Select The Correct Answer.What Is The Factored Form Of B 3 − 1 , 000 B^3 - 1,000 B 3 − 1 , 000 ?A. (b+10)\left(b^2-10b+100\right ] B. (b-10)\left(b^2+10b+100\right ] C. (b-10)\left(10b^2+b+100\right ] D.

Introduction

Factoring is a fundamental concept in algebra that involves expressing an algebraic expression as a product of simpler expressions. In this article, we will focus on factoring the expression $b^3 - 1,000$. This expression can be factored using the difference of cubes formula, which states that $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. We will use this formula to factor the given expression and determine the correct answer.

Understanding the Difference of Cubes Formula

The difference of cubes formula is a powerful tool for factoring expressions of the form $a^3 - b^3$. This formula can be used to factor expressions that involve the difference of cubes, such as $a^3 - b^3$, $a^3 + b^3$, and $a^3 - 2b^3$. The formula is as follows:

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

Factoring the Expression $b^3 - 1,000$

To factor the expression $b^3 - 1,000$, we can use the difference of cubes formula. We can rewrite the expression as follows:

$b^3 - 1,000 = b^3 - 10^3$

Now, we can apply the difference of cubes formula to factor the expression:

$b^3 - 10^3 = (b - 10)(b^2 + 10b + 100)$

Analyzing the Options

Now that we have factored the expression $b^3 - 1,000$, we can analyze the options to determine the correct answer. The options are as follows:

A. $(b+10)\left(b^2-10b+100\right)$ B. $(b-10)\left(b^2+10b+100\right)$ C. $(b-10)\left(10b^2+b+100\right)$ D. $(b+10)\left(b^2+10b+100\right)$

Conclusion

Based on our analysis, we can conclude that the correct answer is option B. $(b-10)\left(b^2+10b+100\right)$. This is because the factored form of the expression $b^3 - 1,000$ is $(b - 10)(b^2 + 10b + 100)$, which matches option B.

Final Answer

The final answer is option B. $(b-10)\left(b^2+10b+100\right)$.

Introduction

In our previous article, we discussed how to factor the expression $b^3 - 1,000$ using the difference of cubes formula. We also analyzed the options to determine the correct answer. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on factoring expressions.

Q: What is the difference of cubes formula?

A: The difference of cubes formula is a powerful tool for factoring 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 to factor an expression?

A: To apply the difference of cubes formula, you need to identify the values of $a$ and $b$ in the expression. Then, you can use the formula to factor the expression. For example, if you have the expression $b^3 - 10^3$, you can rewrite it as $(b - 10)(b^2 + 10b + 100)$.

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

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

• Not identifying the values of $a$ and $b$ correctly
• Not applying the difference of cubes formula correctly
• Not simplifying the expression after factoring
• Not checking the answer for accuracy

Q: How do I check if my answer is correct?

A: To check if your answer is correct, you can use the following steps:

• Simplify the expression after factoring
• Check if the expression can be factored further
• Check if the answer matches one of the options
• Check if the answer is consistent with the original expression

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

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

• Algebraic geometry
• Number theory
• Cryptography
• Optimization problems

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

A: Yes, you can use the difference of cubes formula to factor expressions with negative coefficients. For example, if you have the expression $-b^3 + 10^3$, you can rewrite it as $-(b^3 - 10^3)$ and then apply the difference of cubes formula.

Q: How do I factor expressions with multiple terms?

A: To factor expressions with multiple terms, you can use the following steps:

• Identify the common factors among the terms
• Factor out the common factors
• Simplify the expression after factoring

Q: What are some tips for factoring expressions quickly and accurately?

A: Some tips for factoring expressions quickly and accurately include:

• Practice, practice, practice
• Use the difference of cubes formula regularly
• Simplify the expression after factoring
• Check the answer for accuracy

Conclusion

Factoring expressions is an essential skill in algebra that has many real-world applications. By understanding the difference of cubes formula and practicing factoring expressions, you can become proficient in this skill and apply it to a wide range of problems. We hope this Q&A article has helped clarify any doubts and provided additional information on factoring expressions.