Moving on to real math - and help with an exercise

cdetrio_at_gmail.com
Date: 01/01/05 

Date: 31 Dec 2004 16:36:03 -0800

I am a 3rd year college student with what I now believe to be a poor
preparation in mathematics. Like many others my entire education has
mostly revolved around computational, plug-and-chug problems. For
example, I've taken the standard three semesters of calculus, and
although I can recognize that there is a connection between the mean
value theorem and the fundamental theorem of calculus, integrating and
differentiating are still like magical hand-waving to me. It is moving
symbols around on paper according to formal rules. I'm more
comfortable with basic arithmetic, where I feel I can actually
understand the operations I'm performing. Since I'm hoping to practice
applied mathematics for fun and profit (chemistry, physics, molecular
biology), I think I want to learn higher maths as thoroughly and
rigorously as possible.

So it is apparent that it will benefit me to learn to read and write
proofs. Many of the greatest scientists were also mathematicians, and
I certainly have the desire to study pure mathematics. Perhaps I'll
eventually be able recognize when and how developments in pure
mathematics are applied to solve problems in other areas. Anyway, it
seems at some point in every mathematicians career they made the move.
I'm hoping that the time has come for myself, since studying any
textbook that teaches method over a theoretical development bores me.
In addition to taking the Proofs course at my school this semester,
I've been looking for books aimed at students like myself.

One book I found was recommended by many sources: Polynomials by
Edward Barbeau. Its subject matter is the theory of equations and the
preface appears to indicate that I am indeed the target audience ("high
school and college students who wish to go beyond the usual
curriculum"). It is part of the Problem Books in Mathematics series,
which means it is composed almost entirely of problems. Each section
of problems is preceded by a very short section of definitions with
little explanation. Often times the problems build on each other.

I was able to make decent attempts at the first ten problems. Almost
immediately, however, the style changed. "Show that" and "Prove that"
types begin to predominate. Unfortunately, almost every one of these
is over my head. Worse yet, the accompanying answers and hints are
too. I've browsed a fair amount of math books so I've seen this
before, however those books did not always claim me as the target
audience.

I don't know where to go from here in order to make this material
comprehensible to me. I read one book, Fundamentals of Mathematics by
Moses Richardson (last published 1966), which was the first book I read
that took an axiomatic approach to mathematics. It taught the basics
of logic, set theory, and developed the rules of arithmetic, algebraic
logic, and some other stuff with theorems and proofs. It was the book
that has motivated me to learn pure math. I didn't do many exercises
in it, as most of them seemed relatively easy. Still, I guess it
hasn't been good enough preparation for an exercise like this from
Barbeau's book:

1.1 #12:
a) Is it possible to find a polynomial, apart from the constant 0
itself, which is identically equal to 0 (i.e., a polynomial p(t) with
some nonzero coefficient such that p(c) = 0 for each number c)? Try to
justify your answer. [This is not an easy question, although the
answer is not surprising. Examine your justification carefully to see
what you are assuming about polynomials; can you explain why it is
valid to use the properties you think you need?]
b) Use your answer to (a) to deduce that, if two polynomials assume
exactly the same values for all values of the variable, then their
respective coefficients are equal. [Thus, there is only one way, up to
order of wriing down the terms, of presenting a polynomial as a sum of
monomials.]

It comes with a hint as well:

1.12 (a) The constant term is the value of the polynomial at 0.
(b) The difference of the two polynomials is identically zero.

At this point, I would have no idea of how to proceed. How do I
develop this amazing skill, to produce something like this, the given
answer:

1.12 (a) Let p(t) = an*t^n + ... + a1*t + a0 be such a polynomial.
Then a0 = p(0) = 0. For any real nonzero c, an*c^n-1 + ... + a1 =
p(c)/c = 0. (We do not know that the left side vanishes at c = 0
without further development; in order to avoid this issue, we need to
make a more elaborate argument at this point.) Suppose, if possible,
a1 != 0. Choose c such that 0 < c < 1 and
2c*(|a2| + ... + |an|) < |a1|.
Then
|an*c^n-1 + a(n-1)*c^n-2 + ... + a1|
>= |a1| - [|an|*c^n-1 + |a(n-1)|*c^n-2 + ... + |a2|*c]
>= |a1| - c*[|an| + |a(n-1)| + ... + |a2|]
>= |a1| - (1/2)*|a1| > 0,

a contradiction. Hence a1 = 0, and, for all c != 0, an*c^n-2 + ... +
a2 = p(c)/c^2 = 0. Contine on to show in turn that a2, a3, a4, ... an
all vanish. Thus, p(t) must be the zero polynomial.

With more background, other proofs can be given. For example, the
conditions p(0) = p(1) = ... = p(n) = 0 leads to a system of n + 1
linear equations in the unknowns a0, a1, ..., an for which the solution
is unique. Alternatively, the identical vanishing of p(t) implies the
same for all of its derivatives, whence Taylor's Theorem identifies all
the coefficients as 0.

The reader might wish to reflect on the validity of the following
proof, which assumes the Factor Theorem. Let n be the degree of a
nonzero polynomial p which vanishes identically. Since p vanishes at
0, 1, 2, 3, ..., n, we can write
p(t) = t*(t-1)*(t-2)*...*(t-n)*q(t),
for some polynomial q. the degree of the left side is n while that of
the right side is at least n + 1, a contradiction.

Hopefully you can infer the typesetting. I cannot follow this a lick,
but am familiar with the symbols enough to be convinced that it lays
just outside of my grasp. What can I study in order to have a chance
at understanding this answer, seeing the necessity of the various
parts? Only then would I have a chance at producing such an answer
myself.

If anyone has an idea of how to expand on an answer like this to a
relative beginner, to make it more comprehensible to somebody with my
background, and is feeling generous enough to do so I would greatly
appreciate it. I desperately want to understand how the given answer
to part (a) reflects the note in brackets in the question, how the
given hints are relevant to the problem, and how part (b) could be
answered from the given answer to part (a). I have no idea how I can
get closer to understanding those things. Even after browsing the book
on proofs for the class I will be taking it seems a long way off.
Thanks

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