Definition:Polynomial over Ring/Informal Definition

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Informal Definition

Let $R$ be a commutative ring with unity.

Informally, a polynomial over $R$ is an expression of the form:

$a_0 + a_1X + a_2 X^2 +\cdots + a_n X^n$

where:

  • $n \in \Z_{\ge 0}$
  • $\forall j \in \left\{{0, 1, \ldots, n}\right\}: a_j \in R$
  • $X$ is an object sometimes called indeterminate


Addition

Polynomials can be added:

$\left( a_0 + a_1X + \cdots + a_n X^n \right) + \left( b_0 + b_1X + \cdots + b_n X^n \right) = (a_0 + a_1) + (a_1 + b_1) X + \cdots + (a_n + b_n) X^n$

Similarly, if $n<m$:

$\left( a_0 + \cdots + a_n X^n \right) + \left( b_0 + \cdots + b_m X^m \right) = (a_0 + a_1) + \cdots + (a_n + b_n) X^n + b_{n+1} X^{n+1} + \cdots +b_m X^m$

and if $n>m$:

$\left( a_0 + \cdots + a_n X^n \right) + \left( b_0 + \cdots + b_m X^m \right) = (a_0 + a_1) + \cdots + (a_n + b_n) X^m + a_{m+1} X^{m+1} + \cdots +a_n X^n$


This way, the sum of the polynomials $a_0, a_1X, a_2 X^2, \ldots, a_n X^n$ is the polynomial $a_0 + a_1X + a_2 X^2 +\cdots + a_n X^n$.

Multiplication