Understanding the Concept of Factoring Out the Common Factor

Oct 13, 2025 by windailys.com 145 views

Introduction

Factoring out the common factor is a fundamental concept in algebra that enables us to simplify complex expressions and solve equations more efficiently. In this article, we will delve into the world of factoring and explore how to factor out the common factor from a given expression. We will use the expression $6b^5 + 10b^4 - 10b^2$ as a case study and examine the different options provided.

Understanding the Concept of Factoring Out the Common Factor

Factoring out the common factor involves identifying the greatest common factor (GCF) of the terms in an expression and then expressing each term as a product of the GCF and a remaining factor. This process is essential in simplifying expressions, solving equations, and graphing functions.

The Expression: $6b^5 + 10b^4 - 10b^2$

Let's take a closer look at the given expression: $6b^5 + 10b^4 - 10b^2$ . Our goal is to factor out the common factor from this expression.

Option A: $2b^3(3b^3 + 15b^2 - 25)$

Option A suggests that the common factor is $2b^3$ . To verify this, we need to check if $2b^3$ is indeed a factor of each term in the expression.

$6b^5$ can be factored as $2b^3 \cdot 3b^2$
$10b^4$ can be factored as $2b^3 \cdot 5b$
$-10b^2$ can be factored as $-2b^3 \cdot 5$

As we can see, $2b^3$ is indeed a factor of each term in the expression. Therefore, we can express the given expression as $2b^3(3b^2 + 5b - 5)$ .

Option B: $2b^2(6b^3 + 10b^2 - 10)$

Option B suggests that the common factor is $2b^2$ . To verify this, we need to check if $2b^2$ is indeed a factor of each term in the expression.

$6b^5$ can be factored as $2b^2 \cdot 3b^3$
$10b^4$ can be factored as $2b^2 \cdot 5b^2$
$-10b^2$ can be factored as $-2b^2 \cdot 5$

As we can see, $2b^2$ is indeed a factor of each term in the expression. Therefore, we can express the given expression as $2b^2(3b^3 + 5b^2 - 5)$ .

Option C: $2b^2(3b^3 + 5b^2 - 5)$

Option C suggests that the common factor is $2b^2$ . We have already verified this option in the previous section.

Option D: $2b^3(3b^3 + 5b^2 - 5)$

Option D suggests that the common factor is $2b^3$ . We have already verified this option in the first section.

Conclusion

In conclusion, we have factored out the common factor from the given expression $6b^5 + 10b^4 - 10b^2$ and examined the different options provided. We have verified that the correct option is B) $2b^2(6b^3 + 10b^2 - 10)$ .

Final Answer

The final answer is B.

Discussion

This problem requires a deep understanding of factoring and algebraic expressions. The student needs to identify the common factor and express each term as a product of the common factor and a remaining factor. This problem is an excellent example of how factoring can be used to simplify complex expressions and solve equations.

Additional Resources

For more information on factoring and algebraic expressions, please refer to the following resources:

Khan Academy: Factoring
Mathway: Factoring
Wolfram Alpha: Factoring

Practice Problems

Factor out the common factor from the expression $8z^6 + 12z^5 - 4z^4$ .
Factor out the common factor from the expression $20w^7 + 30w^6 - 10w^5$ .

Conclusion

Q: What is factoring out the common factor?

A: Factoring out the common factor involves identifying the greatest common factor (GCF) of the terms in an expression and then expressing each term as a product of the GCF and a remaining factor.

Q: Why is factoring out the common factor important?

A: Factoring out the common factor is essential in simplifying expressions, solving equations, and graphing functions. It helps to identify the underlying structure of an expression and can make it easier to solve problems.

Q: How do I identify the common factor?

A: To identify the common factor, you need to look for the greatest common factor of the terms in the expression. This can be done by finding the highest power of each variable that appears in all the terms.

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

A: The greatest common factor (GCF) is the largest factor that divides all the terms in an expression without leaving a remainder.

Q: How do I factor out the common factor?

A: To factor out the common factor, you need to express each term as a product of the common factor and a remaining factor. This can be done by dividing each term by the common factor.

Q: What are some common mistakes to avoid when factoring out the common factor?

A: Some common mistakes to avoid when factoring out the common factor include:

Not identifying the greatest common factor (GCF)
Factoring out a factor that is not common to all terms
Not expressing each term as a product of the common factor and a remaining factor

Q: How do I check if I have factored out the common factor correctly?

A: To check if you have factored out the common factor correctly, you need to multiply the common factor by the remaining factors and see if you get the original expression.

Q: What are some real-world applications of factoring out the common factor?

A: Factoring out the common factor has many real-world applications, including:

Simplifying complex expressions in physics and engineering
Solving equations in finance and economics
Graphing functions in computer science and data analysis

Q: How can I practice factoring out the common factor?

A: You can practice factoring out the common factor by working through examples and exercises in your textbook or online resources. You can also try factoring out the common factor from your own expressions and equations.

Q: What are some common expressions that require factoring out the common factor?

A: Some common expressions that require factoring out the common factor include:

Quadratic expressions
Polynomial expressions
Rational expressions

Q: How can I use technology to help me factor out the common factor?

A: You can use technology such as calculators or computer software to help you factor out the common factor. These tools can help you identify the greatest common factor (GCF) and factor out the common factor.

Q: What are some tips for factoring out the common factor?

A: Some tips for factoring out the common factor include:

Take your time and be patient
Read the problem carefully and understand what is being asked
Use a systematic approach to identify the greatest common factor (GCF)
Check your work carefully to ensure that you have factored out the common factor correctly

Conclusion

In conclusion, factoring out the common factor is an essential skill in algebra that can help you simplify complex expressions and solve equations more efficiently. By understanding the concept of factoring out the common factor and practicing it regularly, you can become more confident and proficient in your ability to factor out the common factor.