What Is The Completely Factored Form Of $xy^3 - X^3y$?A. $xy(y+x)(y-x$\] B. $xy(y-x)(y-x$\] C. $xy(x-y)(x^2+xy+y^2$\] D. $xy(x-y)(y^2+xy+x^2$\]

Introduction

In algebra, factoring is a crucial concept that helps us simplify complex expressions and solve equations. Factoring involves expressing an algebraic expression as a product of simpler expressions, known as factors. In this article, we will explore the completely factored form of the expression $xy^3 - x^3y$. We will use various factoring techniques, including the difference of cubes and the distributive property, to arrive at the completely factored form.

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 appropriate factoring techniques. Let's start by examining the expression closely.

Factoring the Expression

We can begin by factoring out the greatest common factor (GCF) of the two terms. The GCF of $xy^3$ and $x^3y$ is $xy$. Factoring out $xy$ from both terms, we get:

$$xy^3 - x^3y = xy(y^2 - x^2)$$

Difference of Squares

Now, we have the expression $y^2 - x^2$. This is a difference of squares, which can be factored as $(y + x)(y - x)$. Therefore, we can rewrite the expression as:

$$xy(y^2 - x^2) = xy(y + x)(y - x)$$

Completely Factored Form

We have now factored the expression $xy^3 - x^3y$ completely. The completely factored form is:

$$xy(y + x)(y - x)$$

Conclusion

In this article, we have explored the completely factored form of the expression $xy^3 - x^3y$. We used the difference of cubes and the distributive property to arrive at the completely factored form. The completely factored form is $xy(y + x)(y - x)$. This form is useful for simplifying complex expressions and solving equations.

Comparison with Options

Let's compare our result with the given options:

• A. $xy(y+x)(y-x)$
• B. $xy(y-x)(y-x)$
• C. $xy(x-y)(x^2+xy+y^2)$
• D. $xy(x-y)(y^2+xy+x^2)$

Our result matches option A. $xy(y+x)(y-x)$.

Final Answer

The completely factored form of $xy^3 - x^3y$ is $xy(y + x)(y - x)$.

Introduction

In our previous article, we explored the completely factored form of the expression $xy^3 - x^3y$. We used various factoring techniques, including the difference of cubes and the distributive property, to arrive at the completely factored form. In this article, we will answer some frequently asked questions (FAQs) about factoring $xy^3 - x^3y$.

Q: What is the greatest common factor (GCF) of $xy^3$ and $x^3y$?

A: The GCF of $xy^3$ and $x^3y$ is $xy$.

Q: How do you factor the difference of squares $y^2 - x^2$?

A: The difference of squares $y^2 - x^2$ can be factored as $(y + x)(y - x)$.

Q: What is the completely factored form of $xy^3 - x^3y$?

A: The completely factored form of $xy^3 - x^3y$ is $xy(y + x)(y - x)$.

Q: Why is it important to factor expressions?

A: Factoring expressions is important because it helps us simplify complex expressions and solve equations. Factoring can also help us identify the underlying structure of an expression, which can be useful in solving problems.

Q: What are some common factoring techniques?

A: Some common factoring techniques include the difference of cubes, the distributive property, and the greatest common factor (GCF) method.

Q: How do you use the distributive property to factor expressions?

A: The distributive property states that for any numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. We can use this property to factor expressions by distributing the terms inside the parentheses.

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 you use the difference of cubes formula to factor expressions?

A: We can use the difference of cubes formula to factor expressions of the form $a^3 - b^3$. We simply substitute $a$ and $b$ into the formula and simplify.

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

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

• Not identifying the greatest common factor (GCF) of the terms
• Not using the distributive property correctly
• Not recognizing the difference of squares or the difference of cubes
• Not simplifying the expression after factoring

Conclusion

In this article, we have answered some frequently asked questions (FAQs) about factoring $xy^3 - x^3y$. We have covered topics such as the greatest common factor (GCF), the difference of squares, and the difference of cubes. We have also discussed common factoring techniques and common mistakes to avoid when factoring expressions.

Final Answer

The completely factored form of $xy^3 - x^3y$ is $xy(y + x)(y - x)$.