Unlike Terms
Unlike terms are algebraic terms that have different variable parts or different exponents, and therefore cannot be combined by addition or subtraction.
Definition
Unlike terms are terms that do not share the same variable part: either the variables themselves differ, or the same variable is raised to different exponents. Because they represent different types of quantities, unlike terms cannot be combined by addition or subtraction and must remain as separate terms in a simplified expression. In the canonical form of a polynomial, unlike terms occupy different positions in the monomial basis, so they cannot be added without altering the polynomial's structure. Every distinct monomial represents an independent dimension in the vector space of polynomials.
Example
$3x$ and $5y$ are unlike terms, and so are $4x^2$ and $4x$, since the exponents on $x$ differ ($2$ vs. $1$); the expression $3x + 5y$ must stay as two separate terms. In $7x^2 + 3x - 5y + 2$, the terms $7x^2$, $3x$, $-5y$, and $2$ are all unlike terms, since each has a distinct variable part. In the polynomial $2x^3 + 5x^2y + 3xy^2 + y^3$, all four terms are unlike because each corresponds to a distinct monomial in the basis for degree-3 polynomials in $x$ and $y$.
Key Insight
Trying to combine unlike terms is like adding apples and oranges and calling the result "apples," the math does not allow it, and mistakenly combining unlike terms is one of the most common errors in introductory algebra. The number of distinct monomials of degree $d$ in $n$ variables is $C(n+d, d)$, a combinatorial fact that drives the complexity of multivariate polynomial manipulation.