Definition:Polynomial Addition

Definition

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:

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

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

The expression $p + q$ is known as the sum of $p$ and $q$.

Polynomial Forms

Let:

$\ds f = \sum_{k \mathop \in Z} a_k \mathbf X^k$

$\ds g = \sum_{k \mathop \in Z} b_k \mathbf X^k$

be polynomials in the indeterminates $\set {X_j: j \in J}$ over $R$.

This article, or a section of it, needs explaining. In particular: What is $Z$ in the above? Presumably the integers, in which case they need to be denoted $\Z$ and limited in domain to non-negative? However, because $Z$ is used elsewhere in the exposition of polynomials to mean something else (I will need to hunt around to find out exactly what), I can not take this assumption for granted. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

The operation polynomial addition is defined as:

$\ds f + g := \sum_{k \mathop \in Z} \paren {a_k + b_k} \mathbf X^k$

The expression $f + g$ is known as the sum of $f$ and $g$.

Polynomials as Sequences

Let:

$f = \sequence {a_k} = \tuple {a_0, a_1, a_2, \ldots}$

and:

$g = \sequence {b_k} = \tuple {b_0, b_1, b_2, \ldots}$

be polynomials over a field $F$.

Then the operation of (polynomial) addition is defined as:

$f + g := \tuple {a_0 + b_0, a_1 + b_1, a_2 + b_2, \ldots}$

Examples

Arbitrary Example $1$

Let:

\(\ds P_1\)

\(=\)

\(\ds x^2 + 2 x + 3\)

\(\ds P_2\)

\(=\)

\(\ds 2 x^2 + x + 5\)

Then:

$P_1 + P_2 = 3 x^2 + 3 x + 8$

Also see

• Results about polynomial addition can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): addition
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): addition