Like Terms
Like 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 share the exact same variable part, meaning the same variables raised to the same powers; their coefficients do not need to match. Because they represent the same "type" of quantity, you can combine like terms by adding or subtracting their coefficients while keeping the variable part unchanged. 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
$3x$ and $7x$ are like terms because both have $x$, so $3x + 7x = 10x$, but $3x$ and $7y$ are not like terms because $x$ and $y$ are different variables. Likewise $5x^2$ and $-2x^2$ combine to $5x^2 - 2x^2 = 3x^2$, while $4xy$ and $4x^2y$ are not like terms because the exponent on $x$ differs. In $R[x, y]$, the terms $6x^2y^3$ and $-4x^2y^3$ are like terms sharing monomial $x^2y^3$, so their sum is $(6-4)x^2y^3 = 2x^2y^3$, whereas $6x^2y^3$ and $6x^3y^2$ are not like terms despite having the same total degree.
Key Insight
Think of it like fruit: $3$ apples plus $7$ apples make $10$ apples, but you cannot add $3$ apples and $7$ oranges into one type of fruit. Combining like terms is really the distributive property in reverse, $5x + 3x = (5 + 3)x = 8x$, and identifying like terms is equivalent to grouping polynomial terms by their monomial basis vector, the fundamental operation that makes polynomial simplification algorithmically tractable.