Polyadditing

Polyadditing is the operation of combining two polynomials by adding their corresponding coefficients. In standard mathematical usage, this is called polynomial addition. The term polyadditing may appear as an informal or colloquial synonym. The operation is defined in the polynomial ring R[x] over a commutative ring R.

Definition: If p(x) = a_0 + a_1 x + ... + a_n x^n and q(x) = b_0 + b_1 x + ... + b_m x^m, then p(x) + q(x) = c_0 + c_1 x + ... + c_k x^k, where k = max(n, m) and c_i = a_i + b_i with a_i = 0 for i > n and b_i = 0 for i > m.

Properties: The polyadditing operation is commutative and associative. The zero polynomial acts as the additive identity, and every polynomial has an additive inverse given by -p(x), since p(x) + (-p(x)) = 0.

Examples: Let p(x) = 3x^2 + 2x + 1 and q(x) = 5x^3 + x^2 + 4. Their sum is p(x) + q(x) = 5x^3 + 4x^2 + 2x + 5. Another example: p(x) = 2x^4 + x and q(x) = 3x^2 + 7. Then p(x) + q(x) = 2x^4 + 3x^2 + x + 7.

Applications and notes: Polynomial addition is a building block of many algebraic procedures and is implemented in computer algebra systems. For multivariate polynomials, addition is performed by summing coefficients of like monomials. In all cases, the operation is performed coefficient-wise, aligning terms by their degree. While “polyadditing” may be used informally, the conventional term is polynomial addition.