What Is The Completely Factored Form Of $xy^3 - X^3y$?A. X Y ( Y + X ) ( Y − X Xy(y+x)(y-x X Y ( Y + X ) ( Y − X ]B. X Y ( Y − X ) ( Y − X Xy(y-x)(y-x X Y ( Y − X ) ( Y − X ]C. X Y ( X − Y ) ( X 2 + X Y + Y 2 Xy(x-y)(x^2 + Xy + Y^2 X Y ( X − Y ) ( X 2 + X Y + Y 2 ]D. X Y ( X − Y ) ( Y 2 + X Y + X 2 Xy(x-y)(y^2 + Xy + X^2 X Y ( X − Y ) ( Y 2 + X Y + X 2 ]
Introduction
In mathematics, factoring is a fundamental concept that involves expressing an algebraic expression as a product of simpler expressions. The completely factored form of an expression is the most simplified form of the expression, where it cannot be factored further. In this article, we will explore the completely factored form of the expression $xy^3 - x^3y$.
Understanding the Expression
The given expression is $xy^3 - x^3y$. To factor this expression, we need to identify the common factors and use the distributive property to simplify it. The expression can be rewritten as $xy^3 - x^3y = xy(y^2 - x^2)$.
Factoring the Difference of Squares
The expression $y^2 - x^2$ is a difference of squares, which can be factored as $(y + x)(y - x)$. Therefore, the expression $xy(y^2 - x^2)$ can be rewritten as $xy(y + x)(y - x)$.
Analyzing the Options
Now that we have factored the expression $xy^3 - x^3y$ as $xy(y + x)(y - x)$, let's analyze the options provided:
A. B. C. D.
Conclusion
Based on our analysis, the completely factored form of the expression $xy^3 - x^3y$ is $xy(y + x)(y - x)$. This matches option A. Therefore, the correct answer is:
The completely factored form of $xy^3 - x^3y$ is , which corresponds to option A.
Final Thoughts
Factoring expressions is an essential skill in mathematics, and it requires a deep understanding of algebraic concepts. In this article, we have explored the completely factored form of the expression $xy^3 - x^3y$ and analyzed the options provided. By following the steps outlined in this article, you can develop your factoring skills and become proficient in simplifying complex algebraic expressions.
Frequently Asked Questions
- What is the completely factored form of $xy^3 - x^3y$?
- How do you factor the difference of squares?
- What is the correct answer among the options provided?
Answers
- The completely factored form of $xy^3 - x^3y$ is $xy(y + x)(y - x)$.
- The difference of squares can be factored as $(y + x)(y - x)$.
- The correct answer is option A: .
Additional Resources
For more information on factoring expressions, you can refer to the following resources:
- Khan Academy: Factoring Expressions
- Mathway: Factoring Calculator
- Wolfram Alpha: Factoring Tool
Conclusion
In conclusion, the completely factored form of the expression $xy^3 - x^3y$ is $xy(y + x)(y - x)$. This article has provided a step-by-step guide on how to factor the expression and analyze the options provided. By following the steps outlined in this article, you can develop your factoring skills and become proficient in simplifying complex algebraic expressions.
Introduction
Factoring expressions is a fundamental concept in mathematics that involves expressing an algebraic expression as a product of simpler expressions. In our previous article, we explored the completely factored form of the expression $xy^3 - x^3y$. In this article, we will address some of the frequently asked questions related to factoring expressions.
Q&A
Q1: What is the completely factored form of $xy^3 - x^3y$?
A1: The completely factored form of $xy^3 - x^3y$ is $xy(y + x)(y - x)$.
Q2: How do you factor the difference of squares?
A2: The difference of squares can be factored as $(y + x)(y - x)$.
Q3: What is the correct answer among the options provided?
A3: The correct answer is option A: .
Q4: How do you factor the sum of cubes?
A4: The sum of cubes can be factored as $(y + x)(y^2 - xy + x^2)$.
Q5: What is the difference between factoring and simplifying an expression?
A5: Factoring involves expressing an expression as a product of simpler expressions, while simplifying involves reducing an expression to its simplest form.
Q6: How do you factor a quadratic expression?
A6: A quadratic expression can be factored as $(y + x)(y - x)$ or $(y + x)(y + x)$, depending on the coefficients.
Q7: What is the completely factored form of $x^2 + 4x + 4$?
A7: The completely factored form of $x^2 + 4x + 4$ is $(x + 2)(x + 2)$.
Q8: How do you factor a polynomial expression?
A8: A polynomial expression can be factored by identifying the common factors and using the distributive property to simplify it.
Q9: What is the difference between factoring and expanding an expression?
A9: Factoring involves expressing an expression as a product of simpler expressions, while expanding involves multiplying an expression to obtain a more complex expression.
Q10: How do you factor a rational expression?
A10: A rational expression can be factored by identifying the common factors and using the distributive property to simplify it.
Conclusion
In conclusion, factoring expressions is a fundamental concept in mathematics that involves expressing an algebraic expression as a product of simpler expressions. This article has addressed some of the frequently asked questions related to factoring expressions, including the completely factored form of $xy^3 - x^3y$, factoring the difference of squares, and factoring a quadratic expression. By following the steps outlined in this article, you can develop your factoring skills and become proficient in simplifying complex algebraic expressions.
Additional Resources
For more information on factoring expressions, you can refer to the following resources:
- Khan Academy: Factoring Expressions
- Mathway: Factoring Calculator
- Wolfram Alpha: Factoring Tool
Final Thoughts
Factoring expressions is an essential skill in mathematics that requires a deep understanding of algebraic concepts. By practicing and mastering factoring techniques, you can simplify complex algebraic expressions and solve a wide range of mathematical problems.