Select The Correct Answer.What Is The Factored Form Of $8x^3 - 27y^3$?A. $(2x + 3y)(4x^2 - 6xy + 9y^2)$B. $ ( 2 X − 3 Y ) ( 4 X 2 + 9 X Y + 6 Y 2 ) (2x - 3y)(4x^2 + 9xy + 6y^2) ( 2 X − 3 Y ) ( 4 X 2 + 9 X Y + 6 Y 2 ) [/tex]C. $(2x - 3y)(4x^2 + 6xy + 9y^2)$D. $(2x + 3y)(4x^2 -

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Introduction

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When it comes to factoring polynomials, one of the most common and useful techniques is the difference of cubes formula. This formula allows us to factor expressions of the form $a^3 - b^3$ into the product of two binomials. In this article, we will explore the difference of cubes formula and apply it to factor the expression $8x^3 - 27y^3$.

The Difference of Cubes Formula

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The difference of cubes formula is a fundamental concept in algebra that allows us to factor expressions of the form $a^3 - b^3$. The formula is as follows:

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

This formula can be applied to any expression of the form $a^3 - b^3$, where $a$ and $b$ are any real numbers.

Factoring the Expression $8x^3 - 27y^3$

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Now that we have the difference of cubes formula, we can apply it to factor the expression $8x^3 - 27y^3$. To do this, we need to identify the values of $a$ and $b$ in the expression.

In this case, we can see that $a = 2x$ and $b = 3y$. Therefore, we can apply the difference of cubes formula as follows:

$$8x^3 - 27y^3 = (2x)^3 - (3y)^3$$

Using the difference of cubes formula, we can factor the expression as follows:

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

Simplifying the expression, we get:

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

Conclusion

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In this article, we have explored the difference of cubes formula and applied it to factor the expression $8x^3 - 27y^3$. We have shown that the factored form of the expression is $(2x - 3y)(4x^2 + 6xy + 9y^2)$. This result is consistent with option C in the given multiple-choice question.

Answer

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The correct answer is:

• C. $(2x - 3y)(4x^2 + 6xy + 9y^2)$
  

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Introduction

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In our previous article, we explored the difference of cubes formula and applied it to factor the expression $8x^3 - 27y^3$. In this article, we will provide a Q&A guide to help you better understand the difference of cubes formula and how to apply it to factor expressions of the form $a^3 - b^3$.

Q&A

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Q: What is the difference of cubes formula?

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A: The difference of cubes formula is a fundamental concept in algebra that allows us to factor expressions of the form $a^3 - b^3$. The formula is as follows:

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

Q: How do I apply the difference of cubes formula?

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A: To apply the difference of cubes formula, you need to identify the values of $a$ and $b$ in the expression. Once you have identified the values of $a$ and $b$, you can plug them into the formula to factor the expression.

Q: What are the steps to factor an expression using the difference of cubes formula?

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A: The steps to factor an expression using the difference of cubes formula are as follows:

1. Identify the values of $a$ and $b$ in the expression.
2. Plug the values of $a$ and $b$ into the difference of cubes formula.
3. Simplify the expression to get the factored form.

Q: Can I use the difference of cubes formula to factor expressions of the form $a^3 + b^3$?

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A: No, the difference of cubes formula can only be used to factor expressions of the form $a^3 - b^3$. If you need to factor an expression of the form $a^3 + b^3$, you will need to use a different formula.

Q: What are some common mistakes to avoid when using the difference of cubes formula?

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A: Some common mistakes to avoid when using the difference of cubes formula include:

• Not identifying the values of $a$ and $b$ correctly.
• Not plugging the values of $a$ and $b$ into the formula correctly.
• Not simplifying the expression correctly.

Q: Can I use the difference of cubes formula to factor expressions with negative coefficients?

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A: Yes, you can use the difference of cubes formula to factor expressions with negative coefficients. The formula will still work as long as the expression is in the form $a^3 - b^3$.

Conclusion

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In this article, we have provided a Q&A guide to help you better understand the difference of cubes formula and how to apply it to factor expressions of the form $a^3 - b^3$. We hope that this guide has been helpful in answering your questions and providing you with a better understanding of the difference of cubes formula.

Additional Resources

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If you are looking for additional resources to help you learn more about the difference of cubes formula, we recommend the following:

• Khan Academy: Difference of Cubes Formula
• Mathway: Difference of Cubes Formula
• Wolfram Alpha: Difference of Cubes Formula

We hope that these resources will be helpful in providing you with a better understanding of the difference of cubes formula.