Match Each Expression To Its Rewritten Form.1. { \frac X^2+4x-7}{x-1}$}$ - Rewritten As { (x+5) + \frac{-2 X-1}$}$2. { \frac{2x^2-3x+7}{x-1}$}$ - Rewritten As { (2x+1) + \frac{-6 {x-1}$}$3.
Introduction
Rational expressions are a fundamental concept in algebra, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying rational expressions, focusing on the concept of factoring and rewriting expressions in a more manageable form. We will examine three specific examples, each with its own unique challenges and opportunities for simplification.
What are Rational Expressions?
A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. Rational expressions can be simplified by factoring the numerator and/or denominator, and then canceling out any common factors.
Example 1: Simplifying a Rational Expression with a Linear Factor
Let's consider the rational expression {\frac{x^2+4x-7}{x-1}$}$. Our goal is to simplify this expression by rewriting it in a more manageable form.
Step 1: Factor the Numerator
To simplify the rational expression, we need to factor the numerator. We can start by looking for two numbers whose product is -7 and whose sum is 4. These numbers are 7 and -1, so we can write the numerator as {(x+7)(x-1)$}$.
Step 2: Rewrite the Expression
Now that we have factored the numerator, we can rewrite the expression as {\frac{(x+7)(x-1)}{x-1}$}$. We can see that the factor {(x-1)$}$ appears in both the numerator and the denominator, so we can cancel it out.
Step 3: Simplify the Expression
After canceling out the common factor, we are left with {x+7$}$. This is the simplified form of the original rational expression.
Example 2: Simplifying a Rational Expression with a Quadratic Factor
Let's consider the rational expression {\frac{2x^2-3x+7}{x-1}$}$. Our goal is to simplify this expression by rewriting it in a more manageable form.
Step 1: Factor the Numerator
To simplify the rational expression, we need to factor the numerator. We can start by looking for two numbers whose product is 2 and whose sum is -3. These numbers are -2 and -1, so we can write the numerator as {(2x-2)(x-1)$}$.
Step 2: Rewrite the Expression
Now that we have factored the numerator, we can rewrite the expression as {\frac{(2x-2)(x-1)}{x-1}$}$. We can see that the factor {(x-1)$}$ appears in both the numerator and the denominator, so we can cancel it out.
Step 3: Simplify the Expression
After canceling out the common factor, we are left with ${2x-2\$}. This is the simplified form of the original rational expression.
Example 3: Simplifying a Rational Expression with a Constant Factor
Let's consider the rational expression {\frac{x^2+4x-7}{x-1}$}$. Our goal is to simplify this expression by rewriting it in a more manageable form.
Step 1: Factor the Numerator
To simplify the rational expression, we need to factor the numerator. We can start by looking for two numbers whose product is -7 and whose sum is 4. These numbers are 7 and -1, so we can write the numerator as {(x+7)(x-1)$}$.
Step 2: Rewrite the Expression
Now that we have factored the numerator, we can rewrite the expression as {\frac{(x+7)(x-1)}{x-1}$}$. We can see that the factor {(x-1)$}$ appears in both the numerator and the denominator, so we can cancel it out.
Step 3: Simplify the Expression
After canceling out the common factor, we are left with {x+7$}$. This is the simplified form of the original rational expression.
Conclusion
Simplifying rational expressions is a crucial skill in algebra, and it requires a deep understanding of factoring and rewriting expressions. By following the steps outlined in this article, you can simplify even the most complex rational expressions and gain a deeper understanding of the underlying mathematics.
Rewritten Forms
Here are the rewritten forms of the three examples:
- {\fracx^2+4x-7}{x-1}$}$ - Rewritten as{x-1}$}$
- {\frac2x^2-3x+7}{x-1}$}$ - Rewritten as{x-1}$}$
- {\fracx^2+4x-7}{x-1}$}$ - Rewritten as{x-1}$}$
Discussion
Simplifying rational expressions is a fundamental concept in algebra, and it has numerous applications in mathematics and science. By mastering the skills outlined in this article, you can simplify even the most complex rational expressions and gain a deeper understanding of the underlying mathematics.
Additional Resources
For further practice and review, we recommend the following resources:
- Khan Academy: Rational Expressions
- Mathway: Simplifying Rational Expressions
- Wolfram Alpha: Rational Expressions
Final Thoughts
Introduction
Simplifying rational expressions is a fundamental concept in algebra, and it can be a challenging task for many students. In this article, we will provide a Q&A guide to help you understand the process of simplifying rational expressions and address some common questions and concerns.
Q: What is a rational expression?
A: A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator.
Q: How do I simplify a rational expression?
A: To simplify a rational expression, you need to factor the numerator and/or denominator, and then cancel out any common factors.
Q: What is factoring?
A: Factoring is the process of expressing a polynomial as a product of simpler polynomials.
Q: How do I factor a polynomial?
A: To factor a polynomial, you need to look for two numbers whose product is the constant term and whose sum is the coefficient of the linear term.
Q: What is a common factor?
A: A common factor is a factor that appears in both the numerator and the denominator of a rational expression.
Q: How do I cancel out common factors?
A: To cancel out common factors, you need to divide both the numerator and the denominator by the common factor.
Q: What are some common mistakes to avoid when simplifying rational expressions?
A: Some common mistakes to avoid when simplifying rational expressions include:
- Not factoring the numerator and/or denominator
- Not canceling out common factors
- Not simplifying the expression completely
Q: How do I know when to simplify a rational expression?
A: You should simplify a rational expression when:
- The numerator and/or denominator can be factored
- There are common factors that can be canceled out
- The expression can be simplified to a more manageable form
Q: What are some real-world applications of simplifying rational expressions?
A: Simplifying rational expressions has numerous real-world applications, including:
- Calculating probabilities and statistics
- Modeling population growth and decay
- Analyzing financial data and making investment decisions
Q: How can I practice simplifying rational expressions?
A: You can practice simplifying rational expressions by:
- Working through example problems
- Using online resources and tools
- Practicing with real-world applications
Q: What are some common types of rational expressions?
A: Some common types of rational expressions include:
- Linear rational expressions
- Quadratic rational expressions
- Polynomial rational expressions
Q: How do I simplify a linear rational expression?
A: To simplify a linear rational expression, you need to factor the numerator and/or denominator, and then cancel out any common factors.
Q: How do I simplify a quadratic rational expression?
A: To simplify a quadratic rational expression, you need to factor the numerator and/or denominator, and then cancel out any common factors.
Q: How do I simplify a polynomial rational expression?
A: To simplify a polynomial rational expression, you need to factor the numerator and/or denominator, and then cancel out any common factors.
Conclusion
Simplifying rational expressions is a fundamental concept in algebra, and it requires a deep understanding of factoring and rewriting expressions. By following the steps outlined in this article and practicing with real-world applications, you can simplify even the most complex rational expressions and gain a deeper understanding of the underlying mathematics.
Additional Resources
For further practice and review, we recommend the following resources:
- Khan Academy: Rational Expressions
- Mathway: Simplifying Rational Expressions
- Wolfram Alpha: Rational Expressions
Final Thoughts
Simplifying rational expressions is a crucial skill in algebra, and it requires a deep understanding of factoring and rewriting expressions. By mastering the skills outlined in this article, you can simplify even the most complex rational expressions and gain a deeper understanding of the underlying mathematics.