Perhaps more interesting to students would be to take a more historical
approach and talk about how finding generic solutions to polynomials
was a major challenge. The quadratic equation was something of a break
through. Going on to tackle equations of the 3rd and 4th degree required
even more spectacular feats (like scaling a mountain).
Working a few quadratics, building a little computer program around
it and so on, makes sense, as well as going through the proof at
least once, step by step:
http://www.math.utep.edu/sosmath/algebra/quadraticeq/quadraformula/quadraformula.html
Then show how easy it is today, to solve these things.
I simply type a random equation in the 3rd degree (!) into MathCad:
2x^3 + x^2 - 4x + 9 = 0
and ask it to solve for x. I get back one real root (2 complex):
- A - 25/(36*A)-(1/6) where A = 3rd root of (523/216 + sqrt(199)/6)
A = approx -2.263
Wow.
This discussion leads naturally to the complex numbers (C) which
subsume the reals (R) -- stuff about how C was needed to get n
roots of any nth degree equation.
Transcendental numbers are not roots of any polynomial with
rational coefficients.
http://www.langara.bc.ca/mathstats/resource/onWeb/precalculus/reals/transcendentals.htm
Some factoring makes sense I suppose, but why drill and kill?
Saying the physicists use 2nd or 3rd degree polys needing roots
is one thing, but claiming that many of them use a lot of paper
and pencil time, when MathCad is sitting right there, is not
my experience of the discipline these days.
So I guess my idea of a useful lesson would be: use the web
to find out what mathematicians first cracked the 2nd, 3rd,
and 4th degree equations, to give generic solutions.
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Quadratic_etc_equations.html
is the best I could find in my search. It's a complicated story,
full of secrets, broken promises, competition -- and a broadening
of the rules to accommodate what we today call "complex numbers".
MathCad also factors 2nd degree trinomials BTW.
Kirby
PS: another interesting topic, is to introduce students to graph
theory and have them start a worksheet called "who knew who?".
Mathematicians will be the nodes, and if they were contemporaries
who came into contact in some way, draw an edge between these
nodes. In a "directed graph" you define a relationship with
direction, and you might give it some properties e.g.
A --- was a student of ---> B
C --- was a rival of ---> D
B --- had a high regard for ---> A
and so on. As students do the web searches into historical
topics, the graph will grow more complicated.
Another related approach is to define "schools of thought"
and then connect various mathematicians via a graph of
"affiliations".
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