# Is the constant term a coefficient?

I'm a baby boomer who was taught that the constant term of a polynomial is a coefficient, being the constant factor for the x^0 term.

That's not what's taught today.

Current text books are vague on their definition of coefficient, but, when they ask for a student to list all coefficient in a given polynomial, they do not include the constant term in their answer keys.

The Kahn Academy video on the topic does include the constant term as a coefficient, as does the related Wikipedia article.

However, all modern text books written to the Common Core that I have seen imply that the constant term is not a coefficient. A lot of web sites created by today's teachers who are teaching to the Common Core explicitly state that the constant term is not a coefficient.

If you have a polynomial of one variable, aren't the coefficients supposed to uniquely determine that polynomial? Aren't they all a computer program needs to know to work with polynomials? Without the constant term, you get the related polynomial that represents a vertical displacement of the original polynomial by the opposite of the constant term. But, you don't get the original polynomial.

Further, if we are talking about natural number constants, isn't each digit, including the ones' place, a coefficient of a power of 10? We don't segregate that digit by calling it something else; it's a digit similar to all the other digits.

Isn't mathematics supposed to be non-arbitrary and consistent?

I would like to see this taught as: after combining like terms, all constant factors, including the constant term, are coefficients. Two of the coefficients are special. The coefficient of the term with the largest exponent is the leading coefficient. The coefficient of the term with the smallest exponent (aka zero) is the constant term. That shouldn't be too much for eighth graders who were taught exponent properties, including the zero exponent property, in seventh grade.

Is this going to be yet another contributor to generational divide?

• @JoelReyesNoche: That just gives another reason why $0^0$ has to be defined to be 1. – Daniel Hast Nov 1 '18 at 1:02
• @ThomasMartin In general, $\left[f(x)\right]^{g(x)}$ where $f(x),g(x) \to 0$ as $x \to 0$, may different answers depending on the particular choice of $f$ and $g$. – Adam Nov 1 '18 at 1:34
• It would be interesting to have a link to sources where the constant term is not treated as a coefficient. i.e. It would be nice to know if it is a case of a (overly-)simplified definition which omits the constant term or an explicit statement that the constant term definitely is not a coefficient. – Adam Nov 1 '18 at 1:38
• What modern text books do is state explicitly the constant factors in front of explicit powers of one or more variables are coefficients. They then call the constant term the constant term, never explicitly declaring it to be a coefficient or not. However, in the answer keys to problems that ask students to list the coefficients, the answer key excludes the constant term. It seems they want to avoid having to state that x^0 is its associated power of the variable, perhaps thinking it is beyond the students' abilities to comprehend. Too bad. Anything to the zero power is zero. Easy peasy. – Thomas Martin Nov 1 '18 at 2:41
• I do agree that x ^ y is undefined as x -> 0 and y ->0. More germane to this discussion, since the power of the constant term is x ^ 0, that is definitely 1 as x -> 0 because the exponent is a constant 0. It's 1 for all x not equal to 0; the limit is therefore 1 at 0. – Thomas Martin Nov 1 '18 at 2:52

Is the constant term a "coefficient"

and one about a philosophy, which I perhaps find to be a more important question to answer.

Isn't mathematics supposed to be non-arbitrary and consistent?

Different fields of mathematics have different conventions; this can lead to some mathematicians using "sigma" for "standard deviation" in statistics and other mathematicians using "sigma" for a map defining a simplex in algebraic topology, and others using it to represent a group element in a permutation group. Even more egregiously, some people consider "Natural numbers" to include zero, and others consider "natural numbers" to be only positive integers. Even in higher mathematics, the definition of a word, or the statement of a theorem, may differ slightly between two textbooks. What is important is that 1) their definitions are clearly stated when they are introduced, and 2) that when a single mathematician uses the word in one body of work, that they are consistent. We are capable of handling the fact that different texts have slightly different definitions of what, exactly, a "coefficient" is, because we know how to interpret mathematical definitions and classify objects according to whether or not they fit that definition. That skill -- determining whether an object satisfies certain properties, or classifying mathematical objects based on definition, and recognizing properties of objects satisfying a particular definition -- is often an unstated goal in teaching mathematical vocabulary.

In that sense, I do not think that the definition of "coefficient" needs a universal definition for schoolchildren. When we teach students the meaning of mathematical words, the intent is often to build in them the ability to think about different "parts" of what they are learning and clearly articulate their thinking mathematically. In this sense, teachers and students being consistent with themselves and their textbook matters. However, being consistent across different states, or even different school districts, seems unimportant. Ultimately, they may grow up and realize that what their school called a coefficient differs slightly from what their college called a coefficient. This realization may help them understand the importance of an "operational definition" in many contexts!

• Thanks for your thoughtful reply! I guess my concern is that communication built upon very basic/low level definitions benefits greatly from consistency at that basic level. For example, I would love to be able to say, "The coefficients of a polynomial, together with their places in the polynomial, uniquely determine the polynomial, similar to the way the digits in natural number together with their places in the number uniquely determine the number." If some students are taught that the constant term is not a coefficient, while others are taught that they are, what do I say? – Thomas Martin Nov 1 '18 at 0:01
• I would say something like: >Some people don't use the word coefficient to include this, and some people do. When I use the word coefficient, I mean ____. This definition lets us use the constant term as a coefficient, and lets me say what I've said without very many words. If I were to use ___ meaning of coefficient, I would have to say "the coefficients, together with the constant term, determine the entire polynomial." Can you see why I choose to use this meaning for coefficient? – Opal E Nov 1 '18 at 18:52

It sometimes happens that slightly different definitions of the same word each have advantages and disadvantages. In such cases, I wouldn't be surprised to see some people supporting one definition and other people supporting a different definition. In the case at hand, though, I can't think of any advantages for defining "coefficient" to exclude the constant term. It seems to me that, if one adopts this definition, one will repeatedly have to say "coefficients and constant term". In addition to your example, "a polyomial is determined by its coefficients and its constant term", we'd have "to multiply a polynomial by $$7$$, multiply all its coefficients and its constant term by $$7$$" and "to add two polynomials, add the corresponding coefficients and add the constant terms." All of these become simpler and clearer if we just include the constant terms among the coefficients instead of treating them separately. (And I refuse to even think about multiplying two polynomials if I'm expected to think about constant terms separately from the other coefficients.)

• Thanks! Oh, and I fervently believe the natural numbers do not include 0. The natural numbers are so named because they come naturally to us (at least 1, 2, and 3 do). When we count, the first number we say is 1. We now know that before we say 1, we have 0--we start with nothing. But we start our count with 1. Formalizing 0 took centuries. I'm ok with educators creating the whole numbers to be the natural numbers plus 0. Educators can use that do clarify counting and distance on the number line without trying to redefine the foundational natural numbers. – Thomas Martin Nov 1 '18 at 1:05
• @ThomasMartin If you exclude $0$ from the natural numbers, then the convention that constant terms aren't coefficients becomes a little bit more reasonable than it is for me: One could say that the coefficients are the numbers that multiply $x^n$ for $n$ a natural number. – Andreas Blass Nov 1 '18 at 1:20

I have to admit I was skeptical of the OP's claim that contemporary textbooks do not identify the constant term as a coefficient, so I checked the first book that I had handy -- and indeed it does seem to be the case, in at least my sample of 1. Here is some evidence:

(Source: McDougal Littell Algebra 2, 2004.)

Note however that 2004 precedes the Common Core by about 6 years, so this cannot be laid at the feet of the CCSS.

• Example 2 there seems to have another somewhat odd feature. Normally, for a polynomial, we might talk about things like the coefficient of $x^2$. In the example, they list $1$ and $-3$ separately. It isn't wrong, just a bit weird. Perhaps the point is so that they can talk about combining the coefficients of like terms on the next page. – Adam Nov 2 '18 at 14:01
• @Adam I noticed that too, but note also that the question asks for the coefficients of the expression. The distinction between "expression" and "polynomial" is sometimes useful -- for example $3(x+2)^2$ and $3x^2+12x+12$ are two different expressions for the same polynomial. (Of course in this example it's not at all clear to me what we would call the "coefficients of the expression" $3(x+2)^2$. Maybe just $3$? – mweiss Nov 2 '18 at 20:40

Let's keep in mind that "mathematics" and "mathematics education" are different subjects. This question brings forth this distinction. At one end of the "Piaget" spectrum of mathematical stages of development, explaining that the constant term $$c$$ is the same as $$cx^0$$ might be too much cognitive overload. However, towards the other end of the spectrum, say in calculus, the construct $$\sum_{n=0}^N c_nx^n$$ needs to be a primary object.