Select The Correct Answer.What Is The Factored Form Of This Expression? X 9 − 1 , 000 X^9 - 1,000 X 9 − 1 , 000 A. ( X 3 − 10 ) ( X 6 + 10 X 3 + 100 (x^3 - 10)(x^6 + 10x^3 + 100 ( X 3 − 10 ) ( X 6 + 10 X 3 + 100 ] B. ( X 3 − 10 ) ( X 3 + 10 ) ( X 2 − 10 X + 100 (x^3 - 10)(x^3 + 10)(x^2 - 10x + 100 ( X 3 − 10 ) ( X 3 + 10 ) ( X 2 − 10 X + 100 ] C. ( X − 10 ) ( X + 10 ) ( X 3 − 10 X 2 + 100 (x - 10)(x + 10)(x^3 - 10x^2 + 100 ( X − 10 ) ( X + 10 ) ( X 3 − 10 X 2 + 100 ] D.

Introduction

Factoring expressions is a fundamental concept in algebra that helps us simplify complex expressions and solve equations. In this article, we will focus on factoring the expression $x^9 - 1,000$. We will explore the different methods of factoring and provide a step-by-step guide on how to factor this particular expression.

Understanding the Expression

The given expression is $x^9 - 1,000$. This expression can be rewritten as $x^9 - 10^3$. We can see that the expression is in the form of a difference of cubes, where $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.

Factoring the Expression

To factor the expression $x^9 - 1,000$, we can use the difference of cubes formula. We can rewrite the expression as $(x^3)^3 - 10^3$. Now, we can apply the difference of cubes formula:

$(x^3)^3 - 10^3 = (x^3 - 10)((x^3)^2 + x^3(10) + 10^2)$

Simplifying the expression, we get:

$(x^3 - 10)(x^9 + 10x^3 + 100)$

However, this is not the only possible factorization. We can also factor the expression as:

$(x^3 - 10)(x^3 + 10)(x^3 - 10x^2 + 100)$

But, this is not the correct answer. We need to find the correct factorization.

Analyzing the Options

Let's analyze the options given:

A. $(x^3 - 10)(x^6 + 10x^3 + 100)$

B. $(x^3 - 10)(x^3 + 10)(x^2 - 10x + 100)$

C. $(x - 10)(x + 10)(x^3 - 10x^2 + 100)$

D. (None of the above)

We can see that option A is a possible factorization, but it is not the correct answer. Option B is also a possible factorization, but it is not the correct answer either. Option C is not a possible factorization.

Conclusion

After analyzing the options, we can conclude that the correct answer is:

A. $(x^3 - 10)(x^6 + 10x^3 + 100)$

This is the correct factorization of the expression $x^9 - 1,000$.

Step-by-Step Solution

Here is the step-by-step solution:

1. Rewrite the expression as $(x^3)^3 - 10^3$.
2. Apply the difference of cubes formula: $(x^3 - 10)((x^3)^2 + x^3(10) + 10^2)$.
3. Simplify the expression: $(x^3 - 10)(x^9 + 10x^3 + 100)$.
4. Factor the expression as $(x^3 - 10)(x^6 + 10x^3 + 100)$.

Final Answer

The final answer is:

A. $(x^3 - 10)(x^6 + 10x^3 + 100)$

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Introduction

In our previous article, we explored the concept of factoring expressions and provided a step-by-step guide on how to factor the expression $x^9 - 1,000$. We also analyzed the options and concluded that the correct answer is:

A. $(x^3 - 10)(x^6 + 10x^3 + 100)$

In this article, we will provide a Q&A guide to help you understand the concept of factoring expressions and how to apply it to different types of expressions.

Q: What is factoring an expression?

A: Factoring an expression is the process of expressing it as a product of simpler expressions, called factors. Factoring an expression can help us simplify complex expressions and solve equations.

Q: What are the different types of factoring?

A: There are several types of factoring, including:

• Difference of squares: This type of factoring involves expressing an expression as the difference of two squares, such as $a^2 - b^2 = (a - b)(a + b)$.
• Difference of cubes: This type of factoring involves expressing an expression as the difference of two cubes, such as $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.
• Sum of cubes: This type of factoring involves expressing an expression as the sum of two cubes, such as $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.
• Factoring by grouping: This type of factoring involves grouping terms in an expression and factoring out common factors.

Q: How do I factor an expression?

A: To factor an expression, you can follow these steps:

1. Identify the type of factoring: Determine whether the expression is a difference of squares, difference of cubes, sum of cubes, or factoring by grouping.
2. Apply the factoring formula: Use the corresponding factoring formula to factor the expression.
3. Simplify the expression: Simplify the factored expression to obtain the final answer.

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

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

• Not identifying the type of factoring: Failing to identify the type of factoring can lead to incorrect factorization.
• Not applying the correct factoring formula: Applying the wrong factoring formula can lead to incorrect factorization.
• Not simplifying the expression: Failing to simplify the factored expression can lead to incorrect answers.

Q: How do I check my work when factoring expressions?

A: To check your work when factoring expressions, you can follow these steps:

1. Multiply the factors: Multiply the factors to obtain the original expression.
2. Check for errors: Check for errors in the factored expression.
3. Simplify the expression: Simplify the factored expression to obtain the final answer.

Conclusion

Factoring expressions is an important concept in algebra that helps us simplify complex expressions and solve equations. By understanding the different types of factoring and how to apply them, you can become proficient in factoring expressions and solve a wide range of problems. Remember to identify the type of factoring, apply the correct factoring formula, and simplify the expression to obtain the final answer.

Final Tips

• Practice, practice, practice: The more you practice factoring expressions, the more comfortable you will become with the different types of factoring.
• Use online resources: There are many online resources available that can help you learn and practice factoring expressions.
• Seek help when needed: Don't be afraid to seek help when you need it. Ask your teacher or tutor for assistance, or seek help from online resources.

Final Answer

The final answer is:

A. $(x^3 - 10)(x^6 + 10x^3 + 100)$

This is the correct factorization of the expression $x^9 - 1,000$.