Factor The Expression: ${ 4m^{10} - 81n^2 }$

Mar 7, 2025 by ADMIN 46 views

**Factor the Expression: A Comprehensive Guide**

Introduction

Factoring expressions is a fundamental concept in mathematics, particularly in algebra. It involves expressing an algebraic expression as a product of simpler expressions, called factors. In this article, we will focus on factoring the expression $4m^{10} - 81n^2$ . We will explore various techniques and methods to factor this expression, and provide a step-by-step guide to help you understand the process.

Understanding the Expression

Before we dive into factoring, let's analyze the given expression. The expression is $4m^{10} - 81n^2$ . We can see that it consists of two terms: $4m^{10}$ and $-81n^2$ . The first term is a power of $m$ , while the second term is a power of $n$ . The coefficients of the two terms are $4$ and $-81$ , respectively.

Factoring by Difference of Squares

One of the most common techniques for factoring expressions is the difference of squares. This technique involves factoring an expression of the form $a^2 - b^2$ as $(a + b)(a - b)$ . Let's see if we can apply this technique to the given expression.

4m^{10} - 81n^2 = (2m^5)^2 - (9n)^2

Now, we can apply the difference of squares formula:

(2m^5)^2 - (9n)^2 = (2m^5 + 9n)(2m^5 - 9n)

So, we have factored the expression $4m^{10} - 81n^2$ as $(2m^5 + 9n)(2m^5 - 9n)$ .

Factoring by Greatest Common Factor (GCF)

Another technique for factoring expressions is the greatest common factor (GCF). This technique involves factoring out the greatest common factor of the terms in the expression. Let's see if we can apply this technique to the given expression.

4m^{10} - 81n^2 = 4m^{10} - 81n^2

We can see that the greatest common factor of the two terms is $1$ . Therefore, we cannot factor out any common factor from the expression.

Factoring by Grouping

Factoring by grouping is another technique for factoring expressions. This technique involves grouping the terms in the expression into pairs and factoring out common factors from each pair. Let's see if we can apply this technique to the given expression.

4m^{10} - 81n^2 = (4m^{10} + 0) - (0 + 81n^2)

We can see that the two terms are already grouped into pairs. Now, we can factor out common factors from each pair:

(4m^{10} + 0) - (0 + 81n^2) = 4m^{10} - 81n^2

We can see that we cannot factor out any common factors from the expression.

Conclusion

In this article, we have explored various techniques and methods for factoring the expression $4m^{10} - 81n^2$ . We have applied the difference of squares, greatest common factor (GCF), and factoring by grouping techniques to factor the expression. We have also provided a step-by-step guide to help you understand the process.

Final Answer

The final answer is:

\boxed{(2m^5 + 9n)(2m^5 - 9n)}

References

[1] "Algebra" by Michael Artin
[2] "Calculus" by Michael Spivak
[3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Glossary

Difference of squares: A technique for factoring expressions of the form $a^2 - b^2$ as $(a + b)(a - b)$ .
Greatest common factor (GCF): A technique for factoring out the greatest common factor of the terms in an expression.
Factoring by grouping: A technique for factoring expressions by grouping the terms into pairs and factoring out common factors from each pair.
Factor the Expression: A Comprehensive Guide =====================================================

Q&A: Frequently Asked Questions

Q: What is factoring in mathematics?

A: Factoring is a fundamental concept in mathematics, particularly in algebra. It involves expressing an algebraic expression as a product of simpler expressions, called factors.

Q: Why is factoring important in mathematics?

A: Factoring is important in mathematics because it allows us to simplify complex expressions and solve equations. It is also a crucial step in many mathematical proofs and theorems.

Q: What are the different techniques for factoring expressions?

A: There are several techniques for factoring expressions, including:

Difference of squares
Greatest common factor (GCF)
Factoring by grouping
Factoring by substitution
Factoring by elimination

Q: How do I know which technique to use for factoring an expression?

A: To determine which technique to use, you need to analyze the expression and identify its structure. Look for patterns such as the difference of squares, greatest common factor, or grouping.

Q: What is the difference of squares technique?

A: The difference of squares technique involves factoring an expression of the form $a^2 - b^2$ as $(a + b)(a - b)$ .

Q: How do I apply the difference of squares technique?

A: To apply the difference of squares technique, you need to identify the two terms in the expression that are being subtracted. Then, you need to factor out the greatest common factor of the two terms.

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

A: The greatest common factor (GCF) technique involves factoring out the greatest common factor of the terms in an expression.

Q: How do I apply the GCF technique?

A: To apply the GCF technique, you need to identify the greatest common factor of the terms in the expression. Then, you need to factor out the GCF.

Q: What is factoring by grouping?

A: Factoring by grouping involves factoring an expression by grouping the terms into pairs and factoring out common factors from each pair.

Q: How do I apply the factoring by grouping technique?

A: To apply the factoring by grouping technique, you need to group the terms in the expression into pairs. Then, you need to factor out common factors from each pair.

Q: What is factoring by substitution?

A: Factoring by substitution involves substituting a variable or expression into another expression to simplify it.

Q: How do I apply the factoring by substitution technique?

A: To apply the factoring by substitution technique, you need to identify a variable or expression that can be substituted into the original expression.

Q: What is factoring by elimination?

A: Factoring by elimination involves eliminating one or more terms in an expression to simplify it.

Q: How do I apply the factoring by elimination technique?

A: To apply the factoring by elimination technique, you need to identify one or more terms in the expression that can be eliminated.