Which Of The Following Statements About $42xy - 49x + 30y - 35$ Are True? Check All Of The Boxes That Apply.- One Of The Factors Is $(6y + 7)$.- One Of The Factors Is \$(7x + 5)$[/tex\].- One Of The Factors Is
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Understanding the Problem
When it comes to factoring algebraic expressions, we often encounter expressions that can be factored into simpler expressions. In this article, we will explore the process of factoring and apply it to the given expression $42xy - 49x + 30y - 35$. Our goal is to determine which of the given statements about this expression are true.
The Given Expression
The given expression is $42xy - 49x + 30y - 35$. To begin factoring, we need to identify any common factors among the terms. In this case, we can see that the terms $42xy$ and $-49x$ have a common factor of $7x$, while the terms $30y$ and $-35$ have a common factor of $5$.
Factoring by Grouping
One way to factor the given expression is by grouping. We can group the terms as follows:
Now, we can factor out the common factors from each group:
Factoring the Common Binomial
We can see that both groups have a common binomial factor of $(6y - 7)$. We can factor this out to get:
Evaluating the Statements
Now that we have factored the given expression, we can evaluate the statements:
- One of the factors is $(6y + 7)$. This statement is false, as we have factored the expression to $(7x + 5)(6y - 7)$, not $(6y + 7)$.
- One of the factors is $(7x + 5)$. This statement is true, as we have factored the expression to $(7x + 5)(6y - 7)$.
- One of the factors is $(7x - 5)$. This statement is false, as we have factored the expression to $(7x + 5)(6y - 7)$, not $(7x - 5)$.
Conclusion
In conclusion, the given expression $42xy - 49x + 30y - 35$ can be factored into $(7x + 5)(6y - 7)$. We have evaluated the statements and found that only one of them is true: One of the factors is $(7x + 5)$.
Tips and Tricks
When factoring algebraic expressions, it's essential to identify any common factors among the terms. We can use the distributive property to factor out common factors. Additionally, we can use the commutative property to rearrange the terms and make factoring easier.
Common Mistakes
When factoring algebraic expressions, it's easy to make mistakes. One common mistake is to forget to factor out common factors. Another mistake is to factor out the wrong factor. To avoid these mistakes, it's essential to carefully read the expression and identify any common factors.
Real-World Applications
Factoring algebraic expressions has many real-world applications. For example, in physics, we use factoring to solve problems involving motion and energy. In engineering, we use factoring to design and optimize systems. In economics, we use factoring to model and analyze economic systems.
Final Thoughts
In conclusion, factoring algebraic expressions is a powerful tool that can be used to solve a wide range of problems. By identifying common factors and using the distributive property, we can factor expressions into simpler expressions. We have evaluated the statements and found that only one of them is true: One of the factors is $(7x + 5)$. We hope that this article has provided you with a better understanding of factoring algebraic expressions and how to apply it to real-world problems.
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Understanding the Basics
Factoring algebraic expressions is a fundamental concept in mathematics that can be used to solve a wide range of problems. In this article, we will provide a Q&A guide to help you understand the basics of factoring and how to apply it to real-world problems.
Q: What is factoring?
A: Factoring is the process of expressing an algebraic expression as a product of simpler expressions. This is done by identifying common factors among the terms and factoring them out.
Q: Why is factoring important?
A: Factoring is important because it allows us to simplify complex expressions and solve problems more easily. It is also a fundamental concept in many areas of mathematics, including algebra, geometry, and calculus.
Q: How do I factor an algebraic expression?
A: To factor an algebraic expression, you need to identify any common factors among the terms. You can use the distributive property to factor out common factors and the commutative property to rearrange the terms.
Q: What are some common mistakes to avoid when factoring?
A: Some common mistakes to avoid when factoring include:
- Forgetting to factor out common factors
- Factoring out the wrong factor
- Not using the distributive property to factor out common factors
- Not using the commutative property to rearrange the terms
Q: How do I know if an expression can be factored?
A: To determine if an expression can be factored, you need to look for any common factors among the terms. If you can identify any common factors, you can use the distributive property to factor them out.
Q: What are some real-world applications of factoring?
A: Factoring has many real-world applications, including:
- Solving problems involving motion and energy in physics
- Designing and optimizing systems in engineering
- Modeling and analyzing economic systems in economics
- Solving problems involving finance and investments
Q: How do I use factoring to solve problems?
A: To use factoring to solve problems, you need to identify any common factors among the terms and factor them out. You can then use the distributive property to simplify the expression and solve the problem.
Q: What are some tips and tricks for factoring?
A: Some tips and tricks for factoring include:
- Using the distributive property to factor out common factors
- Using the commutative property to rearrange the terms
- Identifying any common factors among the terms
- Using factoring to simplify complex expressions
Q: How do I know if I have factored an expression correctly?
A: To determine if you have factored an expression correctly, you need to check your work and make sure that you have factored out all of the common factors. You can also use the distributive property to check your work and make sure that the expression is simplified correctly.
Conclusion
In conclusion, factoring algebraic expressions is a fundamental concept in mathematics that can be used to solve a wide range of problems. By understanding the basics of factoring and how to apply it to real-world problems, you can become a more confident and proficient problem-solver. We hope that this Q&A guide has provided you with a better understanding of factoring and how to use it to solve problems.
Common Factoring Techniques
There are several common factoring techniques that you can use to factor algebraic expressions. Some of these techniques include:
- Factoring out common factors: This involves identifying any common factors among the terms and factoring them out.
- Using the distributive property: This involves using the distributive property to factor out common factors and simplify the expression.
- Using the commutative property: This involves using the commutative property to rearrange the terms and simplify the expression.
- Factoring quadratic expressions: This involves factoring quadratic expressions into the product of two binomials.
Real-World Applications of Factoring
Factoring has many real-world applications, including:
- Solving problems involving motion and energy in physics: Factoring can be used to solve problems involving motion and energy in physics, such as calculating the trajectory of a projectile or the energy of a system.
- Designing and optimizing systems in engineering: Factoring can be used to design and optimize systems in engineering, such as calculating the stress on a beam or the efficiency of a machine.
- Modeling and analyzing economic systems in economics: Factoring can be used to model and analyze economic systems in economics, such as calculating the demand for a product or the supply of a resource.
- Solving problems involving finance and investments: Factoring can be used to solve problems involving finance and investments, such as calculating the return on investment or the risk of a portfolio.
Conclusion
In conclusion, factoring algebraic expressions is a fundamental concept in mathematics that can be used to solve a wide range of problems. By understanding the basics of factoring and how to apply it to real-world problems, you can become a more confident and proficient problem-solver. We hope that this Q&A guide has provided you with a better understanding of factoring and how to use it to solve problems.