Combining Like Terms
Combining like terms is the process of adding or subtracting terms that have the same variable part to simplify an algebraic expression.
Definition
Combining like terms means adding or subtracting terms that have the same variable: you only change the numbers in front (the coefficients), while the variable part stays the same, since terms with different variable parts cannot be combined. This is the key step in simplifying polynomial expressions and is really the distributive property applied in reverse, $ax + bx = (a+b)x$, with every combination step implicitly using that factoring. In a polynomial ring $R[x_1, \ldots, x_n]$, combining like terms corresponds to summing the coefficients of identical monomials to produce the canonical polynomial form: for $p = \sum a_\alpha x^\alpha$ and $q = \sum b_\alpha x^\alpha$, $p + q = \sum (a_\alpha + b_\alpha) x^\alpha$, where $\alpha$ is a multi-index and $x^\alpha$ is the corresponding monomial.
Example
$4x + 7x = 11x$: you add the coefficients $4$ and $7$, and the $x$ stays; similarly $9y - 3y = 6y$. Simplify $5x^2 + 3x - 2x^2 + 7 - x$: combine $x^2$ terms to get $3x^2$, combine $x$ terms to get $2x$, leaving $3x^2 + 2x + 7$. In $\mathbb{Z}[x, y]$: $(3x^2y + 2xy - 5) + (x^2y - 3xy + 7) = 4x^2y - xy + 2$.
Key Insight
$4$ apples plus $7$ apples equals $11$ apples: the fruit type (the variable) does not change, only the count (the coefficient) does. The polynomial ring $R[x]$ is a free $R$-module with the monomials $\{1, x, x^2, \ldots\}$ as a basis, so combining like terms is really vector addition in this module, adding coefficient vectors component by component.