Category:Definitions/Polynomial Addition

This category contains definitions related to Polynomial Addition.
Related results can be found in Category:Polynomial Addition.

Let $\struct {R, +, \circ}$ be a ring.

Let $\struct {S, +, \circ}$ be a subring of $R$.

For arbitrary $x \in R$, let $S \sqbrk x$ be the set of polynomials in $x$ over $S$.

Let $p, q \in S \sqbrk x$ be polynomials in $x$ over $S$:

$\ds p = \sum_{k \mathop = 0}^m a_k \circ x^k$

$\ds q = \sum_{k \mathop = 0}^n b_k \circ x^k$

where:

$(1): \quad a_k, b_k \in S$ for all $k$

$(2): \quad m, n \in \Z_{\ge 0}$.

The operation polynomial addition is defined as:

$\ds p + q := \sum_{k \mathop = 0}^{\map \max {m, n} } \paren {a_k + b_k} x^k$

where:

$\forall k \in \Z: k > m \implies a_k = 0$

$\forall k \in \Z: k > n \implies b_k = 0$