Like terms
In algebra, like terms are monomial expressions that contain identical variables raised to the same powers, allowing their numerical coefficients to be added or subtracted to simplify algebraic expressions.[1][2] For example, terms such as $4x^2 and -3x^2 are like terms because they share the variable x with exponent 2, enabling combination into x^2.[1][3] This concept is fundamental to manipulating polynomials and solving equations, as it reduces complexity while preserving the expression's value.[4] Identifying like terms involves comparing the variable parts of each term in an expression, ignoring the coefficients. Terms are unlike if their variables differ or if exponents vary, such as $5xy and $2x^2 y, which cannot be combined directly.[2][5] In practice, combining like terms is applied in operations like addition, subtraction, and distribution within parentheses, forming the basis for more advanced algebraic techniques including factoring and equation solving.[1][4] The process of combining like terms enhances efficiency in algebraic problem-solving and is introduced early in pre-algebra curricula to build foundational skills. For instance, an expression like $7a + 2b + 4a - 3b simplifies to $11a - b by grouping and adding coefficients of like terms separately.[5] This simplification is crucial in fields such as physics and engineering, where algebraic models require concise representations for analysis.[4]Fundamentals
Definition
In algebra, a term is a component of an expression separated by addition or subtraction signs, consisting of a coefficient multiplied by a variable or variables raised to specific powers.[6] Like terms are algebraic terms that contain identical variable parts, meaning they feature the same variables raised to the same exponents, differing only in their numerical coefficients.[1][7] For instance, terms such as $3x^2 and $7x^2 are like terms because both involve the variable x raised to the power of 2.[6] In contrast, unlike terms lack this identical variable structure; for example, $2x and $3y differ in their variables, while $4x and $5x^2 differ in the exponents on x.[1][8] This distinction is fundamental, as like terms share a common form that allows for specific algebraic operations, such as addition of their coefficients while preserving the variable part.[6]Identification Criteria
Like terms are identified based on the matching of their variable components and exponents, serving as the practical extension of their conceptual definition. Specifically, two terms are like if they contain identical variables raised to precisely the same powers, while their numerical coefficients may vary. For instance, $3x^2 and -5x^2 are like terms because both feature the variable x with an exponent of 2, but $4xand2x^2$ are not, as the exponents differ.[9][10] Constants, which are terms without variables and thus considered to have degree 0, form another category of like terms among themselves regardless of their signs or magnitudes. Examples include $5, -2, and $7.3, all of which can be grouped together since they lack variable factors. This uniformity allows constants to be treated analogously to variable terms in identification processes.[10] In edge cases involving advanced exponent forms, terms with negative or fractional exponents qualify as like if the variables and exponents match exactly. For example, x^{-1} and $2x^{-1} are like terms, as both have the variable x raised to the power of -1, enabling recognition even in expressions with reciprocals or rational powers. Similarly, \frac{1}{2}x^{1/2} and $3x^{1/2} share the same structure. These criteria ensure consistent identification across varied algebraic contexts.[11]Operations
Combining Like Terms
Combining like terms involves adding or subtracting the numerical coefficients of terms that share the same variable factors raised to identical powers, while preserving the variable part unchanged. For instance, given two like terms ax^n and bx^n, where a and b are coefficients and n is the exponent, the sum is (a + b)x^n and the difference is (a - b)x^n.[12] This rule applies only after confirming that the terms are like, meaning their variable components match exactly in both base and exponent. The underlying principle draws from the distributive property, which allows factoring out the common variable: for example, $3x + 5x = (3 + 5)x = 8x.[1] Similarly, subtraction follows the same form, such as $7y^2 - 2y^2 = (7 - 2)y^2 = 5y^2.[12] These operations maintain the integrity of the algebraic structure by altering only the coefficients. Common examples illustrate the rule directly: $4x + 2x = 6x or $9m^3 - 3m^3 = 6m^3.[1] In each case, the coefficients are combined arithmetically, ensuring no alteration to the variable expression.Simplification Process
The simplification process for algebraic expressions involves a structured approach to identify, group, and combine like terms, thereby reducing the expression to its simplest form while preserving its value. This method relies on the properties of addition and the commutative and associative laws to reorganize terms without altering the overall expression. By systematically applying these steps, one can efficiently simplify complex expressions containing multiple terms. The process follows these key steps:- Identify all terms: Begin by listing every individual term in the expression, including constants, variables, and their coefficients, while noting any signs (positive or negative) associated with them. This ensures no term is overlooked during subsequent grouping.[13]
- Group like terms: Rearrange the terms using the commutative property to place those with identical variable parts—such as the same variables raised to the same powers—adjacent to one another. Unlike terms, which differ in variables or exponents, remain separate. The core operation here is combining like terms by focusing on their shared structure.[1][14]
- Add or subtract coefficients: For each group of like terms, add or subtract their numerical coefficients, retaining the common variable part unchanged. Pay close attention to signs, as subtraction of a positive coefficient is equivalent to adding a negative one.[13][1]
- Write in standard form: Express the simplified result by arranging the remaining terms in a conventional order, such as descending powers of the variable for polynomials or grouping by variable type, to present a clear and organized final expression.[14]