If c is a number and if we let g(X) = (X − c) in the
division algorithm, then we
see immediately (because the degree of g(X) is 1 so that r(X), being of degree
0, must
be a number) that f(c) = 0 if and only if r(X) = 0. Thus f(c) = 0 if and only if
f(X) = Q(X)(X − c), where degree of Q(X) is 1 less than the degree of f(X). We
put
this fact in the context of solving equations by defining a number c to be a
root of the
polynomial form f(X) if f(c) = 0. Then repeated applications of the
preceding fact
yields: if a polynomial form f(X) has degree n, then it has at most n roots.
Along this
line, the rational root theorem for equations with integer coefficients should
be mentioned,
although care must be exercised in the teaching of this theorem, in the
following sense.
Because the needed divisibility property of whole numbers for the proof of the
rational
root theorem is generally not proved in the earlier grades, this theorem tends
to be offered
for rote memorization without proof. As a result, a common pitfall is to mistake
the theorem to be a test of the existence of all real roots rather than merely a
test of the
existence of rational roots.
Many polynomial equations do not have any roots among real numbers, e.g., x4+1
= 0.
By extending the real numbers to complex numbers, the situation changes
drastically. It
is easy to introduce complex numbers if students are used to the number line.
What they
have seen thus far is that it is possible to add and multiply any two points on
the x-axis
of the coordinate plane. Now they learn to do arithmetic with points in the
coordinate
plane, as follows: for all real numbers a, b, c, d, we define
(a, b) + (c, d) = (a + c, b + d)
(a, b) · (c, d) = (ac − bd, ad + bc)
One then proves that the points in the plane can be added, subtracted ,
multiplied, and divided
in exactly the same way as real numbers. In other words, addition and
multiplication
of points in the plane obey the associative, commutative, and distributive laws ,
and for
any nonzero complex number z, there is a complex number z -1
so that zz -1 = z -1z = 1.
The resulting number system is called the complex numbers, to be denoted by C. A
more common notation for complex numbers is to write a+ib for (a, b), so that i
= (0, 1),
by definition, and so that all real numbers t (= t+i0) are just points (t, 0) on
the x-axis.
The definition of multiplication among complex numbers of course implies that i2
= −1,
i.e., (0, 1) · (0, 1) = (−1, 0). Any two complex numbers of the form a + ib and
a − ib are
said to be conjugates of each other. Notice that (a+ib)(a−ib) = a2
+b2, which is real.
Notation: 
.
The representation of complex numbers in terms of polar coordinates and de
Moivre’s
theorem are essential ingredients even in a short account of complex numbers.
The latter
provides a straightforward method of taking an n-th root of a nonzero
complex number
z, i.e., to find a complex number w so that wn = z. This discussion
necessarily brings in
the sine and cosine functions, thereby providing a nice tie-in with the earlier
discussion
of trigonometric functions. Note that de Moivre’s theorem provides n such n-th
roots,
which is in contrast with the case of real numbers where we can specify a unique
positive
n-th root of any positive number.
It was mentioned that the concept of a polynomial form can be expanded to
include
any coefficients whose addition and multiplication obey the associative,
commutative, and
distributive laws. We do so now by allowing the coefficients of polynomial forms
to be
complex numbers, and call such polynomial forms complex polynomial forms.
The
major reason for the introduction of complex numbers can now be stated:
Fundamental Theorem of Algebra Every complex polynomial form of positive
degree has a complex root.
The proof of this theorem is beyond the level of school mathematics, but
students
in algebra can achieve a firm grasp of the significance of this theorem by
exploring its
implications. The first consequence is to expand the previous argument using the
division
algorithm to conclude that every complex polynomial form of degree n can be
expressed
as a product
where 
are complex numbers. In particular,
every complex polynomial form of
degree n has exactly n roots (counting repeated roots). We note that this result
depends
on the validity of the division algorithm for complex polynomial forms, which in
turn relies
on the fact that any nonzero complex number has a multiplicative inverse. In the
case of
a complex polynomial form of degree 2, one can derive this result without
invoking the
Fundamental Theorem of Algebra. Indeed, the usual derivation by completing the
square
and the fact that every nonzero complex number has exactly two complex roots
lead to
an expression of the two roots by the quadratic formula. Incidentally, this
shows that
every real quadratic polynomial always has two roots, if complex numbers are
allowed.
Suppose the coefficients of a polynomial form f(X) are
real numbers. It can be
considered a complex polynomial form, of course, and therefore it is equal to a
product
as before. But since the coefficients of f(X)
are real, a basic
theorem of school algebra states that the roots
must come in conjugate pairs.
The proof of this fact, assuming the Fundamental Theorem of Algebra, is very
instructive
and should be mastered by every student. Because
,
and the coefficients of the latter are all real, we have proved that every
polynomial form
with real coefficients is the product of real linear polynomial forms and real
quadratic polynomial
forms without real roots.
We have thus far concentrated on polynomial forms with one symbol X. There is no
reason not to consider polynomial forms in more than one symbol. A case in point
is the
very natural question of whether there is a formula for (X +Y )n,
where X and Y are two
symbols and n is a positive integer. The answer to this question is given by the
so-called
Binomial Theorem .
The main impetus behind this question is the simple identity
A little bit more labor gives
If one is persistent and computes the 4-th, 5-th, and even 6-th powers of X + Y
, one
would perceive a certain pattern and come up with a guess. A legitimate approach
to the
binomial theorem in school is to take for granted that such a guess has been
made and
proceed to prove it. Define the binomial coefficients for whole numbers 0 ≤
k ≤ n by
where by definition, 0! = 1, and n! = 1 · 2 · 3 · · · (n − 1) · n for n > 0.
Then:
Binomial Theorem For all integers n ≥1,
There are many ways to prove this theorem, and one way is to introduce the
principle
of mathematical induction and use it to give a proof. The teaching of
mathematical
induction should explain and stress the intuitive idea behind it rather than
make it a
mechanical procedure. It is also important to emphasize the fact that
mathematical
induction can be used only when the correct guess of a formula has been made. It
cannot
be used to deduce the formula. Another ingredient of the proof of the Binomial
Theorem
is the identity:
This may be proved directly as an exercise in the addition
of fractions. Once this identity
is available, one must mention the construction of Pascal’s triangle on the
basis of this
identity. The proof of the Binomial Theorem then becomes a good exercise in
mathematical
induction.
As further illustration of the use of mathematical induction, one may use it to
re-prove
the formulas for the arithmetic series and geometric series.
Another proof of the Binomial Theorem can be given by considerations of
permutations
and combinations. We first give a different interpretation of the binomial
coefficients,
which can be proved by standard reasoning:
Now think of (X + Y )n as the multiplication of
n factors each equal to (X + Y ):
When the multiplication is carried out by the distributive
law, each term will contain k
Y ’s (and consequently n−k X’s), where k is equal to 0, 1, 2, 3, . . . , n−1, n
in succession.
For a fixed k, we want to collect all the terms containing k Y ’s, and there are
a total of
such terms because each such term comes from
any k of the factors (X + Y ) and
there are ways to pick these k factors from
the totality of n such factors. Thus when
collecting these terms, we get
This is then the Binomial Theorem.
Combinatorics and Finite Probability
The last proof of the preceding section naturally leads to elementary
considerations of
permutation and combination. The basic problems can be formulated abstractly as
the
following four:
(a) How many ways are there to place k distinctly colored balls in n distinctly
numbered boxes, so that each box holds only one ball?
Answer: 
.
(b) How many ways are there of placing k colored balls into n numbered boxes,
if each box can hold as many balls as we wish?
Answer: nk
(c) How many ways are there to place k balls of the same color in n numbered
boxes, so that each box holds only one ball?
Answer:
(d) How many ways are there of placing k colored balls into n numbered boxes,
where (let us say) p of the balls are of one color, q of the balls are of a
second
color, and r (r = k − p − q) of the balls are of a third color, if each box
holds
only one ball?
Answer: 
Combinations and permutations in turn lead naturally to (finite) probability,
e.g., if I
place 5 distinctly colored balls into 9 numbered boxes at random (each box
holding only
one ball), what is the probability that there are balls in the first two boxes?
The basic
concepts of probability should be discussed, if only lightly, such as the
concept of a sample
space, the fact that probabilities are numbers between 0 and 1, the fact that
the sum of
the probability of an event and the probability of the event not happening is 1,
and the
difference between dependent and independent events.
References
[Kodaira 1] Kunihiko Kodaira, Editor. Mathematics 1: Japanese Grade 10. American
Math. Soc., 1996.
[Kodaira 2] Kunihiko Kodaira, Editor. Mathematics 2: Japanese Grade 11. American
Math. Soc., 1997.
[NRC2001] Adding It Up. National Research Council, Washington D.C., 2001.
[Wu] How to prepare students for algebra, American Educator, Summer 2001,
Vol. 25, No. 2, pp. 10-17.
