Combining Like Terms
Pre-AlgebraCombining 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); the variable part stays the same.
Example
$4x + 7x = 11x$. You added the coefficients $4$ and $7$. The $x$ stays. $9y - 3y = 6y$.
Key Insight
$4$ apples + $7$ apples = $11$ apples. The fruit type (the variable) does not change, only the count (the coefficient) changes.
Definition
Combining like terms is the key step in simplifying polynomial expressions. Identify groups of terms with identical variable parts, then add or subtract their coefficients. Terms with different variable parts cannot be combined.
Example
Simplify $5x^2 + 3x - 2x^2 + 7 - x$. Combine $x^2$ terms: $(5-2)x^2 = 3x^2$. Combine $x$ terms: $(3-1)x = 2x$. The constant $7$ is already simplified. Result: $3x^2 + 2x + 7$.
Key Insight
Combining like terms is the application of the distributive property in reverse: $ax + bx = (a+b)x$. Every combination step implicitly uses this factoring.
Definition
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. Formally, 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
In $\mathbb{Z}[x, y]$: $(3x^2y + 2xy - 5) + (x^2y - 3xy + 7) = (3+1)x^2y + (2-3)xy + (-5+7) = 4x^2y - xy + 2$.
Key Insight
The polynomial ring $R[x]$ is a free $R$-module with the monomials $\{1, x, x^2, \ldots\}$ as a basis. Combining like terms is vector addition in this module: adding coefficient vectors component-by-component.