Factor The Expression: 28 X Y − 7 X − 20 Y 2 + 5 Y 28xy - 7x - 20y^2 + 5y 28 X Y − 7 X − 20 Y 2 + 5 Y A) ( 7 X − 5 Y ) ( 4 Y − 1 (7x - 5y)(4y - 1 ( 7 X − 5 Y ) ( 4 Y − 1 ] B) − Y ( 7 X + 5 Y -y(7x + 5y − Y ( 7 X + 5 Y ] C) − Y ( 7 X − 1 -y(7x - 1 − Y ( 7 X − 1 ] D) ( 7 X − 5 Y ) ( 7 X − 1 (7x - 5y)(7x - 1 ( 7 X − 5 Y ) ( 7 X − 1 ]

Introduction

Factoring an expression is a fundamental concept in algebra that involves expressing a given expression as a product of simpler expressions. In this article, we will focus on factoring the expression $28xy - 7x - 20y^2 + 5y$. We will explore the different methods of factoring and provide a step-by-step guide on how to factor the given expression.

Understanding the Expression

Before we begin factoring the expression, let's take a closer look at the given expression: $28xy - 7x - 20y^2 + 5y$. The expression consists of four terms: $28xy$, $-7x$, $-20y^2$, and $5y$. To factor the expression, we need to identify the common factors among the terms.

Identifying Common Factors

The first step in factoring the expression is to identify the common factors among the terms. We can start by looking for the greatest common factor (GCF) of the coefficients of the terms. In this case, the coefficients are 28, -7, -20, and 5. The GCF of these coefficients is 1.

However, we can also look for common factors among the variables. In this case, we can see that the variables $x$ and $y$ are common to all the terms. We can factor out the common variables to simplify the expression.

Factoring Out Common Variables

Let's factor out the common variables $x$ and $y$ from the expression:

$$28xy - 7x - 20y^2 + 5y = (28xy - 7x) - (20y^2 - 5y)$$

Now, we can factor out the common variables from each group:

$$28xy - 7x = 7x(4y - 1)$$

$$-20y^2 + 5y = -5y(4y - 1)$$

Factoring the Expression

Now that we have factored out the common variables, we can factor the expression further. We can see that the expression can be factored as:

$$(7x - 5y)(4y - 1)$$

Conclusion

In this article, we have factored the expression $28xy - 7x - 20y^2 + 5y$ using the method of factoring out common variables. We have identified the common factors among the terms and factored out the common variables to simplify the expression. Finally, we have factored the expression further to obtain the final result.

Answer

The final answer is:

$$(7x - 5y)(4y - 1)$$

Discussion

The expression $28xy - 7x - 20y^2 + 5y$ can be factored using the method of factoring out common variables. The common factors among the terms are the variables $x$ and $y$. By factoring out the common variables, we can simplify the expression and factor it further to obtain the final result.

Common Mistakes

When factoring an expression, it's easy to make mistakes. Some common mistakes include:

• Not identifying the common factors among the terms
• Not factoring out the common variables
• Not factoring the expression further to obtain the final result

Tips and Tricks

When factoring an expression, here are some tips and tricks to keep in mind:

• Identify the common factors among the terms
• Factor out the common variables
• Factor the expression further to obtain the final result
• Use the method of factoring out common variables to simplify the expression

Real-World Applications

Factoring an expression has many real-world applications. Some examples include:

• Solving systems of equations
• Finding the roots of a polynomial equation
• Simplifying complex expressions

Conclusion

Introduction

In our previous article, we explored the concept of factoring an expression and provided a step-by-step guide on how to factor the expression $28xy - 7x - 20y^2 + 5y$. In this article, we will answer some of the most frequently asked questions about factoring expressions.

Q&A

Q: What is factoring an expression?

A: Factoring an expression is a process of expressing a given expression as a product of simpler expressions.

Q: Why is factoring an expression important?

A: Factoring an expression is important because it allows us to simplify complex expressions and solve equations more easily.

Q: How do I identify the common factors among the terms?

A: To identify the common factors among the terms, look for the greatest common factor (GCF) of the coefficients of the terms. You can also look for common factors among the variables.

Q: How do I factor out the common variables?

A: To factor out the common variables, group the terms that have the common variables and factor out the common variables from each group.

Q: What is the difference between factoring and simplifying an expression?

A: Factoring an expression involves expressing a given expression as a product of simpler expressions, while simplifying an expression involves combining like terms and eliminating any unnecessary terms.

Q: Can I factor an expression that has no common factors among the terms?

A: No, you cannot factor an expression that has no common factors among the terms. In this case, the expression is already in its simplest form.

Q: How do I know if I have factored an expression correctly?

A: To check if you have factored an expression correctly, multiply the factors together and see if you get the original expression.

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

A: Some common mistakes to avoid when factoring an expression include not identifying the common factors among the terms, not factoring out the common variables, and not factoring the expression further to obtain the final result.

Q: Can I use factoring to solve equations?

A: Yes, you can use factoring to solve equations. By factoring an expression, you can simplify the equation and solve for the variables.

Q: What are some real-world applications of factoring expressions?

A: Some real-world applications of factoring expressions include solving systems of equations, finding the roots of a polynomial equation, and simplifying complex expressions.

Conclusion

In conclusion, factoring an expression is a fundamental concept in algebra that involves expressing a given expression as a product of simpler expressions. By identifying the common factors among the terms and factoring out the common variables, we can simplify the expression and factor it further to obtain the final result. We hope that this Q&A guide has provided you with a better understanding of factoring expressions and how to apply it in real-world situations.

Additional Resources

For more information on factoring expressions, check out the following resources:

• Khan Academy: Factoring Expressions
• Mathway: Factoring Expressions
• Wolfram Alpha: Factoring Expressions

Practice Problems

Try factoring the following expressions:

• $12x^2 - 9x - 20$
• $15y^2 + 20y - 12$
• $24x^2 - 16x - 20$

Answer Key

• $12x^2 - 9x - 20 = (3x - 5)(4x + 4)$
• $15y^2 + 20y - 12 = (3y - 2)(5y + 6)$
• $24x^2 - 16x - 20 = (4x - 5)(6x + 4)$