Simplify The Expression:${ Y = \frac{x^2 + 4x - 5}{x^2 + 8x + 15} }$ (2 Points)

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Introduction

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In algebra, simplifying expressions is a crucial skill that helps us solve equations and inequalities more efficiently. It involves manipulating the given expression to make it easier to work with, often by factoring, canceling out common factors, or combining like terms. In this article, we will focus on simplifying the given expression: $y = \frac{x^2 + 4x - 5}{x^2 + 8x + 15}$. We will break down the steps involved in simplifying this expression and provide a clear explanation of each step.

Step 1: Factor the Numerator and Denominator

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The first step in simplifying the expression is to factor the numerator and denominator. Factoring involves expressing a polynomial as a product of simpler polynomials. In this case, we can factor the numerator and denominator as follows:

• Numerator: $x^2 + 4x - 5 = (x + 5)(x - 1)$
• Denominator: $x^2 + 8x + 15 = (x + 5)(x + 3)$

Step 2: Cancel Out Common Factors

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Now that we have factored the numerator and denominator, we can cancel out any common factors. In this case, we can cancel out the common factor $(x + 5)$ from the numerator and denominator:

$$y = \frac{(x + 5)(x - 1)}{(x + 5)(x + 3)}$$

Step 3: Simplify the Expression

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After canceling out the common factor, we are left with:

$$y = \frac{x - 1}{x + 3}$$

This is the simplified expression.

Example

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Let's consider an example to illustrate the concept of simplifying expressions. Suppose we have the expression:

$$y = \frac{x^2 + 6x + 8}{x^2 + 10x + 21}$$

We can simplify this expression by factoring the numerator and denominator:

• Numerator: $x^2 + 6x + 8 = (x + 4)(x + 2)$
• Denominator: $x^2 + 10x + 21 = (x + 7)(x + 3)$

We can then cancel out the common factor $(x + 3)$ from the numerator and denominator:

$$y = \frac{(x + 4)(x + 2)}{(x + 7)(x + 3)}$$

After canceling out the common factor, we are left with:

$$y = \frac{x + 4}{x + 7}$$

This is the simplified expression.

Importance of Simplifying Expressions

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Simplifying expressions is an essential skill in algebra that helps us solve equations and inequalities more efficiently. By simplifying expressions, we can:

• Make it easier to work with complex expressions
• Identify patterns and relationships between variables
• Solve equations and inequalities more efficiently
• Understand the behavior of functions and their graphs

Conclusion

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In conclusion, simplifying expressions is a crucial skill in algebra that helps us solve equations and inequalities more efficiently. By factoring, canceling out common factors, and combining like terms, we can simplify complex expressions and make them easier to work with. In this article, we have focused on simplifying the expression $y = \frac{x^2 + 4x - 5}{x^2 + 8x + 15}$ and provided a step-by-step guide to algebraic manipulation. We have also discussed the importance of simplifying expressions and provided an example to illustrate the concept.

Frequently Asked Questions

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Q: What is the purpose of simplifying expressions?

A: The purpose of simplifying expressions is to make it easier to work with complex expressions, identify patterns and relationships between variables, solve equations and inequalities more efficiently, and understand the behavior of functions and their graphs.

Q: How do I simplify an expression?

A: To simplify an expression, you can factor the numerator and denominator, cancel out common factors, and combine like terms.

Q: What are some common techniques for simplifying expressions?

A: Some common techniques for simplifying expressions include factoring, canceling out common factors, and combining like terms.

Q: Why is simplifying expressions important?

A: Simplifying expressions is important because it helps us solve equations and inequalities more efficiently, identify patterns and relationships between variables, and understand the behavior of functions and their graphs.

Final Thoughts

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Simplifying expressions is a crucial skill in algebra that helps us solve equations and inequalities more efficiently. By factoring, canceling out common factors, and combining like terms, we can simplify complex expressions and make them easier to work with. In this article, we have focused on simplifying the expression $y = \frac{x^2 + 4x - 5}{x^2 + 8x + 15}$ and provided a step-by-step guide to algebraic manipulation. We have also discussed the importance of simplifying expressions and provided an example to illustrate the concept.

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Introduction

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In our previous article, we discussed the importance of simplifying expressions in algebra and provided a step-by-step guide to algebraic manipulation. In this article, we will continue to explore the topic of simplifying expressions and answer some frequently asked questions.

Q&A: Simplifying Expressions

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Q: What is the purpose of simplifying expressions?

A: The purpose of simplifying expressions is to make it easier to work with complex expressions, identify patterns and relationships between variables, solve equations and inequalities more efficiently, and understand the behavior of functions and their graphs.

Q: How do I simplify an expression?

A: To simplify an expression, you can factor the numerator and denominator, cancel out common factors, and combine like terms.

Q: What are some common techniques for simplifying expressions?

A: Some common techniques for simplifying expressions include factoring, canceling out common factors, and combining like terms.

Q: Why is simplifying expressions important?

A: Simplifying expressions is important because it helps us solve equations and inequalities more efficiently, identify patterns and relationships between variables, and understand the behavior of functions and their graphs.

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

A: Yes, you can simplify an expression with a variable in the denominator. However, you must be careful not to divide by zero.

Q: How do I handle expressions with multiple variables?

A: To handle expressions with multiple variables, you can use the same techniques as before, such as factoring, canceling out common factors, and combining like terms.

Q: Can I simplify an expression with a fraction in the numerator or denominator?

A: Yes, you can simplify an expression with a fraction in the numerator or denominator. However, you must be careful to handle the fractions correctly.

Q: How do I know when an expression is simplified?

A: An expression is simplified when it cannot be simplified further using the techniques of factoring, canceling out common factors, and combining like terms.

Example Questions

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Q: Simplify the expression: $y = \frac{x^2 + 6x + 8}{x^2 + 10x + 21}$

A: To simplify this expression, we can factor the numerator and denominator:

• Numerator: $x^2 + 6x + 8 = (x + 4)(x + 2)$
• Denominator: $x^2 + 10x + 21 = (x + 7)(x + 3)$

We can then cancel out the common factor $(x + 3)$ from the numerator and denominator:

$$y = \frac{(x + 4)(x + 2)}{(x + 7)(x + 3)}$$

After canceling out the common factor, we are left with:

$$y = \frac{x + 4}{x + 7}$$

Q: Simplify the expression: $y = \frac{x^2 - 4x + 4}{x^2 - 6x + 9}$

A: To simplify this expression, we can factor the numerator and denominator:

• Numerator: $x^2 - 4x + 4 = (x - 2)^2$
• Denominator: $x^2 - 6x + 9 = (x - 3)^2$

We can then cancel out the common factor $(x - 3)^2$ from the numerator and denominator:

$$y = \frac{(x - 2)^2}{(x - 3)^2}$$

After canceling out the common factor, we are left with:

$$y = \frac{x - 2}{x - 3}$$

Conclusion

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In conclusion, simplifying expressions is a crucial skill in algebra that helps us solve equations and inequalities more efficiently. By factoring, canceling out common factors, and combining like terms, we can simplify complex expressions and make them easier to work with. In this article, we have answered some frequently asked questions and provided examples to illustrate the concept.

Final Thoughts

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Simplifying expressions is a fundamental concept in algebra that helps us solve equations and inequalities more efficiently. By mastering the techniques of factoring, canceling out common factors, and combining like terms, we can simplify complex expressions and make them easier to work with. Whether you are a student or a teacher, understanding how to simplify expressions is essential for success in algebra.

Additional Resources

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For more information on simplifying expressions, check out the following resources:

• Khan Academy: Simplifying Expressions
• Mathway: Simplifying Expressions
• Algebra.com: Simplifying Expressions

Frequently Asked Questions

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Q: What is the purpose of simplifying expressions?

A: The purpose of simplifying expressions is to make it easier to work with complex expressions, identify patterns and relationships between variables, solve equations and inequalities more efficiently, and understand the behavior of functions and their graphs.

Q: How do I simplify an expression?

A: To simplify an expression, you can factor the numerator and denominator, cancel out common factors, and combine like terms.

Q: What are some common techniques for simplifying expressions?

A: Some common techniques for simplifying expressions include factoring, canceling out common factors, and combining like terms.

Q: Why is simplifying expressions important?

A: Simplifying expressions is important because it helps us solve equations and inequalities more efficiently, identify patterns and relationships between variables, and understand the behavior of functions and their graphs.

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

A: Yes, you can simplify an expression with a variable in the denominator. However, you must be careful not to divide by zero.

Q: How do I handle expressions with multiple variables?

A: To handle expressions with multiple variables, you can use the same techniques as before, such as factoring, canceling out common factors, and combining like terms.

Q: Can I simplify an expression with a fraction in the numerator or denominator?

A: Yes, you can simplify an expression with a fraction in the numerator or denominator. However, you must be careful to handle the fractions correctly.

Q: How do I know when an expression is simplified?

A: An expression is simplified when it cannot be simplified further using the techniques of factoring, canceling out common factors, and combining like terms.