like terms

Like Terms in Math: Definition and Application

Definition

Like terms in math refer to terms that have the same variables raised to the same powers. These terms can be combined or simplified by adding or subtracting their coefficients. In other words, like terms have identical variables and exponents, but their coefficients may differ.

History

The concept of like terms has been a fundamental part of algebra for centuries. The ancient Greeks, such as Euclid and Diophantus, laid the foundation for algebraic thinking, which eventually led to the development of the concept of like terms. Over time, mathematicians and scholars refined the understanding and application of like terms, making it an essential concept in modern mathematics.

Grade Level

Like terms are typically introduced in middle school or early high school, around grades 7-9. They serve as a building block for more advanced algebraic concepts and are essential for solving equations and simplifying expressions.

Knowledge Points and Explanation

To understand like terms, it is crucial to grasp the following concepts:

1. Variables: Letters or symbols used to represent unknown quantities or values.
2. Coefficients: Numbers that multiply variables.
3. Exponents: Powers to which variables are raised.

Step-by-step explanation:

1. Identify the variables in each term.
2. Check if the variables are the same and have the same exponents.
3. If the variables and exponents match, combine the coefficients by adding or subtracting them.
4. Keep the variables and exponents unchanged.

Types of Like Terms

There are various types of like terms based on the number of variables and their exponents. Some common types include:

1. Monomials: Terms with a single variable raised to a non-negative integer exponent.
2. Binomials: Terms with two variables raised to non-negative integer exponents.
3. Trinomials: Terms with three variables raised to non-negative integer exponents.
4. Polynomials: Expressions with multiple terms, each potentially having different variables and exponents.

Properties of Like Terms

Like terms exhibit several properties that make them easier to work with:

1. Commutative Property: Like terms can be rearranged without changing their sum or difference.
2. Associative Property: Like terms can be grouped and combined in any order without altering the result.
3. Distributive Property: Like terms can be distributed across parentheses when simplifying expressions.

Finding and Calculating Like Terms

To find and calculate like terms, follow these steps:

1. Identify the variables and exponents in each term.
2. Compare the variables and exponents to determine if they are the same.
3. If the variables and exponents match, combine the coefficients by adding or subtracting them.
4. Keep the variables and exponents unchanged.

Formula or Equation for Like Terms

There is no specific formula or equation for like terms. Instead, the process involves identifying and combining terms with the same variables and exponents.

Application of Like Terms

Like terms are applied in various algebraic operations, such as simplifying expressions, solving equations, and factoring polynomials. By combining like terms, complex expressions can be simplified, making them easier to work with and analyze.

Symbol or Abbreviation for Like Terms

There is no specific symbol or abbreviation for like terms. They are typically referred to as "like terms" or "similar terms."

Methods for Like Terms

The primary method for dealing with like terms is to identify and combine them based on their variables and exponents. This process involves comparing the variables and exponents and then performing the necessary addition or subtraction of coefficients.

Solved Examples on Like Terms

1. Simplify the expression: 3x + 2y - 5x + 4y Solution: Combining like terms, we get: (3x - 5x) + (2y + 4y) = -2x + 6y

2. Simplify the expression: 2a^2b - 3ab^2 + 5a^2b - ab^2 Solution: Combining like terms, we get: (2a^2b + 5a^2b) + (-3ab^2 - ab^2) = 7a^2b - 4ab^2

3. Simplify the expression: 4x^3 - 2x^2 + 3x^3 + 5x^2 Solution: Combining like terms, we get: (4x^3 + 3x^3) + (-2x^2 + 5x^2) = 7x^3 + 3x^2

Practice Problems on Like Terms

1. Simplify the expression: 2x + 3y - 4x - 2y
2. Simplify the expression: 5a^2b - 2ab^2 + 3a^2b - ab^2
3. Simplify the expression: 6x^3 - 3x^2 + 4x^3 + 2x^2

FAQ on Like Terms

Q: What are like terms? A: Like terms are terms in algebraic expressions that have the same variables raised to the same powers.

Q: How do you combine like terms? A: To combine like terms, add or subtract their coefficients while keeping the variables and exponents unchanged.

Q: Can unlike terms be combined? A: No, unlike terms cannot be combined directly. They must be simplified separately.

Q: Why are like terms important? A: Like terms allow us to simplify expressions, solve equations, and perform various algebraic operations more efficiently.

Q: Can like terms have different coefficients? A: No, like terms must have the same variables and exponents, but their coefficients can differ.

In conclusion, understanding like terms is crucial for simplifying expressions, solving equations, and working with polynomials. By identifying and combining terms with the same variables and exponents, complex algebraic problems can be simplified and solved more effectively.