# The Major Topics of School Algebra

Without any knowledge of the historical background of the
logarithm, many students

have come to regard it as another function they need to learn to pass an exam,
with no

appreciation whatsoever of the almost magical property that log x changes
multiplication

to addition.

Fixing a base a, a > 0 and a ≠ 1. Then the functions
and g(x) = a^{x} are

inverse functions, the domain of f being all positive numbers. Thus

for all x

and

for all positive t

These two relations , which characterize the fact that
and a^{x} are inverse functions, are

the key to solving equations of the following type: find x so that
. Equations

of this type come up in applications, because nature seems to dictate that
natural growth

and decay processes be modeled by exponential functions, more specifically, by e^{x}
and

variants thereof. There is potential for real excitement here as simple carbon
dating

problems can be discussed in this context.

Problems about changing bases in logarithms are good for testing students’ basic
understanding

of the definitions, but should not be elevated to the status of a major topic.

(Compare the discussion about factoring quadratic polynomials .)

The last collection of functions to receive attention in school algebra are from
outside

algebra: the trigonometric functions. These are functions that are initially
defined only for

(we use radian measure), so the first order
of business is to extend the definition

of sine, cosine to 0 ≤ x ≤ 2π , and then to all values of x by
demanding **periodicity of
period 2π **, namely, sin(x ± 2π ) = sin x, and the same for cosine.
Thus sine and cosine

become defined on the whole number line . They turn out to be the prototypical periodic

functions, because in advanced mathematics one shows that any function f which satisfies

f(x±2π ) = f(x) for all x can be expressed in terms of sine and cosine in a precise sense .

The restriction to 2π as the period is more apparent than real because, if a positive

number c is given, then the function h(x) = a sin(2π x/c) is periodic of period c (i.e.,

h(x±c) = h(x) for all x), and has maximum value |a| and minimum value −|a|. Students

should have instant recall of the graphs of sine and cosine. In particular, sine is increasing

in the interval and therefore has an inverse function there. Similarly, cosine is

decreasing on [0, π] and it too admits an inverse function there.

One of the reasons that sine and cosine are important in mathematics and science is

that nature is full of periodic phenomena.

Once sine and cosine are fully understood, the study of
the other four trigonometric

functions becomes fairly routine. Tangent is the most important of these four
and its

graph too should be accessible to instant recall. Note that tangent is not
defined at all

odd integer multiples of .

We have mentioned the use of exponential and logarithmic functions to model
growth

and decay and the use of sine and cosine to model periodic phenomena. The use of

linear functions to model certain observable phenomena should also be mentioned
in this

connection. Generically, some observational data, when properly graphed, seem to
suggest

a linear relationship although the data do not appear to be strictly linear due
to inevitable

observational inaccuracies. In such situations, one employs the method of ** least
squares**

to arrive at a linear function that is a “best fit” for the data. The details of
the method are

beyond the level of school mathematics, but the main idea of the method can
nevertheless

be conveyed in a qualitative way. The use of computer software, or even a
programmable

calculator, to produce the “line of best fit” may be used, with guidance, to
advantage to

give an intuitive understanding of this process.

**Algebra of Polynomials**

Algebra at a more advanced level becomes an abstract study of structure. School

algebra, at some point, should introduce students to such abstract
considerations. The

study of polynomials provides a good transition from algebra as generalized
arithmetic to

abstract algebra. Rather than considering a polynomial as the sum of multiples
of powers

of a number x, we now consider sums of multiples of powers of a symbol X,

where the are constants (i = 0, 1, . . . , n), and
≠ 0. To avoid confusion with the

polynomials we have been working with so far, we will call such an f(X) a **
polynomial
form**, but will continue to refer to n as its

**degree**, the as its

**coefficients**, and

as its

**leading coefficient**. The term will also be referred to as its constant term.

As a matter of convention, we will omit the writing of a term if its coefficient is 0. Thus

2X

^{2}+ 0X + 3 is abbreviated to 2X

^{2}+ 3. We define two polynomial forms to be

**equal**

if their corresponding coefficients are pairwise equal. In particular, two equal polynomial

forms must have the same degree, because they must have the same leading coefficient.

Note that the concept of the equality of polynomial forms is a matter of definition and is

not subject to psychological interpretations. This should help clarify the current concern

about the meaning of the equal sign in the mathematics education research literature.

We now single out a special case of the equality of polynomial forms for further discussion

because it is a delicate point that can lead to confusion. Given a polynomial form

f(X), what does it mean to say f(X) ≠ 0? In this instance, the “0” can only mean the

“0 polynomial form”, i.e., “0” here means the polynomial form whose coefficients are all

equal to 0. Therefore, as polynomial forms, the statement f(X) ≠ 0 means that the two

polynomial forms f(X) and 0 are not equal. From the preceding definition of the equality

of polynomial forms, we see that f(X) ≠ 0 means there is at least one coefficient of

f(X) which is nonzero. Thus X

^{2}− 1 ≠ 0. The reason this may be confusing is that, if

one is not careful, one might think of X

^{2}− 1 ≠ 0 as the statement that the polynomial

x

^{2}−1 is never equal to 0 for any number x. Obviously the latter is not true, e.g., x

^{2}−1

is equal to 0 when x = ±1. In this instance, one has to exercise care in distinguishing

between a polynomial and a polynomial form.

We do not assume that we have any prior knowledge of the symbol X. The addition

or multiplication of polynomial forms therefore becomes a matter of definition, i.e., it

is up to us to specify how to do these arithmetic operations among polynomial forms

because such an f(X) is no longer a number. We do so in the most obvious way possible,

which is to define addition and multiplication among polynomial forms by treating X as

if it were a number, so that at least formally, dealing with polynomial forms does not

introduce any surprises. The idea is so simple that, in place of the most general definition

possible (which would require the use of symbolic notations not appropriate for school

mathematics), it suffices to indicate what is intended by a typical example in each case.

Thus let

and

where the and are numbers, and . Then by definition, their sum is

and their product is

In other words, the product of two polynomial forms is obtained by multiplying out all

possible terms and then collecting like terms by their powers. It is immediately seen that,

because the additions and multiplications among the coefficients are associative, commutative,

and distributive, so are the addition and multiplication of polynomial forms.

Why polynomial forms instead of just polynomials? This question cannot be answered

satisfactorily in the setting of school mathematics. The idea roughly is that, since in a

polynomial f(x), the variable x plays the primary role whereas the (constant) coefficients

play a subordinate role, it is conceptually more clear to disengage x from the coefficients

altogether. By singling out x as a symbol X in this manner, we open the way for X to

assume other values distinct from those of the coefficients. For example, in linear algebra,

one allows X to be a square matrix to obtain the so-called characteristic polynomial of the

matrix. Moreover, it will be observed that the addition and multiplication of polynomial

forms depend only on the fact that the coefficients obey the associative, commutative, and

distributive laws . Anything we say about polynomial forms that depends only on addition

and multiplication therefore becomes valid not just for real numbers as coefficients, but

also for any number systems that satisfy these abstract laws, such as the complex numbers

that will be taken up presently. This is an example of the power of abstraction and

generality in algebra.

In a limited way, we can illustrate the advantage of the abstraction by considering the

problem of division among polynomials. Imitating the fact that the division of polynomial

functions leads to rational functions, we introduce a

**rational form**as any expression of

the type

, where f(X) and g(X) are polynomial forms, and g(X) ≠ 0.

(Recall that g(X) ≠ 0 merely means that at least one coefficient of g(X) is not equal to

0.) We also agree to identify every polynomial form f(X) with the rational form .

For example, 0 is identified with . We again emphasize that as it stands, is just

a formal expression, and it is up to us to give it meaning. First, what does it mean that

two such expressions are equal? Given two rational forms and , we define

to mean

Thus, every rational form is equal to 0. In other words, we rely on the crossmultiplication

algorithm in fractions as a guide to define the equality of rational forms.

As a consequence, “equivalent fractions” is automatically
valid among rational forms, in

the sense that for all polynomial forms f(X), g(X), h(X), we have

We also define the addition and multiplication of rational forms by imitating
fractions:

One then shows that these operations are well-defined, and that a rational form
has

the desired property of the division of f(X) by g(X), in the sense that

Note also that every nonzero rational form
has a multiplicative inverse
. This

completes the analogy that the set of rational forms is to the set of polynomial
forms as

the rational numbers are to the integers.

It may be wise to soft-pedal the well-definedness of the addition and
multiplication of

rational forms in a school classroom and use the time instead on more
substantive things,

such as the division algorithm and its consequences (see below).

If we were constrained to discuss polynomials rather than polynomial forms, then
we

would be faced with the awkward situation regarding the domain of a rational
function

: it is not the number line but the points on the number line outside the
zeros of g(x),

so that the domain of a sum is the set of the point on the number line outside the

zeros of both g(x) and v(x), etc. But for polynomial forms, we have seen from
the above

discussion concerning a polynomial form being nonzero that there is no such
awkwardness.

The analogy of polynomial forms with whole numbers leads us to consider the
analog

of division-with-remainder among whole numbers in the context of polynomial
forms.

This is the important **division algorithm** for polynomial forms: given any
polynomial

forms f(X) and g(X), there are polynomial forms Q(X) and r(X) so that

f(X) = Q(X)g(X) + r(X)

where r(X) is either 0 or has a degree < the degree of g(X). The reasoning is
essentially

a repackaging of the familiar procedure of long division among polynomials. This
is a

basic fact about polynomial forms in the school algebra curriculum. We note
explicitly

that for the division algorithm to be valid, it is essential that beyond the
associative,

commutative, and distributive laws , the coefficients have the property that a
nonzero

number has a multiplicative inverse.

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