What Is The Completely Factored Form Of $x Y^3-x^3 Y$?A. $x Y(y+x)(y-x)$B. \$x Y(y-x)(y-x)$[/tex\]C. $x Y(x-y)\left(x^2+x Y+y^2\right)$D. $x Y(x-y)\left(y^2+x Y+x^2\right)$

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Introduction

In mathematics, factoring is a fundamental concept that involves expressing an algebraic expression as a product of simpler expressions. Factoring can be used to simplify complex expressions, identify common factors, and solve equations. In this article, we will explore the completely factored form of the expression $x y3-x3 y$.

Understanding the Expression

The given expression is a difference of two terms, where each term involves the product of two variables, x and y. The first term is $x y^3$, which can be interpreted as x multiplied by y cubed. The second term is $-x^3 y$, which can be interpreted as negative x cubed multiplied by y.

Factoring the Expression

To factor the expression, we can start by identifying the greatest common factor (GCF) of the two terms. In this case, the GCF is x y, which can be factored out from both terms.

import sympy as sp

# Define the variables
x, y = sp.symbols('x y')

# Define the expression
expr = x*y**3 - x**3*y

# Factor the expression
factored_expr = sp.factor(expr)

print(factored_expr)

Factoring the Expression Using Algebraic Manipulation

We can also factor the expression using algebraic manipulation. We can start by factoring out the common factor x y from both terms.

xy3βˆ’x3y=xy(y2βˆ’x2)x y^3-x^3 y = x y (y^2-x^2)

Next, we can factor the difference of squares in the second term.

y2βˆ’x2=(yβˆ’x)(y+x)y^2-x^2 = (y-x)(y+x)

Therefore, the completely factored form of the expression is:

xy(yβˆ’x)(y+x)x y (y-x)(y+x)

Comparing with the Options

Now that we have factored the expression, we can compare it with the options provided.

  • Option A: $x y(y+x)(y-x)$
  • Option B: $x y(y-x)(y-x)$
  • Option C: $x y(x-y)\left(x^2+x y+y^2\right)$
  • Option D: $x y(x-y)\left(y^2+x y+x^2\right)$

Conclusion

Based on our analysis, we can conclude that the completely factored form of the expression $x y3-x3 y$ is:

xy(yβˆ’x)(y+x)x y (y-x)(y+x)

This is the same as Option A.

Final Answer

The final answer is: A\boxed{A}

Introduction

In our previous article, we explored the completely factored form of the expression $x y3-x3 y$. We used algebraic manipulation to factor the expression and arrived at the final answer. In this article, we will answer some frequently asked questions (FAQs) related to the completely factored form of the expression.

Q: What is the greatest common factor (GCF) of the two terms in the expression?

A: The GCF of the two terms is x y.

Q: How do I factor the expression using algebraic manipulation?

A: To factor the expression using algebraic manipulation, you can start by factoring out the common factor x y from both terms. Then, you can factor the difference of squares in the second term.

Q: What is the difference of squares formula?

A: The difference of squares formula is:

a2βˆ’b2=(aβˆ’b)(a+b)a^2-b^2 = (a-b)(a+b)

Q: How do I apply the difference of squares formula to the expression?

A: To apply the difference of squares formula to the expression, you can substitute a = y and b = x into the formula.

Q: What is the completely factored form of the expression?

A: The completely factored form of the expression is:

xy(yβˆ’x)(y+x)x y (y-x)(y+x)

Q: How do I compare the completely factored form of the expression with the options provided?

A: To compare the completely factored form of the expression with the options provided, you can simply look at the factors and see if they match.

Q: What is the final answer?

A: The final answer is:

xy(yβˆ’x)(y+x)x y (y-x)(y+x)

Q: Which option is correct?

A: Option A is correct.

Q: What is the significance of factoring the expression?

A: Factoring the expression is significant because it allows us to simplify complex expressions, identify common factors, and solve equations.

Q: How do I use factoring to solve equations?

A: To use factoring to solve equations, you can start by factoring the expression on one side of the equation. Then, you can set the other side of the equation equal to zero and solve for the variable.

Q: What are some common applications of factoring?

A: Some common applications of factoring include:

  • Simplifying complex expressions
  • Identifying common factors
  • Solving equations
  • Graphing functions

Q: How do I practice factoring?

A: To practice factoring, you can start by working on simple factoring problems. Then, you can gradually move on to more complex problems.

Q: What are some resources for learning factoring?

A: Some resources for learning factoring include:

  • Textbooks
  • Online tutorials
  • Practice problems
  • Video lectures

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to the completely factored form of the expression $x y3-x3 y$. We covered topics such as the greatest common factor (GCF), algebraic manipulation, the difference of squares formula, and the significance of factoring. We also provided some resources for learning factoring.

Final Answer

The final answer is: A\boxed{A}