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

Introduction

Factoring the sum of cubes is a fundamental concept in algebra that allows us to express a sum of cubes as a product of two binomials. In this article, we will explore the factored form of the expression $27m^3 + 125n^3$ and provide a step-by-step guide on how to factor it.

Understanding the Sum of Cubes

The sum of cubes is a mathematical expression that involves the sum of two or more cubes. It is represented by the formula $a^3 + b^3$. The factored form of the sum of cubes is given by the formula $(a + b)(a^2 - ab + b^2)$.

The Expression $27m^3 + 125n^3$

The given expression is $27m^3 + 125n^3$. To factor this expression, we need to identify the two cubes that are being added together. We can see that $27m^3$ is a cube of $3m$ and $125n^3$ is a cube of $5n$.

Factoring the Expression

To factor the expression $27m^3 + 125n^3$, we can use the formula for the sum of cubes. We can rewrite the expression as $(3m)^3 + (5n)^3$. Now, we can apply the formula for the sum of cubes:

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

Simplifying the expression, we get:

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

Evaluating the Options

Now, let's evaluate the options provided:

• Option 1: $3m^2 - 8mn + 5n^2$
• Option 2: $9m^2 + 15mn + 25n^2$
• Option 3: $3m + 5n$

Comparing the factored form of the expression with the options, we can see that option 2 is the correct answer.

Conclusion

In conclusion, the factored form of the expression $27m^3 + 125n^3$ is $(3m + 5n)(9m^2 - 15mn + 25n^2)$. We can see that option 2 is the correct answer, which is $9m^2 + 15mn + 25n^2$. This example illustrates the importance of understanding the formula for the sum of cubes and how to apply it to factor expressions.

Key Takeaways

• The factored form of the sum of cubes is given by the formula $(a + b)(a^2 - ab + b^2)$.
• To factor the expression $27m^3 + 125n^3$, we can use the formula for the sum of cubes.
• The correct answer is option 2, which is $9m^2 + 15mn + 25n^2$.

Final Answer

Introduction

In our previous article, we explored the factored form of the expression $27m^3 + 125n^3$ and provided a step-by-step guide on how to factor it. In this article, we will answer some frequently asked questions about factoring the sum of cubes.

Q&A

Q: What is the formula for the sum of cubes?

A: The formula for the sum of cubes is $(a + b)(a^2 - ab + b^2)$.

Q: How do I identify the two cubes in the expression?

A: To identify the two cubes in the expression, you need to look for the cube root of each term. For example, in the expression $27m^3 + 125n^3$, the cube root of $27m^3$ is $3m$ and the cube root of $125n^3$ is $5n$.

Q: What is the difference between the sum of cubes and the difference of cubes?

A: The sum of cubes is a mathematical expression that involves the sum of two or more cubes, while the difference of cubes is a mathematical expression that involves the difference of two or more cubes. The formula for the difference of cubes is $(a - b)(a^2 + ab + b^2)$.

Q: Can I factor the sum of cubes using the difference of cubes formula?

A: No, you cannot factor the sum of cubes using the difference of cubes formula. The sum of cubes and the difference of cubes have different formulas and require different techniques to factor.

Q: What are some common mistakes to avoid when factoring the sum of cubes?

A: Some common mistakes to avoid when factoring the sum of cubes include:

• Not identifying the two cubes in the expression
• Not using the correct formula for the sum of cubes
• Not simplifying the expression correctly
• Not checking the answer for errors

Q: Can I use the sum of cubes formula to factor expressions with more than two terms?

A: No, the sum of cubes formula is only applicable to expressions with two terms. If you have an expression with more than two terms, you will need to use a different technique to factor it.

Q: How do I know if an expression can be factored using the sum of cubes formula?

A: To determine if an expression can be factored using the sum of cubes formula, you need to check if the expression can be written in the form $a^3 + b^3$. If it can, then you can use the sum of cubes formula to factor it.

Conclusion

In conclusion, factoring the sum of cubes is a fundamental concept in algebra that requires a clear understanding of the formula and the techniques involved. By following the steps outlined in this article and avoiding common mistakes, you can master the art of factoring the sum of cubes.

Key Takeaways

• The formula for the sum of cubes is $(a + b)(a^2 - ab + b^2)$.
• To factor the sum of cubes, you need to identify the two cubes in the expression and use the correct formula.
• The sum of cubes formula is only applicable to expressions with two terms.
• You can use the sum of cubes formula to factor expressions that can be written in the form $a^3 + b^3$.

Final Answer

The final answer is: The sum of cubes formula is $(a + b)(a^2 - ab + b^2)$.