Like Terms
Pre-AlgebraLike terms are terms in an algebraic expression that have the same variable(s) raised to the same exponent(s) and can be combined by adding or subtracting their coefficients.
Definition
Like terms are terms that have the exact same variable part. You can add or subtract like terms because they represent the same "type" of quantity.
Example
$3x$ and $7x$ are like terms because both have $x$. You can add them: $3x + 7x = 10x$. But $3x$ and $7y$ are NOT like terms because $x$ and $y$ are different variables.
Key Insight
Think of it like fruit: $3$ apples + $7$ apples = $10$ apples, but you cannot add $3$ apples and $7$ oranges into one type of fruit.
Definition
Like terms have identical variable parts, including the same variables raised to the same powers. Coefficients do not need to match. To combine like terms, add or subtract their coefficients and keep the variable part unchanged.
Example
$5x^2$ and $-2x^2$ are like terms: $5x^2 - 2x^2 = 3x^2$. The terms $4xy$ and $9xy$ are like terms: $4xy + 9xy = 13xy$. However, $4xy$ and $4x^2y$ are NOT like terms because the exponent on $x$ differs.
Key Insight
Combining like terms is essentially using the distributive property in reverse: $5x + 3x = (5 + 3)x = 8x$.
Definition
Two terms in a multivariate polynomial are like terms if and only if they share the same monomial (same variables raised to the same powers). In a polynomial ring, like terms correspond to the same basis monomial, so combining them is simply adding scalar coefficients in the underlying ring.
Example
In $R[x, y]$, the terms $6x^2y^3$ and $-4x^2y^3$ are like terms sharing monomial $x^2y^3$. Their sum is $(6-4)x^2y^3 = 2x^2y^3$. Terms $6x^2y^3$ and $6x^3y^2$ are not like terms despite having the same total degree.
Key Insight
Identifying like terms is equivalent to grouping elements by their monomial basis vector. This is the fundamental operation that makes polynomial simplification algorithmically tractable.