Perform The Distributive Property On The Expression:${ 3(5x - 2) }$A. ${ 9x }$ B. ${ 15x - 6 }$ C. ${ -9x }$ D. ${ 15x + 6 }$ E. None Of These.
Understanding the Distributive Property
The distributive property is a fundamental concept in algebra that allows us to expand expressions with multiple terms. It states that for any real numbers a, b, and c, the following equation holds:
a(b + c) = ab + ac
In this article, we will apply the distributive property to the expression 3(5x - 2) and explore the different options provided.
Applying the Distributive Property
To apply the distributive property, we need to multiply the term outside the parentheses (3) by each term inside the parentheses (5x and -2).
Using the distributive property, we can rewrite the expression as:
3(5x - 2) = 3(5x) - 3(2)
Simplifying the Expression
Now, we can simplify the expression by multiplying 3 by each term inside the parentheses.
3(5x) = 15x -3(2) = -6
Therefore, the expression 3(5x - 2) can be rewritten as:
15x - 6
Evaluating the Options
Now that we have simplified the expression, let's evaluate the options provided:
A. 9x B. 15x - 6 C. -9x D. 15x + 6 E. None of these
Based on our calculation, we can see that option B (15x - 6) is the correct answer.
Conclusion
In this article, we applied the distributive property to the expression 3(5x - 2) and simplified it to 15x - 6. We also evaluated the options provided and determined that option B is the correct answer. The distributive property is a powerful tool in algebra that allows us to expand expressions with multiple terms. By understanding and applying this concept, we can solve a wide range of algebraic problems.
Common Mistakes to Avoid
When applying the distributive property, it's essential to remember the following common mistakes:
- Forgetting to distribute the term outside the parentheses to each term inside the parentheses.
- Not simplifying the expression after applying the distributive property.
- Not evaluating the options provided carefully.
By avoiding these common mistakes, we can ensure that our calculations are accurate and our answers are correct.
Real-World Applications
The distributive property has numerous real-world applications in various fields, including:
- Physics: The distributive property is used to calculate the force and momentum of objects in motion.
- Engineering: The distributive property is used to design and optimize systems, such as electrical circuits and mechanical systems.
- Economics: The distributive property is used to analyze and model economic systems, including supply and demand curves.
By understanding and applying the distributive property, we can solve a wide range of problems in these fields and make informed decisions.
Practice Problems
To reinforce your understanding of the distributive property, try solving the following practice problems:
- Apply the distributive property to the expression 2(x + 3).
- Simplify the expression 4(2x - 1).
- Evaluate the expression 3(2x + 1) using the distributive property.
By practicing these problems, you can develop your skills and become more confident in applying the distributive property to solve algebraic problems.
Conclusion
Frequently Asked Questions
In this article, we will address some of the most common questions related to the distributive property.
Q: What is the distributive property?
A: The distributive property is a fundamental concept in algebra that allows us to expand expressions with multiple terms. It states that for any real numbers a, b, and c, the following equation holds:
a(b + c) = ab + ac
Q: How do I apply the distributive property?
A: To apply the distributive property, you need to multiply the term outside the parentheses by each term inside the parentheses. For example, if you have the expression 3(5x - 2), you would multiply 3 by 5x and -2 separately.
Q: What are some common mistakes to avoid when applying the distributive property?
A: Some common mistakes to avoid when applying the distributive property include:
- Forgetting to distribute the term outside the parentheses to each term inside the parentheses.
- Not simplifying the expression after applying the distributive property.
- Not evaluating the options provided carefully.
Q: How do I simplify an expression after applying the distributive property?
A: To simplify an expression after applying the distributive property, you need to combine like terms. For example, if you have the expression 15x - 6, you can simplify it by combining the like terms 15x and -6.
Q: What are some real-world applications of the distributive property?
A: The distributive property has numerous real-world applications in various fields, including:
- Physics: The distributive property is used to calculate the force and momentum of objects in motion.
- Engineering: The distributive property is used to design and optimize systems, such as electrical circuits and mechanical systems.
- Economics: The distributive property is used to analyze and model economic systems, including supply and demand curves.
Q: How do I practice the distributive property?
A: To practice the distributive property, try solving the following practice problems:
- Apply the distributive property to the expression 2(x + 3).
- Simplify the expression 4(2x - 1).
- Evaluate the expression 3(2x + 1) using the distributive property.
Q: What are some common misconceptions about the distributive property?
A: Some common misconceptions about the distributive property include:
- Believing that the distributive property only applies to expressions with two terms inside the parentheses.
- Thinking that the distributive property only applies to expressions with positive coefficients.
- Assuming that the distributive property is only used in algebra and not in other mathematical fields.
Q: How do I apply the distributive property to expressions with variables?
A: To apply the distributive property to expressions with variables, you need to multiply the term outside the parentheses by each variable inside the parentheses. For example, if you have the expression 3(2x + 1), you would multiply 3 by 2x and 1 separately.
Q: What are some tips for mastering the distributive property?
A: Some tips for mastering the distributive property include:
- Practicing regularly to develop your skills and confidence.
- Starting with simple expressions and gradually moving on to more complex ones.
- Using visual aids, such as diagrams and charts, to help you understand the concept.
- Breaking down complex expressions into smaller, more manageable parts.
By following these tips and practicing regularly, you can master the distributive property and become proficient in solving algebraic problems.