)=30 x^7+3 x^6+4 x^3-x-56$

• Leading coefficient: 30
• Constant term: -56

Factors of -56 include ±1, ±2, ±4, ±7, ±8, ±14, ±28, ±56. -7 is a factor of -56.

Factors of 30 include ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30. 8 is not a factor of 30.

Therefore, $-\frac{7}{8}$ is not a potential rational root for this function.

Option D

$f(x)=56 x^7+3 x^6+4 x^3-x-30$

• Leading coefficient: 56
• Constant term: -30

Factors of -30 include ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30. -7 is not a factor of -30.

Factors of 56 include ±1, ±2, ±4, ±7, ±8, ±14, ±28, ±56. 8 is a factor of 56.

Therefore, $-\frac{7}{8}$ is not a potential rational root for this function.

Conclusion

After analyzing all the options, we find that only Option A satisfies the conditions of the Rational Root Theorem for $-\frac{7}{8}$ to be a potential rational root. The constant term (-28) has -7 as a factor, and the leading coefficient (24) has 8 as a factor. This comprehensive analysis showcases the practical application of the theorem in narrowing down possible rational roots of polynomial functions. The ability to efficiently identify potential roots is a valuable skill in algebra and higher mathematics, enabling a more streamlined approach to solving complex equations.

Final Answer

Therefore, the correct answer is:

A. $f(x)=24 x^7+3 x^6+4 x^3-x-28$

This step-by-step explanation should provide a clear understanding of how to apply the Rational Root Theorem to solve this type of problem. The systematic approach of identifying factors and checking conditions makes the process more manageable and less prone to errors. Understanding and applying such mathematical theorems and techniques forms a solid foundation for further studies in algebra and beyond.