Not applying the distributive property

Oct 13, 2025 by windailys.com 143 views

Introduction

In mathematics, multiplying expressions is a fundamental operation that involves combining two or more terms using the multiplication operation. In this article, we will focus on multiplying a fraction by a binomial expression, specifically the expression $\frac{1}{5} x (10x - 15)$ . We will break down the problem into manageable steps and provide a clear explanation of each step.

Understanding the Problem

The given expression is $\frac{1}{5} x (10x - 15)$ . To multiply this expression, we need to apply the distributive property, which states that for any real numbers $a$ , $b$ , and $c$ , $a(b + c) = ab + ac$ . In this case, we have a fraction $\frac{1}{5}$ multiplied by a binomial expression $(10x - 15)$ .

Step 1: Apply the Distributive Property

To multiply the fraction by the binomial expression, we need to apply the distributive property. This means that we will multiply the fraction by each term in the binomial expression separately.

$\frac{1}{5} x (10x - 15) = \frac{1}{5} x (10x) - \frac{1}{5} x (15)$

Step 2: Simplify Each Term

Now that we have applied the distributive property, we need to simplify each term. To simplify the first term, we can multiply the fraction by the term $10x$ .

$\frac{1}{5} x (10x) = \frac{1}{5} \times 10x = 2x$

To simplify the second term, we can multiply the fraction by the term $15$ .

$\frac{1}{5} x (15) = \frac{1}{5} \times 15 = 3$

Step 3: Combine Like Terms

Now that we have simplified each term, we can combine like terms. In this case, we have two terms: $2x$ and $-3$ . Since these terms have the same variable $x$ , we can combine them by adding their coefficients.

$2x - 3$

Conclusion

In conclusion, multiplying the expression $\frac{1}{5} x (10x - 15)$ involves applying the distributive property, simplifying each term, and combining like terms. By following these steps, we can simplify the expression and arrive at the final answer.

Final Answer

The final answer is: $\boxed{2x - 3}$

Common Mistakes to Avoid

When multiplying expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

Not applying the distributive property: Make sure to apply the distributive property when multiplying a fraction by a binomial expression.
Not simplifying each term: Make sure to simplify each term after applying the distributive property.
Not combining like terms: Make sure to combine like terms after simplifying each term.

Practice Problems

To practice multiplying expressions, try the following problems:

Multiply the expression $\frac{1}{2} x (6x + 8)$ .
Multiply the expression $\frac{1}{3} x (9x - 12)$ .

Real-World Applications

Multiplying expressions has many real-world applications. Here are a few examples:

Physics: In physics, multiplying expressions is used to calculate the force of a moving object.
Engineering: In engineering, multiplying expressions is used to calculate the stress on a material.
Economics: In economics, multiplying expressions is used to calculate the cost of a product.

Conclusion

Introduction

In our previous article, we explored the concept of multiplying expressions, specifically the expression $\frac{1}{5} x (10x - 15)$ . We broke down the problem into manageable steps and provided a clear explanation of each step. In this article, we will answer some frequently asked questions about multiplying expressions.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in mathematics that states that for any real numbers $a$ , $b$ , and $c$ , $a(b + c) = ab + ac$ . This means that we can multiply a single term by each term in a binomial expression separately.

Q: How do I apply the distributive property?

A: To apply the distributive property, simply multiply the single term by each term in the binomial expression separately. For example, if we have the expression $\frac{1}{5} x (10x - 15)$ , we would multiply the fraction by each term in the binomial expression separately.

Q: What is the difference between multiplying a fraction by a binomial expression and multiplying two binomial expressions?

A: When multiplying a fraction by a binomial expression, we apply the distributive property and multiply the fraction by each term in the binomial expression separately. When multiplying two binomial expressions, we multiply each term in the first expression by each term in the second expression.

Q: How do I simplify each term after applying the distributive property?

A: To simplify each term, simply multiply the fraction by the term. For example, if we have the expression $\frac{1}{5} x (10x - 15)$ , we would multiply the fraction by each term in the binomial expression separately and simplify each term.

Q: What is the difference between combining like terms and combining unlike terms?

A: When combining like terms, we add or subtract terms that have the same variable. When combining unlike terms, we add or subtract terms that have different variables.

Q: How do I combine like terms?

A: To combine like terms, simply add or subtract the coefficients of the terms. For example, if we have the expression $2x + 3x$ , we would combine the like terms by adding the coefficients.

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

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

Not applying the distributive property
Not simplifying each term
Not combining like terms

Q: How can I practice multiplying expressions?

A: To practice multiplying expressions, try the following problems:

Multiply the expression $\frac{1}{2} x (6x + 8)$ .
Multiply the expression $\frac{1}{3} x (9x - 12)$ .

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

A: Multiplying expressions has many real-world applications, including:

Physics: In physics, multiplying expressions is used to calculate the force of a moving object.
Engineering: In engineering, multiplying expressions is used to calculate the stress on a material.
Economics: In economics, multiplying expressions is used to calculate the cost of a product.

Conclusion

In conclusion, multiplying expressions is a fundamental operation in mathematics that involves combining two or more terms using the multiplication operation. By following the steps outlined in this article, we can simplify expressions and arrive at the final answer. Remember to apply the distributive property, simplify each term, and combine like terms to avoid common mistakes. With practice, you'll become proficient in multiplying expressions and be able to apply this skill to real-world problems.