What Is The Factored Form Of This Expression? 27 M 3 + 125 N 3 27 M^3 + 125 N^3 27 M 3 + 125 N 3 Drag The Factors To The Correct Locations On The Image. Not All Factors Will Be Used.Options:- 3 M 2 − 8 M N + 5 N 2 3 M^2 - 8 M N + 5 N^2 3 M 2 − 8 Mn + 5 N 2 - 3 M − 5 N 3 M - 5 N 3 M − 5 N - $9 M^2 + 15 M N + 25

Introduction

In mathematics, factoring is a process of expressing an algebraic expression as a product of simpler expressions. It is an essential concept in algebra and is used to simplify complex expressions and solve equations. In this article, we will explore the factored form of the expression $27m^3 + 125n^3$ and discuss the steps involved in factoring it.

Understanding the Expression

The given expression is $27m^3 + 125n^3$. To factor this expression, we need to identify the greatest common factor (GCF) of the two terms. The GCF is the largest expression that divides both terms without leaving a remainder. In this case, the GCF is 1, which means that there is no common factor other than 1.

Factoring the Expression

To factor the expression $27m^3 + 125n^3$, we can use 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 apply the sum of cubes formula to get:

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

Simplifying the expression, we get:

$27m^3 + 125n^3 = (3m + 5n)(9m^2 - 15mn + 25n^2)$

Dragging the Factors to the Correct Locations

Now that we have factored the expression, we need to drag the factors to the correct locations on the image. The correct locations are:

• $(3m + 5n)$ goes in the first location
• $(9m^2 - 15mn + 25n^2)$ goes in the second location

Options

The options for the factors are:

• $3m^2 - 8mn + 5n^2$
• $3m - 5n$
• $9m^2 + 15mn + 25n^2$

Conclusion

In conclusion, the factored form of the expression $27m^3 + 125n^3$ is $(3m + 5n)(9m^2 - 15mn + 25n^2)$. This expression can be simplified further by factoring out the GCF, but in this case, there is no common factor other than 1.

Discussion

The factored form of an expression is an essential concept in mathematics, and it is used to simplify complex expressions and solve equations. In this article, we have discussed the factored form of the expression $27m^3 + 125n^3$ and the steps involved in factoring it. We have also discussed the options for the factors and the correct locations for the factors on the image.

Frequently Asked Questions

• What is the factored form of the expression $27m^3 + 125n^3$?
• How do you factor the expression $27m^3 + 125n^3$?
• What are the options for the factors of the expression $27m^3 + 125n^3$?
• What are the correct locations for the factors of the expression $27m^3 + 125n^3$?

Answers

• The factored form of the expression $27m^3 + 125n^3$ is $(3m + 5n)(9m^2 - 15mn + 25n^2)$.
• To factor the expression $27m^3 + 125n^3$, you can use the sum of cubes formula.
• The options for the factors of the expression $27m^3 + 125n^3$ are $3m^2 - 8mn + 5n^2$, $3m - 5n$, and $9m^2 + 15mn + 25n^2$.
• The correct locations for the factors of the expression $27m^3 + 125n^3$ are $(3m + 5n)$ in the first location and $(9m^2 - 15mn + 25n^2)$ in the second location.

Final Answer

The final answer is: $\boxed{(3m + 5n)(9m^2 - 15mn + 25n^2)}$

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Introduction

In our previous article, we discussed the factored form of the expression $27m^3 + 125n^3$. We also provided the steps involved in factoring it and the options for the factors. In this article, we will provide a Q&A section to help clarify any doubts and provide further understanding of the concept.

Q&A

Q1: What is the factored form of the expression $27m^3 + 125n^3$?

A1: The factored form of the expression $27m^3 + 125n^3$ is $(3m + 5n)(9m^2 - 15mn + 25n^2)$.

Q2: How do you factor the expression $27m^3 + 125n^3$?

A2: To factor the expression $27m^3 + 125n^3$, you can use 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 apply the sum of cubes formula to get:

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

Simplifying the expression, we get:

$27m^3 + 125n^3 = (3m + 5n)(9m^2 - 15mn + 25n^2)$

Q3: What are the options for the factors of the expression $27m^3 + 125n^3$?

A3: The options for the factors of the expression $27m^3 + 125n^3$ are:

• $3m^2 - 8mn + 5n^2$
• $3m - 5n$
• $9m^2 + 15mn + 25n^2$

Q4: What are the correct locations for the factors of the expression $27m^3 + 125n^3$?

A4: The correct locations for the factors of the expression $27m^3 + 125n^3$ are:

• $(3m + 5n)$ goes in the first location
• $(9m^2 - 15mn + 25n^2)$ goes in the second location

Q5: Can you provide an example of how to use the sum of cubes formula?

A5: Yes, here is an example of how to use the sum of cubes formula:

Suppose we want to factor the expression $x^3 + 27$. We can let $a = x$ and $b = 3$. Then, we can apply the sum of cubes formula to get:

$x^3 + 27 = (x + 3)(x^2 - 3x + 9)$

Simplifying the expression, we get:

$x^3 + 27 = (x + 3)(x^2 - 3x + 9)$

Q6: Can you provide an example of how to factor an expression using the sum of cubes formula?

A6: Yes, here is an example of how to factor an expression using the sum of cubes formula:

Suppose we want to factor the expression $8x^3 + 27y^3$. We can let $a = 2x$ and $b = 3y$. Then, we can apply the sum of cubes formula to get:

$(2x)^3 + (3y)^3 = (2x + 3y)((2x)^2 - (2x)(3y) + (3y)^2)$

Simplifying the expression, we get:

$8x^3 + 27y^3 = (2x + 3y)(4x^2 - 6xy + 9y^2)$

Conclusion

In conclusion, the factored form of the expression $27m^3 + 125n^3$ is $(3m + 5n)(9m^2 - 15mn + 25n^2)$. We have also provided the steps involved in factoring it and the options for the factors. Additionally, we have provided examples of how to use the sum of cubes formula and how to factor an expression using the sum of cubes formula.

Frequently Asked Questions

• What is the factored form of the expression $27m^3 + 125n^3$?
• How do you factor the expression $27m^3 + 125n^3$?
• What are the options for the factors of the expression $27m^3 + 125n^3$?
• What are the correct locations for the factors of the expression $27m^3 + 125n^3$?
• Can you provide an example of how to use the sum of cubes formula?
• Can you provide an example of how to factor an expression using the sum of cubes formula?

Answers

• The factored form of the expression $27m^3 + 125n^3$ is $(3m + 5n)(9m^2 - 15mn + 25n^2)$.
• To factor the expression $27m^3 + 125n^3$, you can use the sum of cubes formula.
• The options for the factors of the expression $27m^3 + 125n^3$ are $3m^2 - 8mn + 5n^2$, $3m - 5n$, and $9m^2 + 15mn + 25n^2$.
• The correct locations for the factors of the expression $27m^3 + 125n^3$ are $(3m + 5n)$ in the first location and $(9m^2 - 15mn + 25n^2)$ in the second location.
• Yes, here is an example of how to use the sum of cubes formula: $x^3 + 27 = (x + 3)(x^2 - 3x + 9)$.
• Yes, here is an example of how to factor an expression using the sum of cubes formula: $8x^3 + 27y^3 = (2x + 3y)(4x^2 - 6xy + 9y^2)$.

Final Answer

The final answer is: $\boxed{(3m + 5n)(9m^2 - 15mn + 25n^2)}$