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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?

Thanks for your thoughts,

• Tom

Your question is kind of two parts: one about a convention

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!

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.)