Simplified Rational Expression

Oct 13, 2025 by recoveryshake.com 43 views

Introduction

Rational expressions are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. In this article, we will focus on finding the simplified quotient and the domain of each expression, specifically for the given rational expression $\frac{1}{x^2+9x} \div \left(\frac{6-x}{3x^2-18x}\right)$ . We will break down the problem step by step, using various mathematical techniques to simplify the expression and determine its domain.

Step 1: Simplify the Dividend and Divisor

To simplify the given rational expression, we need to start by simplifying the dividend and divisor separately.

Simplifying the Dividend

The dividend is $\frac{1}{x^2+9x}$ . To simplify this expression, we can factor the denominator as follows:

x^2+9x = x(x+9)

Therefore, the simplified dividend is $\frac{1}{x(x+9)}$ .

Simplifying the Divisor

The divisor is $\left(\frac{6-x}{3x^2-18x}\right)$ . To simplify this expression, we can factor the denominator as follows:

3x^2-18x = 3x(x-6)

Therefore, the simplified divisor is $\left(\frac{6-x}{3x(x-6)}\right)$ .

Step 2: Simplify the Rational Expression

Now that we have simplified the dividend and divisor, we can simplify the rational expression by dividing the dividend by the divisor.

\frac{\frac{1}{x(x+9)}}{\left(\frac{6-x}{3x(x-6)}\right)}

To simplify this expression, we can multiply the numerator and denominator by the reciprocal of the divisor:

\frac{\frac{1}{x(x+9)}}{\left(\frac{6-x}{3x(x-6)}\right)} \cdot \frac{3x(x-6)}{6-x}

This simplifies to:

\frac{3x(x-6)}{6-x} \cdot \frac{1}{x(x+9)}

We can further simplify this expression by canceling out common factors:

\frac{3(x-6)}{6-x} \cdot \frac{1}{x(x+9)}

Step 3: Determine the Domain

To determine the domain of the simplified rational expression, we need to consider the values of $x$ that make the denominator equal to zero.

The denominator is $6-x$ , which equals zero when $x=6$ . Therefore, the domain of the simplified rational expression is all real numbers except $x=6$ .

Conclusion

In this article, we have simplified the rational expression $\frac{1}{x^2+9x} \div \left(\frac{6-x}{3x^2-18x}\right)$ and determined its domain. We have used various mathematical techniques, including factoring and canceling out common factors, to simplify the expression and determine its domain. The simplified rational expression is $\frac{3(x-6)}{6-x} \cdot \frac{1}{x(x+9)}$ , and its domain is all real numbers except $x=6$ .

Final Answer

The simplified quotient is $\frac{3(x-6)}{6-x} \cdot \frac{1}{x(x+9)}$ , and the domain is all real numbers except $x=6$ .

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "College Algebra" by James Stewart

Glossary

Rational Expression: A mathematical expression that can be written as the ratio of two polynomials.
Simplified Rational Expression: A rational expression that has been simplified by canceling out common factors.
Domain: The set of all possible input values for a function or expression.
Frequently Asked Questions (FAQs) about Simplifying Rational Expressions ====================================================================

Introduction

In our previous article, we discussed how to simplify rational expressions and determine their domains. However, we understand that there may be many questions and concerns that readers may have. In this article, we will address some of the most frequently asked questions (FAQs) about simplifying rational expressions.

Q: What is a rational expression?

A: A rational expression is a mathematical expression that can be written as the ratio of two polynomials. It is a fraction that contains variables and/or constants in the numerator and/or denominator.

Q: Why is it important to simplify rational expressions?

A: Simplifying rational expressions is important because it helps to make the expression easier to work with and understand. It also helps to eliminate any unnecessary complexity, making it easier to solve problems and make calculations.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to follow these steps:

Factor the numerator and denominator.
Cancel out any common factors.
Simplify the resulting expression.

Q: What is the difference between a simplified rational expression and a rational expression?

A: A simplified rational expression is a rational expression that has been simplified by canceling out common factors. A rational expression is a mathematical expression that can be written as the ratio of two polynomials.

Q: How do I determine the domain of a rational expression?

A: To determine the domain of a rational expression, you need to consider the values of the variable that make the denominator equal to zero. These values are not included in the domain of the expression.

Q: What is the domain of a rational expression?

A: The domain of a rational expression is the set of all possible input values for the expression. It is the set of all real numbers except those that make the denominator equal to zero.

Q: Can I simplify a rational expression with a variable in the denominator?

A: Yes, you can simplify a rational expression with a variable in the denominator. However, you need to be careful when canceling out common factors to ensure that you do not introduce any extraneous solutions.

Q: How do I simplify a rational expression with a negative exponent?

A: To simplify a rational expression with a negative exponent, you need to rewrite the expression with a positive exponent. This can be done by taking the reciprocal of the expression.

Q: Can I simplify a rational expression with a fraction in the denominator?

A: Yes, you can simplify a rational expression with a fraction in the denominator. However, you need to be careful when canceling out common factors to ensure that you do not introduce any extraneous solutions.

Q: How do I simplify a rational expression with multiple variables?

A: To simplify a rational expression with multiple variables, you need to follow the same steps as before. However, you need to be careful when canceling out common factors to ensure that you do not introduce any extraneous solutions.

Conclusion

In this article, we have addressed some of the most frequently asked questions (FAQs) about simplifying rational expressions. We hope that this article has provided you with a better understanding of how to simplify rational expressions and determine their domains.

Final Answer

Simplifying rational expressions is an important skill that is used in many areas of mathematics, including algebra, geometry, and calculus. By following the steps outlined in this article, you can simplify rational expressions and determine their domains with ease.

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "College Algebra" by James Stewart

Glossary

Rational Expression: A mathematical expression that can be written as the ratio of two polynomials.
Simplified Rational Expression: A rational expression that has been simplified by canceling out common factors.
Domain: The set of all possible input values for a function or expression.