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Section Day 17
This is an outline of the topics we covered in the seventeenth day of class. The skeleton notes are in a handout, which can be printed out using the printer icon at the top right of its section of the page for filling in during class. Filled notes for each day will be posted after class to Canvas.
Handout Thursday 7/16
Objectives: Advanced Learning Outcomes
During our class meeting, we will work on learning the following. Fluency with these is not expected or required before class.
State and apply the following mathematical results: Division Algorithm for
\(F[x]\text{,}\) Remainder Theorem, Factor Theorem, Polynomials of Degree
\(n\) have at most
\(n\) Zeros over a Field
Use the Division Algorithm to factor and find zeros in
\(F[x]\)
Algebraist of the Day.
Saunders MacLane , 1909-2005
American mathematician in algebra, topology, and one of the founders of
category theory
Co-author of
Survey of Modern Algebra and
Algebra with Garrett Birkhoff
President of MAA and AMS, 41 PhD students
Theorem 175 . Division Algorithm for \(F[x]\) .
Let \(F\) be a field and let \(f(x),g(x)\in F[x]\) with \(g(x)\neq 0\text{.}\) Then there exist unique polynomials \(q(x),r(x)\in F[x]\) such that
\begin{equation*}
f(x)=g(x)q(x)+r(x)
\end{equation*}
and either \(r(x)=0\) or \(\deg(r(x))\lt \deg(g(x))\text{.}\)
Proof.
Corollary 176 . Remainder Theorem.
Let
\(F\) be a field,
\(a\in F\text{,}\) and
\(f(x)\in F[x]\text{.}\) Then
\(f(a)\) is the remainder in the division of
\(f(x)\) by
\(x-a\text{.}\)
Proof.
Corollary 177 . Factor Theorem.
Let
\(F\) be a field,
\(a\in F\text{,}\) and
\(f(x)\in F[x]\text{.}\) Then
\(a\) is a zero of
\(f(x)\) if and only if
\(x-a\) is a factor of
\(f(x)\text{.}\)
Corollary 178 . Polynomials of Degree \(n\) have at most \(n\) Zeros.
A polynomial of degree
\(n\) over a field has at most
\(n\) zeros, counting multiplicity.
Proof.
