Adding Polynomials
Adding polynomials means combining like terms from two or more polynomials to produce a single simplified polynomial.
Definition
Adding polynomials means combining like terms, terms with the same variable raised to the same power, from each polynomial into a single simplified result, whose degree is at most the largest degree among the addends (and can be smaller if the highest-degree terms cancel). Formally, polynomial addition is a vector space operation: $F[x]$ is a vector space over $F$, and addition acts component-wise on the coefficient vectors $(a_n, \ldots, a_0)$, so the set of polynomials of degree at most $n$ forms an $(n+1)$-dimensional subspace.
Example
$(3x^2 + 2x + 1) + (x^2 - 5x + 4) = 4x^2 - 3x + 5$. $(2x^3 - x + 6) + (-2x^3 + 4x^2 + 3x - 1) = 4x^2 + 2x + 5$, note the $x^3$ terms cancel entirely, dropping the degree. Representing $3x^2 + x - 2$ as $(3, 1, -2)$ and $x^2 - 5$ as $(1, 0, -5)$, their sum is $(4, 1, -7)$, corresponding to $4x^2 + x - 7$.
Key Insight
Only like terms can be combined, $x^2$ and $x$ are not like terms since their exponents differ, much like combining apples with apples, not apples with oranges. The coefficient-vector view of addition connects polynomial algebra to linear algebra, letting polynomial operations on a fixed-degree subspace be represented as matrix operations over function spaces.