What Is The Factored Form Of This Expression?$27m^3 + 125n^3$Choose The Correct Factors:A. $3m + 5n$ B. $3m^2 - 8mn + 5n^2$ C. $9m^2 - 15mn + 25n^2$ D. $3m - 5n$

Feb 28, 2025 by ADMIN 165 views

Understanding the Problem

The given expression is $27m^3 + 125n^3$ . We are asked to find the factored form of this expression. To do this, we need to identify the greatest common factor (GCF) of the two terms and then use the sum of cubes formula to factor the expression.

Identifying the Greatest Common Factor (GCF)

The GCF of $27m^3$ and $125n^3$ is $1$ , since there is no common factor other than $1$ that divides both terms.

Using the Sum of Cubes Formula

The sum of cubes formula is:

$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

In this case, we can let $a = 3m$ and $b = 5n$ . Then, we can plug these values into the sum of cubes formula:

$(3m)^3 + (5n)^3 = (3m + 5n)((3m)^2 - (3m)(5n) + (5n)^2)$

Simplifying the Expression

Now, we can simplify the expression by expanding the squared terms:

$(3m + 5n)(9m^2 - 15mn + 25n^2)$

Comparing with the Answer Choices

We can compare our factored form with the answer choices:

A. $3m + 5n$ B. $3m^2 - 8mn + 5n^2$ C. $9m^2 - 15mn + 25n^2$ D. $3m - 5n$

Our factored form is:

$(3m + 5n)(9m^2 - 15mn + 25n^2)$

This matches answer choice C.

Conclusion

The factored form of the expression $27m^3 + 125n^3$ is $(3m + 5n)(9m^2 - 15mn + 25n^2)$ . Therefore, the correct answer is:

C. $9m^2 - 15mn + 25n^2$

Why is this Important?

Factoring expressions is an important skill in algebra, as it allows us to simplify complex expressions and solve equations more easily. In this case, we used the sum of cubes formula to factor the expression $27m^3 + 125n^3$ . This is a common technique used in algebra to factor expressions of the form $a^3 + b^3$ .

Real-World Applications

Factoring expressions has many real-world applications, such as:

Simplifying complex equations: Factoring expressions can help us simplify complex equations and solve them more easily.
Optimizing systems: Factoring expressions can help us optimize systems by identifying the underlying structure of the system.
Modeling real-world phenomena: Factoring expressions can help us model real-world phenomena, such as population growth or chemical reactions.

Common Mistakes to Avoid

When factoring expressions, there are several common mistakes to avoid:

Not identifying the GCF: Failing to identify the GCF of the two terms can lead to incorrect factoring.
Not using the sum of cubes formula: Failing to use the sum of cubes formula when factoring expressions of the form $a^3 + b^3$ can lead to incorrect factoring.
Not simplifying the expression: Failing to simplify the expression after factoring can lead to incorrect solutions.

Tips and Tricks

Here are some tips and tricks for factoring expressions:

Use the GCF to simplify the expression: Using the GCF to simplify the expression can make it easier to factor.
Use the sum of cubes formula: Using the sum of cubes formula when factoring expressions of the form $a^3 + b^3$ can make it easier to factor.
Simplify the expression after factoring: Simplifying the expression after factoring can help us identify the underlying structure of the system.

Conclusion

In conclusion, factoring expressions is an important skill in algebra that has many real-world applications. By identifying the GCF, using the sum of cubes formula, and simplifying the expression after factoring, we can factor expressions more easily and solve equations more efficiently.

Q: What is factoring an expression?

A: Factoring an expression is the process of expressing it as a product of simpler expressions, called factors. This is done by identifying the greatest common factor (GCF) of the terms and then using various factoring techniques to break down the expression into its constituent parts.

Q: What are the different types of factoring techniques?

A: There are several types of factoring techniques, including:

Greatest Common Factor (GCF) factoring: This involves identifying the GCF of the terms and factoring it out.
Difference of Squares factoring: This involves factoring expressions of the form $a^2 - b^2$ .
Sum of Cubes factoring: This involves factoring expressions of the form $a^3 + b^3$ .
Factoring by Grouping: This involves factoring expressions by grouping terms together.

Q: How do I identify the GCF of two terms?

A: To identify the GCF of two terms, you need to find the largest expression that divides both terms evenly. For example, the GCF of $12x$ and $18x$ is $6x$ , since $6x$ is the largest expression that divides both terms evenly.

Q: What is the difference of squares formula?

A: The difference of squares formula is:

$a^2 - b^2 = (a + b)(a - b)$

This formula can be used to factor expressions of the form $a^2 - b^2$ .

Q: What is the sum of cubes formula?

A: The sum of cubes formula is:

$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

This formula can be used to factor expressions of the form $a^3 + b^3$ .

Q: How do I factor an expression by grouping?

A: To factor an expression by grouping, you need to group the terms together in a way that allows you to factor out a common expression. For example, the expression $x^2 + 5x + 6x + 15$ can be factored by grouping as follows:

$x^2 + 5x + 6x + 15 = (x^2 + 5x) + (6x + 15)$

$= x(x + 5) + 3(2x + 5)$

$= (x + 3)(x + 5)$

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

A: Some common mistakes to avoid when factoring expressions include:

Not identifying the GCF: Failing to identify the GCF of the terms can lead to incorrect factoring.
Not using the correct factoring technique: Using the wrong factoring technique can lead to incorrect factoring.
Not simplifying the expression after factoring: Failing to simplify the expression after factoring can lead to incorrect solutions.

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

A: To determine which factoring technique to use, you need to examine the expression and identify the type of terms it contains. For example, if the expression contains terms of the form $a^2 - b^2$ , you can use the difference of squares formula. If the expression contains terms of the form $a^3 + b^3$ , you can use the sum of cubes formula.

Q: Can I use factoring to solve equations?

A: Yes, factoring can be used to solve equations. By factoring the equation, you can identify the solutions and solve for the variable.

Q: What are some real-world applications of factoring expressions?

A: Factoring expressions has many real-world applications, including:

Simplifying complex equations: Factoring expressions can help us simplify complex equations and solve them more easily.
Optimizing systems: Factoring expressions can help us optimize systems by identifying the underlying structure of the system.
Modeling real-world phenomena: Factoring expressions can help us model real-world phenomena, such as population growth or chemical reactions.

Q: How can I practice factoring expressions?

A: You can practice factoring expressions by working through examples and exercises. You can also use online resources, such as factoring calculators or worksheets, to help you practice. Additionally, you can try factoring expressions on your own, using different techniques and strategies to see what works best for you.