Simplify The Expression: $\frac{4x^2 + 20x}{x^3 + 4x^2} \cdot (x^2 + 8x + 16$\]

Feb 26, 2025 by ADMIN 80 views

Simplify the Expression: A Step-by-Step Guide to Algebraic Manipulation

Introduction

Algebraic manipulation is a crucial skill in mathematics, and simplifying expressions is an essential part of it. In this article, we will focus on simplifying the given expression: $\frac{4x^2 + 20x}{x^3 + 4x^2} \cdot (x^2 + 8x + 16)$ . We will break down the problem into manageable steps, and by the end of this article, you will have a clear understanding of how to simplify complex algebraic expressions.

Understanding the Expression

Before we dive into simplifying the expression, let's take a closer look at it. The given expression is a product of two fractions: $\frac{4x^2 + 20x}{x^3 + 4x^2}$ and $(x^2 + 8x + 16)$ . To simplify this expression, we need to first factorize the numerator and denominator of the first fraction.

Factorizing the Numerator and Denominator

Let's start by factorizing the numerator and denominator of the first fraction.

Factorizing the Numerator

The numerator of the first fraction is $4x^2 + 20x$ . We can factor out the greatest common factor (GCF) of the two terms, which is $4x$ . Factoring out $4x$ , we get:

$4x^2 + 20x = 4x(x + 5)$

Factorizing the Denominator

The denominator of the first fraction is $x^3 + 4x^2$ . We can factor out the greatest common factor (GCF) of the two terms, which is $x^2$ . Factoring out $x^2$ , we get:

$x^3 + 4x^2 = x^2(x + 4)$

Simplifying the Expression

Now that we have factorized the numerator and denominator of the first fraction, we can simplify the expression. We can cancel out the common factors between the numerator and denominator.

Canceling Out Common Factors

The numerator of the first fraction is $4x(x + 5)$ , and the denominator is $x^2(x + 4)$ . We can cancel out the common factor of $x$ between the two expressions. We are left with:

$\frac{4(x + 5)}{x(x + 4)} \cdot (x^2 + 8x + 16)$

Further Simplification

Now that we have canceled out the common factor of $x$ , we can further simplify the expression. We can factorize the numerator and denominator of the second fraction.

Factorizing the Numerator

The numerator of the second fraction is $x^2 + 8x + 16$ . We can factor out the greatest common factor (GCF) of the two terms, which is $(x + 4)$ . Factoring out $(x + 4)$ , we get:

$x^2 + 8x + 16 = (x + 4)(x + 4)$

Factorizing the Denominator

The denominator of the second fraction is $x(x + 4)$ . We have already factorized this expression in the previous step.

Final Simplification

Now that we have factorized the numerator and denominator of the second fraction, we can simplify the expression. We can cancel out the common factors between the numerator and denominator.

Canceling Out Common Factors

The numerator of the second fraction is $(x + 4)(x + 4)$ , and the denominator is $x(x + 4)$ . We can cancel out the common factor of $(x + 4)$ between the two expressions. We are left with:

$\frac{4(x + 5)}{x} \cdot (x + 4)$

Final Answer

The final simplified expression is $\frac{4(x + 5)}{x} \cdot (x + 4)$ . This expression cannot be simplified further.

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics. By breaking down the problem into manageable steps and using factorization, we can simplify complex expressions. In this article, we have simplified the given expression: $\frac{4x^2 + 20x}{x^3 + 4x^2} \cdot (x^2 + 8x + 16)$ . We have factorized the numerator and denominator of the first fraction, canceled out common factors, and further simplified the expression. By following these steps, you can simplify complex algebraic expressions and become proficient in algebraic manipulation.

Frequently Asked Questions

What is algebraic manipulation?
How do I simplify algebraic expressions?
What are the steps involved in simplifying algebraic expressions?

Step-by-Step Guide to Simplifying Algebraic Expressions

Factorize the numerator and denominator of the expression.
Cancel out common factors between the numerator and denominator.
Further simplify the expression by factorizing the numerator and denominator.
Cancel out common factors between the numerator and denominator.
The final simplified expression is the answer.

Tips and Tricks

Factorize the numerator and denominator of the expression to simplify it.
Cancel out common factors between the numerator and denominator to simplify the expression.
Use factorization to simplify complex expressions.
Practice simplifying algebraic expressions to become proficient in algebraic manipulation.

Real-World Applications

Simplifying algebraic expressions has numerous real-world applications. It is used in various fields such as physics, engineering, and economics. Algebraic manipulation is used to solve complex problems and make predictions.

Conclusion

Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it has numerous real-world applications. In our previous article, we provided a step-by-step guide to simplifying the expression: $\frac{4x^2 + 20x}{x^3 + 4x^2} \cdot (x^2 + 8x + 16)$ . In this article, we will answer some of the most frequently asked questions about simplifying algebraic expressions.

Q&A

Q: What is algebraic manipulation?

A: Algebraic manipulation is the process of simplifying or transforming algebraic expressions using various techniques such as factorization, cancellation, and rearrangement.

Q: How do I simplify algebraic expressions?

A: To simplify algebraic expressions, you need to follow these steps:

Factorize the numerator and denominator of the expression.
Cancel out common factors between the numerator and denominator.
Further simplify the expression by factorizing the numerator and denominator.
Cancel out common factors between the numerator and denominator.
The final simplified expression is the answer.

Q: What are the steps involved in simplifying algebraic expressions?

A: The steps involved in simplifying algebraic expressions are:

Factorize the numerator and denominator of the expression.
Cancel out common factors between the numerator and denominator.
Further simplify the expression by factorizing the numerator and denominator.
Cancel out common factors between the numerator and denominator.
The final simplified expression is the answer.

Q: How do I factorize the numerator and denominator of an expression?

A: To factorize the numerator and denominator of an expression, you need to identify the greatest common factor (GCF) of the two terms and factor it out.

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) is the largest factor that divides both terms of an expression.

Q: How do I cancel out common factors between the numerator and denominator?

A: To cancel out common factors between the numerator and denominator, you need to identify the common factors and cancel them out.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions are:

Not factorizing the numerator and denominator of the expression.
Not canceling out common factors between the numerator and denominator.
Not further simplifying the expression by factorizing the numerator and denominator.
Not canceling out common factors between the numerator and denominator.

Q: How do I practice simplifying algebraic expressions?

A: To practice simplifying algebraic expressions, you can:

Start with simple expressions and gradually move on to more complex ones.
Use online resources and practice problems to help you practice.
Join a study group or find a study partner to help you practice.
Take online courses or watch video tutorials to help you learn.

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it has numerous real-world applications. In this article, we have answered some of the most frequently asked questions about simplifying algebraic expressions. We hope that this article has provided you with a better understanding of how to simplify algebraic expressions and has helped you to become proficient in algebraic manipulation.

Tips and Tricks

Factorize the numerator and denominator of the expression to simplify it.
Cancel out common factors between the numerator and denominator to simplify the expression.
Use factorization to simplify complex expressions.
Practice simplifying algebraic expressions to become proficient in algebraic manipulation.

Real-World Applications

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics, and it has numerous real-world applications. By following the steps outlined in this article, you can simplify complex algebraic expressions and become proficient in algebraic manipulation.