Expressing Rational Functions in a Specific Form

Oct 13, 2025 by windailys.com 72 views

Introduction

Rational functions are a fundamental concept in mathematics, and expressing them in a specific form can be a crucial step in solving various mathematical problems. In this article, we will focus on reformulating the given rational function $g(x) = \frac{x-9}{x}$ in the form $g(x) = c + \frac{r}{x}$ , where $c$ and $r$ are constants. We will also address a discussion category in mathematics.

Expressing Rational Functions in a Specific Form

To express the rational function $g(x) = \frac{x-9}{x}$ in the form $g(x) = c + \frac{r}{x}$ , we need to follow a series of steps. The first step is to perform polynomial long division or synthetic division to divide the numerator by the denominator.

Step 1: Perform Polynomial Long Division

To perform polynomial long division, we divide the numerator $x-9$ by the denominator $x$ . The result of this division will be a quotient and a remainder.

import sympy as sp

# Define the variables
x = sp.symbols('x')

# Define the numerator and denominator
numerator = x - 9
denominator = x

# Perform polynomial long division
quotient, remainder = sp.div(numerator, denominator)

print("Quotient:", quotient)
print("Remainder:", remainder)

The result of this division is a quotient of $1$ and a remainder of $-9$ .

Step 2: Express the Rational Function in the Desired Form

Now that we have the quotient and remainder, we can express the rational function $g(x) = \frac{x-9}{x}$ in the form $g(x) = c + \frac{r}{x}$ . The constant $c$ is equal to the quotient, which is $1$ . The constant $r$ is equal to the remainder, which is $-9$ .

# Define the constants
c = 1
r = -9

# Express the rational function in the desired form
g_x = c + r / x

print("g(x) =", g_x)

The final expression for the rational function $g(x)$ is $g(x) = 1 - \frac{9}{x}$ .

Conclusion

In this article, we reformulated the given rational function $g(x) = \frac{x-9}{x}$ in the form $g(x) = c + \frac{r}{x}$ , where $c$ and $r$ are constants. We performed polynomial long division to divide the numerator by the denominator and obtained a quotient and a remainder. We then expressed the rational function in the desired form using the quotient and remainder. The final expression for the rational function $g(x)$ is $g(x) = 1 - \frac{9}{x}$ .

Discussion Category: Mathematics

The discussion category in mathematics is a crucial aspect of mathematical education. It provides a platform for students to discuss and share their ideas, thoughts, and experiences related to mathematics. The discussion category can be used to:

Discuss mathematical concepts and theories
Share mathematical problems and solutions
Collaborate on mathematical projects and assignments
Provide feedback and support to peers

The discussion category can be used in various ways, including:

Online forums and discussion boards
In-person discussions and group work
Video conferencing and online meetings
Social media groups and online communities

Given: c = 1

The given value of $c$ is $1$ . This value is used to express the rational function $g(x)$ in the form $g(x) = c + \frac{r}{x}$ . The constant $c$ is equal to the quotient obtained from the polynomial long division, which is $1$ .

Example Use Cases

The rational function $g(x) = 1 - \frac{9}{x}$ can be used in various mathematical applications, including:

Calculus: The rational function can be used to find the derivative and integral of the function.
Algebra: The rational function can be used to solve equations and inequalities involving the function.
Geometry: The rational function can be used to find the area and perimeter of shapes involving the function.

Conclusion

$In conclusion, we reformulated the given rational function $g(x) = \frac{x-9}{x}$ in the form $g(x) = c + \frac{r}{x}$, where $c$ and $r$ are constants. We performed polynomial long division to divide the numerator by the denominator and obtained a quotient and a remainder. We then expressed the rational function in the desired form using the quotient and remainder. The final expression for the rational function $g(x)$ is $g(x) = 1 - \frac{9}{x}$.<br/> **Q&A: Rational Functions and Polynomial Long Division**$

Introduction

In our previous article, we reformulated the given rational function $g(x) = \frac{x-9}{x}$ in the form $g(x) = c + \frac{r}{x}$ , where $c$ and $r$ are constants. We performed polynomial long division to divide the numerator by the denominator and obtained a quotient and a remainder. In this article, we will answer some frequently asked questions related to rational functions and polynomial long division.

Q: What is polynomial long division?

A: Polynomial long division is a mathematical process used to divide a polynomial by another polynomial. It is similar to long division in arithmetic, but it is used for polynomials instead of numbers.

Q: How do I perform polynomial long division?

A: To perform polynomial long division, you need to follow these steps:

Divide the leading term of the numerator by the leading term of the denominator.
Multiply the entire denominator by the result from step 1.
Subtract the result from step 2 from the numerator.
Repeat steps 1-3 until the degree of the remainder is less than the degree of the denominator.

Q: What is the quotient and remainder in polynomial long division?

A: The quotient is the result of the division, and the remainder is the amount left over after the division. The quotient is a polynomial, and the remainder is a polynomial or a constant.

Q: How do I express a rational function in the form $g(x) = c + \frac{r}{x}$ ?

A: To express a rational function in the form $g(x) = c + \frac{r}{x}$ , you need to perform polynomial long division to divide the numerator by the denominator. The constant $c$ is equal to the quotient, and the constant $r$ is equal to the remainder.

Q: What are some common applications of rational functions?

A: Rational functions have many applications in mathematics, including:

Calculus: Rational functions can be used to find the derivative and integral of a function.
Algebra: Rational functions can be used to solve equations and inequalities involving the function.
Geometry: Rational functions can be used to find the area and perimeter of shapes involving the function.

Q: How do I use polynomial long division to solve equations and inequalities?

A: To use polynomial long division to solve equations and inequalities, you need to follow these steps:

Divide the numerator by the denominator to obtain a quotient and a remainder.
Set the remainder equal to zero and solve for the variable.
Use the solution to the remainder to solve the original equation or inequality.

Q: What are some common mistakes to avoid when performing polynomial long division?

A: Some common mistakes to avoid when performing polynomial long division include:

Not following the order of operations.
Not multiplying the entire denominator by the result from step 1.
Not subtracting the result from step 2 from the numerator.
Not repeating steps 1-3 until the degree of the remainder is less than the degree of the denominator.

Conclusion

In conclusion, we answered some frequently asked questions related to rational functions and polynomial long division. We discussed the process of polynomial long division, the quotient and remainder, and how to express a rational function in the form $g(x) = c + \frac{r}{x}$ . We also discussed some common applications of rational functions and how to use polynomial long division to solve equations and inequalities.