Adding Polynomials
AlgebraAdding polynomials means combining like terms from two or more polynomials to produce a single simplified polynomial.
Definition
Adding polynomials means combining two polynomials by adding together the terms that have the same variable and exponent (called like terms).
Example
$(3x^2 + 2x + 1) + (x^2 - 5x + 4) = (3+1)x^2 + (2-5)x + (1+4) = 4x^2 - 3x + 5$.
Key Insight
Only like terms can be combined. $x^2$ and $x$ are NOT like terms (different exponents). Think of it like combining apples with apples, not apples with oranges.
Definition
To add polynomials, identify and combine all like terms (terms with the same variable raised to the same power). The result is a new polynomial whose degree is at most the maximum degree of the addends.
Example
$(2x^3 - x + 6) + (-2x^3 + 4x^2 + 3x - 1) = (2-2)x^3 + 4x^2 + (-1+3)x + (6-1) = 4x^2 + 2x + 5$.
Key Insight
Adding polynomials can reduce the degree if the highest-degree terms cancel. The sum of two degree-$3$ polynomials can be degree $2$, $1$, $0$, or even the zero polynomial.
Definition
Polynomial addition is a vector space operation: $F[x]$ is a vector space over $F$ (and a ring under multiplication). Addition is component-wise on the coefficient vectors $(a_n, \ldots, a_0)$. The set of polynomials of degree at most $n$ forms an $(n+1)$-dimensional subspace. Adding two polynomials corresponds to adding their coefficient vectors.
Example
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
The coefficient-vector representation of polynomial addition connects algebra to linear algebra. Polynomial operations on a fixed degree subspace can be represented as matrix operations, enabling algebraic computation over function spaces.