A question last week (Hi, Zahraa!) led me to dig up some old discussions of how we define a polynomial (or monomial, or term) and, specifically, why the exponents have to be non-negative integers. Why can we only multiply, and not divide by, variables? Since we’ve been looking at polynomials, let’s continue.
What is a polynomial?
I’ll start with an introductory question from Erin in 1998:
Polynomial Basics and Terms I'm having a lot of trouble understanding polynomials. My teacher goes through a lesson one chapter a day so it's hard to keep up. If you could, I'd like you to help to explain them to me.
Doctor Santu explained all about them; I’ll excerpt the parts of interest here:
Polynomials are just combinations of powers of x, like: 7x^5 + x^4 - 4x^3 + 15x^2 + 11 That's basically all there is to what they are. If you want more, here's some vocabulary that goes with polynomials. The highest power of x in the polynomial is called the degree of the polynomial. The polynomial I wrote above has degree 5. I wrote the polynomial in descending order of powers. Not everyone does that; so in your homework, look carefully to see if there's a high power of x in the middle of the polynomial somewhere.
He didn’t explicitly state the fact that those powers of x must have positive, integer exponents. This is often omitted at first, because students may not even know yet about negative or fractional exponents; but it is what we mean.
We commonly write polynomials in descending order in order to make the degree easy to see (because the first term has that exponent); this also makes many things we do with polynomials easier. Some textbooks ask you to always do this; sometimes they will not do so in an exercise, to make sure you are paying attention!
Doctor Santu is talking only about polynomials in one variable, which is what the word often means; we’ll be occasionally seeing one with two variables, like $$5x^2y-2xy^2-x^2+3xy+2y^2+2x-3y+8.$$ Note that there is no standard way to define “descending order” in such a case; but what I have done here is a common way to attempt it: I put all the terms with total degree 3 first, and arranged those with the higher powers of x first; then I repeated with all terns with total degree 2, and so on. The degree of this polynomial is 3, the highest sum of the exponents on all variables in a term.
Each separate part of the polynomial that is combined using + and - signs to make up the whole thing is called a term. The above polynomial has 5 terms, starting with 7x^5, sometimes called the leading term because it's in front, then x^4, then -4x^3, and so on. (Minus signs are considered to belong to the term immediately following them.)
What he called a “combination of powers” is, specifically, a sum of multiples of powers of the variable. When there are more than one variable, each term is, say, a number times a power of x times a power of y.
Terms are the things that we are adding together. The leading term is the term we write first when we put them in descending order, not just whichever term we happen to write first. And the leading coefficient is the coefficient in the leading term.
It’s important to notice that we think of subtractions in terms of addition here, by attaching the subtraction sign to the number (called the coefficient) as a negative sign. Here, we are viewing $$7x^5+x^4-4x^3+15x^2+11$$ as $$(\mathbf{7}x^5)+(\mathbf{1}x^4)+(\mathbf{-4}x^3)+(\mathbf{15}x^2)+(\mathbf{11}),$$ where I put the coefficients in bold.
There are special names for low-powered terms. The term without any x at all, in the above case the 11, is called the constant term. The term with only x is called the linear term (and our example just didn't have one). The term with x^2 is called the quadratic term, (15x^2 for our example). The x^3 term is called the cubic term (-4x^3 in our example), and the fourth-degree term is called the quartic term (x^4 in our example).
Of these, “constant term” is the most common.
What about negative exponents, or division?
Sometimes you’ll see “polynomials” with negative exponents. like \(3x^2-4x+1+4x^{-1}-x^{-2}\). A 2004 question from a teacher raised that issue:
Definition of Monomial Do monomials include quotients of variables? For example, would x/y or xy^-2 be considered a monomial? I have always thought that monomials included quotients of variables. However, the text book I am teaching from this year says that x/y is not a monomial, because monomials only include the products of variables or variables and constants. Any division of variables or variables with negative exponents would not be a monomial. Which definition is correct? Also, if x/y is not a monomial, does that mean that x/y + 3 is not a binomial? Thanks!
A monomial is a polynomial with one term; a binomial is a polynomial with two terms. The word “monomial” is sometimes also used to mean “term”. Division by a variable, as Marcy points out, is equivalent to a negative power.
Note that these examples are polynomials in two variables; that doesn’t significantly affect the question.
I answered, first stating the proper definition:
Hi, Marcy. A polynomial is a sum of terms, each of which is a PRODUCT of constants and variables, and therefore can be written as a coefficient (possibly 1) times a product of (positive) powers of variables (possibly none). So xy is a monomial, but x/y is not; and likewise x/y + 3 is not a binomial.
Because the question was about negative exponents, I didn’t specifically mention here that the exponents must in fact be non-negative integers. Zero is allowed (which is what I meant by “none”), but fractions are not. More on that as we proceed!
The main reason for this, I think (apart from the historical fact that when polynomials were first studied, negative exponents were somewhat disreputable, if not entirely unknown) is that many important theorems depend on having only positive exponents. If we allowed negative exponents, polynomials would not be continuous and defined everywhere, which is an important property. Other theorems about polynomials, such as those concerning the number of zeros, would similarly fail under the broader definition. Also, this definition works in many situations where the variables can take on values that are not numbers, and which can't be raised to negative powers, making it meaningful in abstract algebra.
Just as students often first learn about polynomials before learning about negative exponents (reciprocals) and fractional exponents (radicals), so that we don’t need (then) to explicitly exclude those possibilities, when polynomials were first studied, such ideas were barely conceivable!
The fact that all polynomials are continuous everywhere, and have all real numbers as their domain (or, in fact, all complex numbers) is part of what makes them as useful as they are.
The reference to variables not representing numbers is to abstract algebra; for example, in linear algebra we can apply a polynomial to matrices (arrays of numbers), which can be multiplied but not, in general, divided. This doesn’t directly affect the definition in ordinary algebra, but is yet another reason the definition is useful at higher levels.
But it is easy to forget this and accidentally extend the definition to allow negative exponents, since we often just talk about terms as products of powers (without specifying "positive" powers); and some books or teachers probably do that. In fact, it's been done within the Dr. Math archives! Functions like polynomials, but with negative powers, are called rational functions, and can always be written as the quotient of two polynomials.
The example of \(xy^{-2}\) could be written as \(\frac{x}{y^2}\) as a rational function; as another example, \(2x^2+x-1+2x^-1\) can be written as \(\frac{2x^3+x^2-x+2}{x}\).
Why, again?
The previous month I had answered a similar question, this one from a student who knew the definition, but not why:
Definitions of Monomials and Polynomials Why is 1/x considered not to be a monomial? It could be written as x^(-1) which is just one term. My math text says that it is NOT a monomial but it does not say why.
My answer to this amounts to a first draft of what we’ve just seen, so I’ll just quote a couple paragraphs I like:
Now, if you are asking why we define a monomial this way, excluding negative powers of variables, there may be more to discuss; but it's basically because defining polynomials that way makes for nice math; polynomials as defined have useful properties that would be lost if we changed the definition. When we start dividing by variables (rational functions), we find we can always put the expression in the form of a ratio of polynomials (with positive powers); if polynomials included negative powers it would be harder to describe the standard form of either.
The same is true for fractional powers, which produce radical equations.
I also quoted the following, which was originally in a private message to another Math Doctor who was pondering the issue:
Why do we make this restriction? That's just the definition, and it can be hard to say exactly why a definition is what it is. But I would guess that it arises from (a) the historical origin of polynomials, and (b) the usefulness of such a definition in various contexts. I'm not prepared to give a full defense of the definition, and don't even find any mention of its origin in the few math history books I have at my disposal. But my suspicion is that (a) it originated before negative exponents were considered, so that exponents were thought of as merely a shorthand for multiplication, and a monomial can be considered as involving multiplication ONLY; and (b) it is useful in contexts in abstract algebra where division and negative exponents need not even be defined! Moreover, many theorems involving polynomials would not apply to the broader definition. This is often what motivates a definition: it ties together a set of objects that belong in the same theorems.
Definitions are what they are because we have a reason to make them!
What makes a term a term?
Here is a 2007 question from a different perspective:
The Number of Terms in an Expression My text book says 4x + 4y + 4z is three terms, but 4(x + y + z) is only one term. I thought the rule that terms are separated by + or - would apply if after multiplying you still had terms separated by + or -. Other than somehow combining these unlike terms, what would cause this to become one term? I understand that 4xyz would be one term because it's all multiplied, not separated by + or -, but why would 4(x + y + z) be one term instead of three?
If you expand \(4(x+y+z)\), it becomes three terms; and it has additions inside; what makes it different?
I answered:
Hi, Michelle. Terms are things that are added. In splitting an expression into terms, we look only at the "top level"--the last step in the order of operations. So we see 4x + 4y + 4z as a sum of three products, (4x) + (4y) + (4z) The two additions are the last thing done.
So the last thing we do in evaluating the expression is to add the three terms.
In 4(x + y + z) the parentheses force you to do the additions FIRST, and the last operation done is the multiplication. That means there is no final addition step, and the whole thing is thought of as one term, which is formed as a product just as 4x is. We don't look inside the parentheses to see if there is any addition in there; we think of the whole parenthesized part just as one chunk that is being multiplied.
Terms are a matter of how the expression is written, not of the function itself; so rewriting it changes the order of operations. Only the last operation, as written, determines how many terms there are.
What if there’s only one term?
Here’s another quick question, from 1997:
Definition of a Polynomial My daughter had a problem that said "are the following expressions polynomials?" Most were straightforward but one was just the number 5. She said no and I agreed. She got it wrong. I called a professor of math at a local university, who explained that it is a polynomial because it is 5 times x^0 (x to zero power.) Okay, I get that. But one of the other problems was just x. The answer in the back of the book says no, x alone is not a polynomial. How is just x different from just 5?
Because the word “polynomial” basically means “many terms”, or a “sum of terms”, some people think one with only one term doesn’t deserve the name; but this is typical of mathematical vocabulary: A “sum of terms” can be a “sum” of just one term!
But that clearly isn’t the issue here, because both \(5\) and \(x\) have one term. It appears to be just an error in the book.
Doctor Pete answered:
Hi, Richard.
I'd say the book was wrong. For a more detailed answer, most mathematicians would define a polynomial this way:
A polynomial of one variable is a function P[x] of the form a[0] + a[1]x^1 + a[2]x^2 + ..., where the coefficients a[0], a[1], ... may be 0. The largest value of n for which a[n] is not zero is called the degree of the polynomial. More generalized polynomials occur when we replace x with some function of x, say Sin[x] or Exp[x]. There are also polynomials of more than one variable, which you can guess have individual terms of the form
a[k] x1^(p1) x2^(p2) ... xm^(pm).
Clearly, the value x falls under this category (a[1] = 1, a[k] = 0 otherwise), and so does 5 (a[0] = 5, a[k] = 0 otherwise). So does
1 + x + x^2 + x^3 + x^4 + ... = 1/(1-x),
which we usually don't like to call a polynomial since it has infinite degree. Rather, we call it a power series.
The usual way we state the definition doesn’t give room for infinitely many terms, but his approach requires saying that explicitly.
Form, not function
I’ll close with a tricky question from 2002:
Is (x^(1/2))^4 a polynomial? Is (x^(1/2))^4 a polynomial or not? The head of my math department said it is not since it is under the square root sign, but I said we have to simplify first to make it a polynomial. What is really the convention for deciding whether something is a polynomial? Thanks!
It would be obvious, from the definition we’ve given, that \(\sqrt{x}\), which could be written as \(x^{\frac{1}{2}}\), is not a polynomial, because the exponent is not an integer. But we can simplify \(\left(x^{\frac{1}{2}}\right)^4\) to \(x^2\). That looks like a polynomial, doesn’t it?
I answered:
Hi, Mario.
You can't really simplify this expression to a polynomial; in doing so, you lose the fact that it is not defined for negative x. (That is true if you are working with real numbers, because you can't take the square root; and also if you allow complex numbers, because then "the" square root is not defined; there are two roots, and no way to define a single principal root.) So your function does not have the right domain to be a polynomial. The same would be true for
x^2 - 1
-------
x - 1
In simplifying our expression, we need to explicitly state the domain: $$\left(x^{\frac{1}{2}}\right)^4\text{ becomes }x^2,\;x\ge0$$ With that domain restriction, it is not a polynomial!
One of the key reasons we gave for defining a polynomial as we do is that the domain always consists of all real numbers. Anything we do to it that reduces the domain must prevent it from being a polynomial.
My new example involves a division rather than a radical, but it, too, simplifies to a restricted polynomial: $$\frac{x^2-1}{x-1}=\frac{(x+1)(x-1)}{x-1}=x+1,\;x\ne1.$$ In doing this simplification, we need to explicitly state that the domain (implicit in the original form, but not in the new form) is \(x\ne1\).
Strictly speaking, a polynomial is a specific _form_ of expression, not just any function equal to such an expression; I wouldn't even say in this strict sense that x(x+1) is a polynomial, because it is not written in polynomial form. In cases like this the work is so trivial that we usually don't bother to distinguish between "being a polynomial" and "being able to be written as a polynomial", but there is still a difference. See the definition here, which calls a polynomial an expression of a certain form, not a function with a certain behavior:
http://mathworld.wolfram.com/Polynomial.html
The definition there is:
A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. A polynomial in one variable (i.e., a univariate polynomial) with constant coefficients is given by
\(a_nx^2+\dots+a_2x^2+a_1x+a_0.\)
Next week we’ll continue with a few questions that would have made this post too long, focusing on the degree of a polynomial.