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Section Daily Prep 17

You will become familiar with the definition and basic properties of a polynomial ring, including how to compute in a polynomial ring. In class, we will explore the division algorithm for polynomial rings and its consequences.

Subsection Resources for Learning

Use these resources to prepare for class and answer the questions below.

Subsection Important Terms

Definition 171. Ring of Polynomials over \(R\).

Let \(R\) be a commutative ring. The set of formal symbols
\begin{equation*} R[x]= \{ a_n x^n+a_{n-1}x^{n-1}+\dots+a_1 x + a_0 \mid a_i\in R, n\geq 0 \in \Z\} \end{equation*}
is called the ring of polynomials over \(R\) in the indeterminate \(x\). Two elements
\begin{equation*} a_n x^n+a_{n-1}x^{n-1}+\dots+a_1 x + a_0 \end{equation*}
and
\begin{equation*} b_n x^n+b_{n-1}x^{n-1}+\dots+b_1 x + b_0 \end{equation*}
of \(R[x]\) are considered equal if and only if \(a_i=b_i\) for all nonnegative integers \(i\text{.}\) (Define \(a_i=0\) when \(i\gt n\) and \(b_i=0\) when \(i\gt m\text{.}\))

Definition 172. Addition and Multiplication in \(R[x]\).

Let \(R\) be a commutative ring and let
\begin{equation*} f(x)=a_n x^n+a_{n-1}x^{n-1}+\dots+a_1 x + a_0 \end{equation*}
and
\begin{equation*} g(x)=b_n x^n+b_{n-1}x^{n-1}+\dots+b_1 x + b_0 \end{equation*}
belong to \(R[x]\text{.}\) Then
\begin{equation*} f(x)+g(x) := (a_s+b_s)x^s + (a_{s-1}+b_{s-1})x^{s-1}+\dots+ (a_1+b_1)x+(a_0+b_0)\text{,} \end{equation*}
where \(s\) is the maximum of \(m\) and \(n\text{,}\) \(a_i=0\) for \(i\gt n\text{,}\) and \(b_i=0\) for \(i\gt m\text{.}\) Also,
\begin{equation*} f(x)g(x)= c_{m+n}x^{m+n} + c_{m+n-1}x^{m+n-1}+\dots+c_1x+c_0\text{,} \end{equation*}
where
\begin{equation*} c_k=a_kb_0+a_{k-1}b_1+\dots+a_1b_{k-1}+a_0b_k \end{equation*}
for \(k=0,\dots,m+n\text{.}\)
Note that these rules are the familiar rules for adding and multiplying polynomials by distributing and collecting like terms, defined in terms of the operations in the ring \(R\text{.}\)

Definition 173. Polynomial Properties.

Let
\begin{equation*} f(x)=a_n x^n+a_{n-1}x^{n-1}+\dots+a_1 x + a_0 \end{equation*}
in \(R[x]\) with \(a_n\neq 0\text{.}\) Then we say that \(f(x)\) has degree \(n\text{,}\) denoted \(\deg(f)=n\text{,}\) the ring element \(a_n\) is the leading coefficient of \(f(x)\text{,}\) and if \(a_n=1_R\text{,}\) we say that \(f(x)\) is a monic polynomial. The polynomial \(f(x)=0\) has no degree and is called the zero polynomial. A polynomial of degree zero is called a constant polynomial.
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