What Is The Factorization Of $1,000 X^6 - 27$?A. $(10 X^2 - 3)(100 X^2 + 30 X^2 + 9)$B. \$(10 X^2 - 3)(100 X^4 + 30 X^2 + 9)$[/tex\]C. $(10 X^3 - 3)(100 X^2 + 30 X^3 + 9)$D. $(10 X^3 - 3)(100 X^6 + 30

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Introduction to Factorization

Factorization is a fundamental concept in mathematics that involves expressing an algebraic expression as a product of simpler expressions. It is a crucial technique used to simplify complex expressions, solve equations, and identify patterns in mathematics. In this article, we will explore the factorization of the expression $1,000 x^6 - 27$ and determine the correct factorization among the given options.

Understanding the Expression

The given expression is $1,000 x^6 - 27$. To factorize this expression, we need to identify the greatest common factor (GCF) of the two terms. The GCF of $1,000 x^6$ and $27$ is $1$, as there is no common factor other than $1$.

Using the Difference of Squares Formula

We can rewrite the expression as $1,000 x^6 - 27 = (10 x3)2 - 3^2$. This is a classic example of the difference of squares formula, which states that $a^2 - b^2 = (a + b)(a - b)$. Applying this formula, we get:

(10x3)2−32=(10x3+3)(10x3−3)(10 x^3)^2 - 3^2 = (10 x^3 + 3)(10 x^3 - 3)

Simplifying the Expression

Now, we can simplify the expression further by factoring out the common term $10 x^3$ from the first term:

(10x3+3)(10x3−3)=(10x3+3)(10x3−3)(10 x^3 + 3)(10 x^3 - 3) = (10 x^3 + 3)(10 x^3 - 3)

Comparing with the Options

Now, let's compare the simplified expression with the given options:

A. $(10 x^2 - 3)(100 x^2 + 30 x^2 + 9)$ B. $(10 x^2 - 3)(100 x^4 + 30 x^2 + 9)$ C. $(10 x^3 - 3)(100 x^2 + 30 x^3 + 9)$ D. $(10 x^3 - 3)(100 x^6 + 30 x^3 + 9)$

Conclusion

Based on the factorization of the expression $1,000 x^6 - 27$, we can conclude that the correct factorization is:

(10x3+3)(10x3−3)(10 x^3 + 3)(10 x^3 - 3)

However, this option is not available in the given choices. Let's re-examine the options and try to match the factorization with one of the given options.

Re-examining the Options

Upon re-examining the options, we can see that option C is close to the correct factorization:

(10x3−3)(100x2+30x3+9)(10 x^3 - 3)(100 x^2 + 30 x^3 + 9)

However, the term $100 x^2 + 30 x^3 + 9$ does not match the correct factorization. Let's try to simplify this term further.

Simplifying the Term

We can simplify the term $100 x^2 + 30 x^3 + 9$ by factoring out the common term $10 x^3$:

100x2+30x3+9=10x3(10x2+3)+9100 x^2 + 30 x^3 + 9 = 10 x^3 (10 x^2 + 3) + 9

Re-examining the Options Again

Now, let's re-examine the options again:

A. $(10 x^2 - 3)(100 x^2 + 30 x^2 + 9)$ B. $(10 x^2 - 3)(100 x^4 + 30 x^2 + 9)$ C. $(10 x^3 - 3)(100 x^2 + 30 x^3 + 9)$ D. $(10 x^3 - 3)(100 x^6 + 30 x^3 + 9)$

Conclusion

Based on the re-examination of the options, we can conclude that the correct factorization is:

(10x3−3)(100x2+30x3+9)(10 x^3 - 3)(100 x^2 + 30 x^3 + 9)

However, this option is not the correct factorization. Let's try to simplify the term $100 x^2 + 30 x^3 + 9$ further.

Simplifying the Term Again

We can simplify the term $100 x^2 + 30 x^3 + 9$ by factoring out the common term $10 x^3$:

100x2+30x3+9=10x3(10x2+3)+9100 x^2 + 30 x^3 + 9 = 10 x^3 (10 x^2 + 3) + 9

Re-examining the Options Again

Now, let's re-examine the options again:

A. $(10 x^2 - 3)(100 x^2 + 30 x^2 + 9)$ B. $(10 x^2 - 3)(100 x^4 + 30 x^2 + 9)$ C. $(10 x^3 - 3)(100 x^2 + 30 x^3 + 9)$ D. $(10 x^3 - 3)(100 x^6 + 30 x^3 + 9)$

Conclusion

Based on the re-examination of the options, we can conclude that the correct factorization is:

(10x3−3)(100x6+30x3+9)(10 x^3 - 3)(100 x^6 + 30 x^3 + 9)

This option matches the correct factorization.

The final answer is: D\boxed{D}

Introduction

In our previous article, we explored the factorization of the expression $1,000 x^6 - 27$ and determined the correct factorization among the given options. In this article, we will answer some frequently asked questions related to the factorization of this expression.

Q1: What is the greatest common factor (GCF) of the two terms in the expression $1,000 x^6 - 27$?

A1: The GCF of the two terms is $1$, as there is no common factor other than $1$.

Q2: How can we rewrite the expression $1,000 x^6 - 27$ to apply the difference of squares formula?

A2: We can rewrite the expression as $(10 x3)2 - 3^2$.

Q3: What is the difference of squares formula?

A3: The difference of squares formula states that $a^2 - b^2 = (a + b)(a - b)$.

Q4: How can we simplify the expression $(10 x3)2 - 3^2$ using the difference of squares formula?

A4: We can simplify the expression as $(10 x^3 + 3)(10 x^3 - 3)$.

Q5: How can we compare the simplified expression with the given options?

A5: We can compare the simplified expression with the given options by looking for the correct factorization.

Q6: What is the correct factorization of the expression $1,000 x^6 - 27$?

A6: The correct factorization is $(10 x^3 - 3)(100 x^6 + 30 x^3 + 9)$.

Q7: Why is option C not the correct factorization?

A7: Option C is not the correct factorization because the term $100 x^2 + 30 x^3 + 9$ does not match the correct factorization.

Q8: How can we simplify the term $100 x^2 + 30 x^3 + 9$ further?

A8: We can simplify the term by factoring out the common term $10 x^3$.

Q9: What is the simplified form of the term $100 x^2 + 30 x^3 + 9$?

A9: The simplified form of the term is $10 x^3 (10 x^2 + 3) + 9$.

Q10: How can we compare the simplified term with the given options?

A10: We can compare the simplified term with the given options by looking for the correct factorization.

Q11: What is the correct factorization of the expression $1,000 x^6 - 27$?

A11: The correct factorization is $(10 x^3 - 3)(100 x^6 + 30 x^3 + 9)$.

Conclusion

In this article, we answered some frequently asked questions related to the factorization of the expression $1,000 x^6 - 27$. We hope that this article has provided a better understanding of the factorization of this expression and has helped to clarify any doubts that readers may have had.

The final answer is: D\boxed{D}