Write each expression with the LCD

Oct 13, 2025 by windailys.com 127 views

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Introduction

Rational expressions are a fundamental concept in algebra, and they play a crucial role in various mathematical operations, including addition and subtraction. In this article, we will explore the process of adding and subtracting rational expressions, with a focus on simplifying complex expressions.

What are Rational Expressions?

Rational expressions are fractions that contain polynomials in the numerator and denominator. They can be written in the form of $\frac{p(x)}{q(x)}$ , where $p(x)$ and $q(x)$ are polynomials. Rational expressions can be added, subtracted, multiplied, and divided, just like regular fractions.

Adding Rational Expressions

To add rational expressions, we need to follow a few simple steps:

Find the least common denominator (LCD): The LCD is the smallest polynomial that can be divided evenly by both denominators.
Write each expression with the LCD: Multiply the numerator and denominator of each expression by the necessary factors to obtain the LCD.
Add the numerators: Combine the numerators of the two expressions.
Simplify the result: Simplify the resulting expression by canceling out any common factors.

Subtracting Rational Expressions

Subtracting rational expressions is similar to adding them, with a few key differences:

Find the LCD: The LCD is the smallest polynomial that can be divided evenly by both denominators.
Write each expression with the LCD: Multiply the numerator and denominator of each expression by the necessary factors to obtain the LCD.
Subtract the numerators: Combine the numerators of the two expressions, but subtract the second numerator from the first.
Simplify the result: Simplify the resulting expression by canceling out any common factors.

Simplifying Complex Expressions

Simplifying complex expressions is a crucial step in adding and subtracting rational expressions. To simplify an expression, we need to:

Factor the numerator and denominator: Factor the numerator and denominator of the expression to identify any common factors.
Cancel out common factors: Cancel out any common factors between the numerator and denominator.
Simplify the result: Simplify the resulting expression by combining any like terms.

Example Problem

Let's consider the following example problem:

Which expression is equivalent to $\frac{\frac{3}{x-2}-5}{2-\frac{4}{x-2}}$ ?

A. $\frac{13-5x}{2x-8}$ B. $\frac{2x-8}{13-5x}$ C. $\frac{-5x-8}{2x-8}$ D. $\frac{-5x-4}{x-2}$

Solution

To solve this problem, we need to simplify the complex expression using the steps outlined above.

Step 1: Simplify the numerator

The numerator of the expression is $\frac{3}{x-2}-5$ . To simplify this expression, we need to find a common denominator and combine the terms.

import sympy as sp
x = sp.symbols('x')
numerator = 3/(x-2) - 5
simplified_numerator = sp.simplify(numerator)
print(simplified_numerator)

Step 2: Simplify the denominator

The denominator of the expression is $2-\frac{4}{x-2}$ . To simplify this expression, we need to find a common denominator and combine the terms.

denominator = 2 - 4/(x-2)
simplified_denominator = sp.simplify(denominator)
print(simplified_denominator)

Step 3: Combine the numerator and denominator

Now that we have simplified the numerator and denominator, we can combine them to obtain the final expression.

expression = simplified_numerator / simplified_denominator
simplified_expression = sp.simplify(expression)
print(simplified_expression)

Conclusion

In this article, we explored the process of adding and subtracting rational expressions, with a focus on simplifying complex expressions. We also provided a step-by-step solution to the example problem, using Python code to simplify the expressions.

The final answer to the example problem is:

$\boxed{\frac{13-5x}{2x-8}}$

This expression is equivalent to the original complex expression, and it can be obtained by simplifying the numerator and denominator using the steps outlined above.

Final Answer

The final answer is $\boxed{\frac{13-5x}{2x-8}}$ .

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Introduction

In our previous article, we explored the process of adding and subtracting rational expressions, with a focus on simplifying complex expressions. In this article, we will provide a Q&A section to help you better understand the concepts and techniques involved.

Q&A

Q: What is the least common denominator (LCD) of two rational expressions?

A: The least common denominator (LCD) of two rational expressions is the smallest polynomial that can be divided evenly by both denominators.

Q: How do I find the LCD of two rational expressions?

A: To find the LCD of two rational expressions, you need to factor the denominators and identify the common factors. Then, multiply the factors together to obtain the LCD.

Q: What is the difference between adding and subtracting rational expressions?

A: The main difference between adding and subtracting rational expressions is the operation performed on the numerators. When adding rational expressions, you combine the numerators by adding them together. When subtracting rational expressions, you combine the numerators by subtracting the second numerator from the first.

Q: How do I simplify a complex rational expression?

A: To simplify a complex rational expression, you need to factor the numerator and denominator, cancel out any common factors, and simplify the resulting expression.

Q: What is the final answer to the example problem?

A: The final answer to the example problem is $\boxed{\frac{13-5x}{2x-8}}$ .

Q: Can I use a calculator to simplify rational expressions?

A: Yes, you can use a calculator to simplify rational expressions. However, it's always a good idea to check your work by hand to ensure that the calculator is giving you the correct answer.

Q: How do I know if a rational expression is equivalent to another rational expression?

A: To determine if a rational expression is equivalent to another rational expression, you need to simplify both expressions and compare them. If the simplified expressions are the same, then the original expressions are equivalent.

Q: Can I add or subtract rational expressions with different denominators?

A: Yes, you can add or subtract rational expressions with different denominators. However, you need to find the least common denominator (LCD) of the two expressions before performing the operation.

Q: How do I handle fractions with negative exponents?

A: To handle fractions with negative exponents, you need to rewrite the fraction as a positive exponent and then simplify the expression.

Q: Can I use rational expressions in real-world applications?

A: Yes, rational expressions are used in a variety of real-world applications, including physics, engineering, and economics.

Conclusion

In this Q&A article, we provided answers to common questions about adding and subtracting rational expressions. We hope that this article has helped you better understand the concepts and techniques involved.

Final Tips

Always simplify rational expressions before performing operations.
Use the least common denominator (LCD) when adding or subtracting rational expressions with different denominators.
Factor the numerator and denominator to simplify complex rational expressions.
Check your work by hand to ensure that the calculator is giving you the correct answer.

Final Answer

The final answer is $\boxed{\frac{13-5x}{2x-8}}$ .