Select The Correct Answer.What Is The Factored Form Of $1,458 X^3 - 2$?A. $2(9x-1)(81x^2+9x+1)$ B. $ 2 ( 9 X + 1 ) ( 81 X 2 − 9 X + 1 ) 2(9x+1)(81x^2-9x+1) 2 ( 9 X + 1 ) ( 81 X 2 − 9 X + 1 ) [/tex] C. $(9x-2)(81x^2+18x+4)$ D. $(9x+2)(81x^2-18x+4)$
Introduction
Factoring a cubic expression can be a challenging task, but with the right approach, it can be broken down into manageable steps. In this article, we will explore the factored form of the given cubic expression $1,458 x^3 - 2$ and determine the correct answer among the options provided.
Understanding the Expression
The given expression is a cubic expression in the form of $ax^3 + bx^2 + cx + d$. In this case, the expression is $1,458 x^3 - 2$, where $a = 1,458$, $b = 0$, $c = 0$, and $d = -2$. Our goal is to factor this expression into a product of three binomials.
Factoring the Expression
To factor the expression, we can start by factoring out the greatest common factor (GCF) of the two terms. In this case, the GCF is 2, so we can write the expression as $2(729x^3 - 1)$.
Next, we can recognize that $729x^3 - 1$ is a difference of cubes, which can be factored as $(9x-1)(81x^2+9x+1)$. Therefore, the factored form of the expression is $2(9x-1)(81x^2+9x+1)$.
Evaluating the Options
Now that we have factored the expression, we can evaluate the options provided to determine the correct answer.
- Option A: $2(9x-1)(81x^2+9x+1)$
- Option B: $2(9x+1)(81x^2-9x+1)$
- Option C: $(9x-2)(81x^2+18x+4)$
- Option D: $(9x+2)(81x^2-18x+4)$
Based on our factoring, we can see that the correct answer is Option A: $2(9x-1)(81x^2+9x+1)$.
Conclusion
In conclusion, factoring a cubic expression requires a step-by-step approach, starting with the greatest common factor and recognizing patterns such as the difference of cubes. By following these steps, we can determine the factored form of the expression and evaluate the options provided to determine the correct answer.
Key Takeaways
- Factoring a cubic expression requires a step-by-step approach.
- The greatest common factor (GCF) should be factored out first.
- Recognize patterns such as the difference of cubes to simplify the expression.
- Evaluate the options provided to determine the correct answer.
Additional Resources
For more information on factoring cubic expressions, check out the following resources:
- Khan Academy: Factoring Cubic Expressions
- Mathway: Factoring Cubic Expressions
- Wolfram Alpha: Factoring Cubic Expressions
Introduction
Factoring cubic expressions can be a challenging task, but with the right approach, it can be broken down into manageable steps. In this article, we will explore the factored form of the given cubic expression $1,458 x^3 - 2$ and determine the correct answer among the options provided. We will also answer some frequently asked questions about factoring cubic expressions.
Q&A: Factoring Cubic Expressions
Q: What is the greatest common factor (GCF) of the two terms in the expression $1,458 x^3 - 2$?
A: The GCF of the two terms is 2.
Q: How do I factor out the GCF from the expression?
A: To factor out the GCF, we can divide each term by the GCF. In this case, we can write the expression as $2(729x^3 - 1)$.
Q: What is the difference of cubes formula?
A: The difference of cubes formula is $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.
Q: How do I recognize the difference of cubes pattern in the expression?
A: To recognize the difference of cubes pattern, we need to look for two terms that are cubes. In this case, we can see that $729x^3$ is a cube, and $1$ is also a cube. Therefore, we can write the expression as $(9x-1)(81x^2+9x+1)$.
Q: What is the factored form of the expression $1,458 x^3 - 2$?
A: The factored form of the expression is $2(9x-1)(81x^2+9x+1)$.
Q: How do I evaluate the options provided to determine the correct answer?
A: To evaluate the options, we need to compare the factored form of the expression with each option. In this case, we can see that option A matches the factored form of the expression.
Q: What are some common mistakes to avoid when factoring cubic expressions?
A: Some common mistakes to avoid when factoring cubic expressions include:
- Not factoring out the greatest common factor (GCF) first.
- Not recognizing the difference of cubes pattern.
- Not evaluating the options provided to determine the correct answer.
Conclusion
In conclusion, factoring cubic expressions requires a step-by-step approach, starting with the greatest common factor and recognizing patterns such as the difference of cubes. By following these steps and avoiding common mistakes, we can determine the factored form of the expression and evaluate the options provided to determine the correct answer.
Key Takeaways
- Factoring a cubic expression requires a step-by-step approach.
- The greatest common factor (GCF) should be factored out first.
- Recognize patterns such as the difference of cubes to simplify the expression.
- Evaluate the options provided to determine the correct answer.
- Avoid common mistakes such as not factoring out the GCF, not recognizing the difference of cubes pattern, and not evaluating the options provided.
Additional Resources
For more information on factoring cubic expressions, check out the following resources:
- Khan Academy: Factoring Cubic Expressions
- Mathway: Factoring Cubic Expressions
- Wolfram Alpha: Factoring Cubic Expressions
By following these steps and using the resources provided, you can master the art of factoring cubic expressions and become a math whiz!