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Mar 8, 2025 by ADMIN 126 views

**Distribute and Combine Like Terms: A Comprehensive Guide**

Introduction

In mathematics, distributing and combining like terms is a fundamental concept that helps simplify complex algebraic expressions. It is a crucial skill that students and professionals alike need to master in order to solve equations, inequalities, and other mathematical problems efficiently. In this article, we will delve into the world of distributing and combining like terms, exploring various types of expressions, and providing step-by-step examples to help you understand the concept better.

What are Like Terms?

Like terms are algebraic expressions that have the same variable(s) raised to the same power. They can be added or subtracted from each other, but not multiplied or divided. For example, in the expression 2x + 3x, 2x and 3x are like terms because they both have the variable x raised to the power of 1.

Distributing Like Terms

Distributing like terms involves multiplying each term in an expression by a common factor. This is done to simplify the expression and make it easier to work with. For instance, in the expression (2x + 3x) * 4, we can distribute the common factor 4 to each term inside the parentheses:

(2x + 3x) * 4 = 2x * 4 + 3x * 4

= 8x + 12x

= 20x

Combining Like Terms

Combining like terms involves adding or subtracting terms that have the same variable(s) raised to the same power. This is done to simplify the expression and make it easier to work with. For example, in the expression 2x + 3x + 5x, we can combine the like terms 2x, 3x, and 5x:

2x + 3x + 5x = (2 + 3 + 5)x

= 10x

Types of Expressions

There are several types of expressions that we need to consider when distributing and combining like terms. Let's explore each of them in detail:

1. Fractions

Fractions are expressions that have a numerator and a denominator. When distributing and combining like terms with fractions, we need to consider the numerator and denominator separately. For example, in the expression 2x/y + 3x/y, we can combine the like terms 2x/y and 3x/y:

2x/y + 3x/y = (2 + 3)x/y

= 5x/y

2. Functions

Functions are expressions that have a variable as the input and a value as the output. When distributing and combining like terms with functions, we need to consider the function notation and the variable(s) involved. For example, in the expression f(x) + f(x), we can combine the like terms f(x) and f(x):

f(x) + f(x) = 2f(x)

3. Radicals

Radicals are expressions that have a root or a power of a variable. When distributing and combining like terms with radicals, we need to consider the index of the radical and the variable(s) involved. For example, in the expression √x + √x, we can combine the like terms √x and √x:

√x + √x = 2√x

4. Absolute Values

Absolute values are expressions that have a variable enclosed in absolute value brackets. When distributing and combining like terms with absolute values, we need to consider the variable(s) involved and the absolute value notation. For example, in the expression |x| + |x|, we can combine the like terms |x| and |x|:

|x| + |x| = 2|x|

5. Inequalities

Inequalities are expressions that have a variable and a value separated by an inequality symbol. When distributing and combining like terms with inequalities, we need to consider the inequality symbol and the variable(s) involved. For example, in the expression x ≤ 2 + x, we can combine the like terms x and x:

x ≤ 2 + x

= x ≤ 2 + x

6. Subscripts

Subscripts are expressions that have a variable with a subscript. When distributing and combining like terms with subscripts, we need to consider the subscript notation and the variable(s) involved. For example, in the expression x_n + x_n, we can combine the like terms x_n and x_n:

x_n + x_n = 2x_n

Conclusion

Distributing and combining like terms is a fundamental concept in mathematics that helps simplify complex algebraic expressions. By understanding the different types of expressions and how to distribute and combine like terms, you can solve equations, inequalities, and other mathematical problems efficiently. Remember to always consider the variable(s) involved, the power of the variable(s), and the notation used in the expression. With practice and patience, you will become proficient in distributing and combining like terms, and you will be able to tackle even the most challenging mathematical problems with confidence.

Examples

Here are some examples of distributing and combining like terms:

2x + 3x + 5x = (2 + 3 + 5)x = 10x
2x/y + 3x/y = (2 + 3)x/y = 5x/y
f(x) + f(x) = 2f(x)
√x + √x = 2√x
|x| + |x| = 2|x|
x ≤ 2 + x = x ≤ 2 + x
x_n + x_n = 2x_n

Practice Problems

Here are some practice problems to help you understand distributing and combining like terms:

Distribute and combine like terms in the expression 2x + 3x + 5x.
Distribute and combine like terms in the expression 2x/y + 3x/y.
Distribute and combine like terms in the expression f(x) + f(x).
Distribute and combine like terms in the expression √x + √x.
Distribute and combine like terms in the expression |x| + |x|.
Distribute and combine like terms in the expression x ≤ 2 + x.
Distribute and combine like terms in the expression x_n + x_n.

Solutions

Here are the solutions to the practice problems:

2x + 3x + 5x = (2 + 3 + 5)x = 10x
2x/y + 3x/y = (2 + 3)x/y = 5x/y
f(x) + f(x) = 2f(x)
√x + √x = 2√x
|x| + |x| = 2|x|
x ≤ 2 + x = x ≤ 2 + x
x_n + x_n = 2x_n

Final Thoughts

Q: What are like terms?

A: Like terms are algebraic expressions that have the same variable(s) raised to the same power. They can be added or subtracted from each other, but not multiplied or divided.

Q: How do I distribute like terms?

A: To distribute like terms, you need to multiply each term in an expression by a common factor. This is done to simplify the expression and make it easier to work with.

Q: Can I distribute like terms with fractions?

A: Yes, you can distribute like terms with fractions. When distributing like terms with fractions, you need to consider the numerator and denominator separately.

Q: Can I distribute like terms with functions?

A: Yes, you can distribute like terms with functions. When distributing like terms with functions, you need to consider the function notation and the variable(s) involved.

Q: Can I distribute like terms with radicals?

A: Yes, you can distribute like terms with radicals. When distributing like terms with radicals, you need to consider the index of the radical and the variable(s) involved.

Q: Can I distribute like terms with absolute values?

A: Yes, you can distribute like terms with absolute values. When distributing like terms with absolute values, you need to consider the variable(s) involved and the absolute value notation.

Q: Can I distribute like terms with inequalities?

A: Yes, you can distribute like terms with inequalities. When distributing like terms with inequalities, you need to consider the inequality symbol and the variable(s) involved.

Q: Can I distribute like terms with subscripts?

A: Yes, you can distribute like terms with subscripts. When distributing like terms with subscripts, you need to consider the subscript notation and the variable(s) involved.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract terms that have the same variable(s) raised to the same power.

Q: Can I combine like terms with fractions?

A: Yes, you can combine like terms with fractions. When combining like terms with fractions, you need to consider the numerator and denominator separately.

Q: Can I combine like terms with functions?

A: Yes, you can combine like terms with functions. When combining like terms with functions, you need to consider the function notation and the variable(s) involved.

Q: Can I combine like terms with radicals?

A: Yes, you can combine like terms with radicals. When combining like terms with radicals, you need to consider the index of the radical and the variable(s) involved.

Q: Can I combine like terms with absolute values?

A: Yes, you can combine like terms with absolute values. When combining like terms with absolute values, you need to consider the variable(s) involved and the absolute value notation.

Q: Can I combine like terms with inequalities?

A: Yes, you can combine like terms with inequalities. When combining like terms with inequalities, you need to consider the inequality symbol and the variable(s) involved.

Q: Can I combine like terms with subscripts?

A: Yes, you can combine like terms with subscripts. When combining like terms with subscripts, you need to consider the subscript notation and the variable(s) involved.

Q: What are some common mistakes to avoid when distributing and combining like terms?

A: Some common mistakes to avoid when distributing and combining like terms include:

Not considering the variable(s) involved
Not considering the power of the variable(s)
Not considering the notation used in the expression
Not distributing or combining like terms correctly
Not simplifying the expression correctly

Q: How can I practice distributing and combining like terms?

A: You can practice distributing and combining like terms by:

Working on practice problems
Solving equations and inequalities
Simplifying complex algebraic expressions
Using online resources and tools
Asking for help from a teacher or tutor

Q: What are some real-world applications of distributing and combining like terms?

A: Some real-world applications of distributing and combining like terms include:

Solving equations and inequalities in physics and engineering
Simplifying complex algebraic expressions in computer science and programming
Working with financial models and data analysis
Solving optimization problems in business and economics
Working with statistical models and data analysis