Very early in these notes we found that the natural numbers are inadequate for dealing with many situations encountered in everyday life (e.g., bank overdrafts), and our solution to that problem was to enlarge that collection to the system of integers. But we subsequently discovered deficiencies in this new collection (e.g., one cannot always divide), so we again added on, this time achieving the rationals. Low and behold, problems still occurred (e.g., very few square roots), and as a result we created the real numbers (in which, at least, all positive reals have square roots).
Even the reals have problems, the most obvious being that negative numbers still do not have square roots. This particular deficiency can be overcome by extending the number system yet one more time, in this instance achieving the complex number system, but we are not going to worry about the specifics of that construction^{43}. What we would like the reader to remember is that when some mathematical deficiency is found in a ``number system'', the problem can often be overcome by enlarging the system.
From our point of view the collection of polynomials amounts to an enlargement of the real number system for the purpose of giving a very simple way to describe many physical phenomena which ``change over time'', e.g., the height of a balloon in flight, or the amount of carbon 14 in a dinosaur bone^{44}. (In particular, the enlargement is for reasons other than to solve the ``no square root'' problem, and in that sense the construction of polynomials is an extension of the real numbers in a completely different direction than is the construction of the complex numbers.)
The definition of a polynomial found in standard elementary texts is usually along the following lines: a polynomial is an expression of the form
where and are real numbers, e.g., . (If you are comfortable with this approach to polynomials, fine! Skip the next two paragraphs, the subsequent sample test question, and begin again with the paragraph beginning ``As was the case ... ''.) So what is the problem? Answer: it is a ``definition in terms of notation'', i.e., it attempts to define something not by specifying the appropriate characteristics but by describing how ``it'' (whatever ``it'' is) is to be represented notationally. That's analogous to defining the number three (which lives in the untouchable mathematical world) to be the numeral 3 (which lives in the real world of death and taxes). Or, to dredge up a prior analogy, it's akin to confusing a picture of George Washington with George Washington. Until we are told the meaning of ``'' this is not a definition; it is slight-of-hand.
We intend to give a definition of ``polynomial'' more in the spirit late twentieth century mathematics. But you'll need to put your thinking caps on^{45}.
Let A denote the system^{46} of integers, or the system of rational
numbers, or the system of real numbers and let B denote a system
larger than A, e.g., the system of rational numbers, real
numbers or complex numbers respectively. (We will illustrate
parenthetically with the following example: A is the collection of rational numbers; B is the
collection of real numbers.) Choose an element of and consider the collection (read ``A square
brackets B'') of all non negative
integer powers of ; all multiples
of
such powers by numbers contained within A; and all (finite) sums and
differences of such multiples. Elements of
are
called
polynomials in with coefficients in A. (Example: Let
. Then
,
and (
) would be
typical elements of
, and therefore would be called ``polynomials in
with rational coefficients''. The coefficients of
the first polynomial are
and
, those of the second are and , and
that of the third is .) We say that
is algebraic over A if at least one of these polynomials, with at
least one non zero coefficient, is zero^{47}. (Example:
is algebraic over the
integers, because
.) When is not
algebraic over A it is (a) transcendental ( element) over A, e.g.,
is transcendental over the integers, and is also
transcendental over the rationals^{48}.
Transcendental elements (over A) are also called indeterminants (over A) or variables^{49}. When one speaks of a polynomial in an
indeterminant (or variable) (with coefficients in
A) one means a polynomial in (with coefficients in
A) in the sense just described, always with
the understanding that is a symbol
representing some otherwise unspecified^{50}
transcendental element over
A. Examples:
is a polynomial over the
integers, i.e., a polynomial with integer coefficients;
is a polynomial over the rationals, i.e., a
polynomial with rational coefficients; and is a
polynomial over the reals, i.e., a polynomial with real
coefficients. Note that
may also be considered a
polynomial over the rationals or reals, and that
may also be considered a polynomial over the reals. On the
other hand, is neither a polynomial over the integers
nor a polynomial over the rationals.
Sample Test Question : What is ?
Sample Responses :
The first answer fails to satisfy my need to have
mathematical symbols represent specific entities in the mathematical
world^{51}. (Moreover, the
``unknown'' terminology is a bit frightening, conjuring
up images of ``secret initiation rites'' which must
be attended before one can understand .) The
second answer does fill the need, because I have the freedom to
view as representing a number if I so desire, e.g., I could
let represent .
As was the case with real numbers we define to be . An integer (rational number, real number) may then be imagined as the polynomial with integer (rational, real) coefficients; polynomials of this form are said to be constant. With this ``think of as '' interpretation as our justification we henceforth regard the collection of polynomials in with integer (rational, real) coefficients as an enlargement of the system of integers (rational numbers, real numbers).
The reader needs to be aware of certain terminology associated with polynomials.
Notice that is a monomial: take in the definition. All other monomials are called non zero monomials.
For examples of polynomials which are not monomials consider and .
The exponent in a non zero monomial is the degree of that monomial. (The degree of the zero monomial is not defined^{52}.) Example: has degree ; has degree (since ).
In all subsequent definitions relating to non zero polynomials we always regard such a polynomial as having been rearranged (if necessary) into a sum of monomials of different degrees. An example will be seen in the very next item, wherein we think of as the polynomial .
The reader is assumed familiar with the arithmetical manipulation of polynomials, i.e., with addition, subtraction and multiplication thereof, but perhaps not with formal statements of the properties being used. These are now listed: the similarity with the formal laws of addition and subtraction of natural numbers, integers, rational and real numbers should not be ignored. Here are denote typical polynomials.
These laws are so familiar in algebraic calculations that they are usually applied unconsciously. For example, ``factoring out'' the so as to justify writing is a straightforward application of the distributive law (e), although this fact is somewhat obscured by the omission of the central equality in the unabbreviated computation .
We further illustrate the use of laws with a discussion of the example used
in (d), which (without comment) made use of the equality
How do we know this is true? Here is an explanation with more detail than you probably wish to see. The lines are numbered for reference in the discussion which follows the calculations.
The calculations are justified as follows.
Similarly, we choose to think of as added to the sum of and .
Note that we have not bothered with parentheses in connection with addition, i.e., precision is somewhat lacking in our explanation^{53}.
As a consequence of the fact that polynomials obey the same rules are real numbers we could also do the same calculation in the style of elementary arithmetic, i.e.,
Indeed, this is more likely the way the reader would have
done it. But it is simply an alternate way of writing the same thing,
although the placement of parentheses is a bit different.
The points to remember are: the laws (a)-(h) are what guarantee the
correctness of such calculations; wrong answers (which are not simply
careless mistakes) are usually due to
incorrect applications and/or misunderstandings of these
laws^{54}.
A Common Mistake : The assertion
is wrong. Or, what amounts to the same thing: the formula is correct. Students who feel the expressions are ``different'' would probably not question the (correct) equality , but they do not seem to appreciate the fact that manipulations of the indeterminant are governed by the the same rules which govern the manipulation of real numbers. Here's another example which can ``trip'' students. Is a correct formula? It certainly is. Why? Because and .
Here are some further examples of correct formulas:
;
;
(because
).
When doing polynomial arithmetic there are two important properties of degrees to keep in mind:
Powers of polynomials are defined exactly as were powers of real
numbers, e.g.,
means ``write down
five times and multiply all the entries
together^{55}''. And there is nothing new about the laws
of exponents:
they work exactly as before. Specifically, if and
are polynomials, and if and are non
negative integers, then
Examples^{56}: Let , , and . Then
The first and third laws of exponents also generalize as in (1.4) and (1.5), but for the sake of saving space we will omit examples.
In reading the next few paragraphs the reader should be reminded of a similar discussion in §1.
A polynomial divides a polynomial , in which case we write , if there is a polynomial such that . When this is not the case we say that does not divide and we write . Note from the additive property of degrees under multiplication that is impossible when the degree of is less than that of . Examples: (because , i.e., ); (because , i.e., ); .
When as in the previous paragraph one writes and refers to as the result of dividing by . The first two examples of the previous paragraph could thus take the form: and , whereas expressing the third example in the form would make no sense (at this point) because we have yet to discuss quotients of polynomials^{57}.
Notice that
makes sense
when and are monomials, say
and
, provided
. Indeed, under these assumptions we have
When a polynomial is a product of other polynomials we call that collection a factorization of and refer to the as factors of . In particular, is a factor of if , for in that case we have a factorization . Here is a concrete example: , which would more likely be written , is a factorization of
the factors are and .
A polynomial is irreducible if in any factorization it must be the case that at least one of and is a real number; otherwise is reducible, i.e., it admits a factorization in which both and have positive degree^{58}. Example: Any linear polynomial is irreducible. (Recall that a linear polynomial is one of the form , where . This can be factored as , or as , or as , etc., but by the additive property of degrees it cannot be the case that each of two factors involves .)
As will soon be seen, monic irreducible polynomials assume the role of prime numbers within the system of polynomials. But they are much easier to identify.
Examples: , and are irreducible, and the same is true of (because ) and (because ).
The number which arises in the proposition statement is called the discriminant of the polynomial . More generally, the discriminant of a quadratic polynomial is , but in elementary courses the factor is usually omitted, and we will follow this custom ^{59}.
The proposition is a bit deceptive: it can tell you if a given
polynomial can be factored, but when this is the case it is of no help in
finding a factorization^{60}. For
example, it tells us immediately that
can be factored, but it takes some
fiddling^{61} to come up
with
An exception to the difficulties alluded to in the previous paragraph
occurs with monic quadratic polynomials
: if
Proposition 6.3(b) indicates that factorization is possible,
i.e., if the discriminant is non negative, then one can
immediately write down that factorization. Indeed, simply check in
that case that
Examples: (i) For we have and therefore . From the formula, or by straightforward verification, we see that . (ii) .
Example (ii) of the previous paragraph is usually viewed from a
different perspective, i.e., as a particular case of the formula
As an immediate consequence we see that to factor , when it is known from Proposition 6.3(b) that factorization is possible, we simply look for two numbers which multiply to and add up to . For ``simple'' quadratic polynomials this method of factoring is generally faster than appealing to (6.5). Examples: (a) To factor look for numbers which provide a factorization of and sum to . Since we might try and , but then , and we thus move on to other combinations. Eventually we find that works, and therefore . Of course this polynomial could also be factored using (6.5). (b) (because and ). (c) (because and ).
When is a real number and
formula (6.5) generalizes to
Indeed, simply note that and then apply (6.5), with and in that formula replaced by and respectively. Example: .
There are similar (but less well-known) methods for factoring third and fourth degree equations, and one can actually prove (but the methods are quite sophisticated) that there is no analogous method for factoring arbitrary fifth degree equations^{62}.
But there are tricks that enable one to factor some high degree
polynomials. To illustrate these we first point out that formulas
(2.4) hold, precisely as written, when and
are regarded as polynomials, e.g., in polynomial form
(2.4c) gives
Taking and it follows, for example, that
Of course Proposition 6.3(b) guarantees that each of the polynomials on the right-hand-side can in turn be factored, but none of the formulas (2.4) will do the job, and we therefore leave the task unfinished^{63}.
This phrase is used when a term of the form is introduced into an expression without changing the value (of the original expression). Perhaps the most common example is the replacement of within an equation by , which would usually be summarized by the statement: ``To complete the square divide the coefficient of by two, square it, and add it''. But keep in mind that you must also subtract that quantity to keep things in balance. Example: .
Why would we ever want to do this? Because it often enables us to ``see'' a factorization (``factoring by completing the square''). Indeed, note that has the form of (2.4c), and therefore factors as , i.e., as . Thus factors as . Of course this particular factorization is more easily achieved using (6.6), but the general technique applies to a much wider class of examples.
Here's a more sophisticated application of the idea. Notice that we introduce rather than , and that we do so by adding and subtracting the quantity rather than dividing some coefficient by two and then squaring and adding.
This is, in fact, how we obtained (6.4).
The reader should note that the last few examples have simply been variations on a theme, i.e., that of , whereas (2.4) involves many other formulas. These are also available for use: ``completing the square'' is just the tip of the iceberg. Here is an example of ``completing the cube'' using (g) and (d) of (2.4):
We now present the polynomial analogue of the Fundamental Theorem of Arithmetic^{64} (Theorem 1.6).
Example:
is the
factorization of the monic polynomial
into irreducible monic polynomials. Here ( is simply the number of irreducible polynomials involved in the factorization), . Readers without computer access are may not wish to verify the example, but they should certainly recognize the analogy with the examples following Theorem 1.6.
Continuing with analogies from §1 we define the greatest common divisor or ``g.c.d.'' of a collection of monic polynomials to be the monic polynomial which divides all these polynomials, and is divisible by any other monic polynomial which has this property. Example: is the greatest common divisor of the three monic polynomials and . (To see that it divides all these polynomials simply note that and . That it is the greatest common divisor can be established by comparing degrees.) Notice that and also divide each of these polynomials, and that both also divide . Two monic polynomials are relatively prime when their g.c.d. is , e.g., and . (Remember that real numbers also qualify as polynomials.) The least common multiple or ``l.c.m.'' of a collection of monic polynomials is the the polynomial which is divisible by all these polynomials and which divides any other polynomial with this property. Example: The l.c.m. of the monic polynomials and is , i.e., the product .
When and are monic polynomials and the best one can do is the following^{66}.
The polynomial in the statement is called the remainder when dividing by . Examples: For and we have and ; for and we have and . The reader can verify both examples simply by working out . But to see how they were computed we need to review long division.
The calculations involved in working out the last example of the previous paragraph would usually appear on paper as
wherein and appear on the second line and
and appear in the first and last lines respectively.
This is an example of (``polynomial'') ``long division'', which is
assumed familiar to the reader. Nevertheless, recalling the
procedure might be worthwhile.
Step I : Write down and as shown, i.e., the first step is to write out the second line of the calculation shown above.
Step II : Divide the highest degree term of by the highest degree term of (i.e., calculate ) and write down the answer (i.e., ) above the highest order term of as shown next.
(The placement of
is simply to
remind us that we have now ``finished'' with the term it sits
above.)
Step III : Multiply by the term calculated in Step II (in this case ) and write the answer on Line 3 as shown. (Here and in all subsequent steps ``like powers'' of are aligned in columns as indicated.)
Step IV : Subtract Line 3 from , obtaining
Step V : Now repeat Steps II-IV, with replaced by what is in Line 4, leaving the term already in Line 1 (i.e., ) exactly where it is, but otherwise ignoring it. Specifically:
Step VI : Again repeat Steps II-IV, this time with replaced by what is in Line 6, leaving the terms already in Line 1 (i.e., ) exactly where they are, but otherwise ignoring them.
Specifically:
Step VII : Observe that we can no longer repeat Steps II-IV, with replaced by Line 8, because we cannot divide the highest order term on Line by the highest order term of . We have thus finished: the bottom line must be the remainder. Indeed, we have achieved the schematic which opened the discussion of long division.
When is a monic linear polynomial, i.e., has the form
, there is a much easier way to divide by .
An explicit example should suffice to convey this method of synthetic
division. We choose
and .
Step I : We begin by writing
Step II : We augment the table with the addition of two additional terms, as indicated.
Step III : We enter two more terms, as shown.
Step IV : By this point the reader should be catching on to the pattern. The next two entries bring us to
We note that if the final entry in a synthetic division calculation is zero (i.e., the right entry on the bottom line), then the linear polynomial ``evenly divides'' , i.e., there is no remainder, and therefore . For example, from the calculation