Is it mathematically proper to call the constant term in a polynomial a coefficient?

Question

Main character Althea asks about the square root of -1, and her mom (who's a math professor) tells the story of how the solution of cubic equations led to the creation of complex numbers.

One reader said I was using the word coefficients incorrectly and should fix it. That would make it more wordy. My question to you. Is the constant in an equation also considered one of the coefficients?

From the story:

Luca Pacioli was saying that there was no way, if you had somewhat random numbers as co-efficients, to solve things like this.”

Mom writes it all on the whiteboard, and I stare at it all.

(The text has examples here, but I don't see how to make Latex work here.)

“So the 2, 3, and 4 are the … co-efficients? And the a, b, and c mean you could have any numbers there?”

After this dialogue, Mom says "Yep". She could instead say "Mostly. We actually call that 4 and the c the constant." But I think that's clunky.

So. Is it mathematically proper to call the constant a coefficient?

Answer 1

There are, I think, two questions here:

1. Is it correct to say that the constant term in a polynomial is a "coefficient"? and
2. Is it reasonable for a fictional character in a YA novel to refer to the constant term of a polynomial as a "coefficient"?

The answers are, I think, "probably yes" and "absolutely yes".

---

Is the constant term a coefficient?

Whenever one asks "Is a [foo] a [bar]?", it is necessary to first look at what the definitions of a [foo] and [bar] are. In this case, there are a number of places we might look (basic dictionaries of English, elementary texts on these topics, as well as more advanced texts, etc). For reference, a few definitions:

1. From Oxford languages via Google:

   a numerical or constant quantity placed before and multiplying the variable in an algebraic expression (e.g. $4$ in $4x^y$).

2. From Wikipedia:

   In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or any expression; it is usually a number, but may be any expression (including variables such as $a$, $b$ and $c$). When the coefficients are themselves variables, they may also be called parameters.

   For example, the polynomial $2x^2 - x + 3$ has coefficients $2$, $−1$, and $3$...

   The constant coefficient is the coefficient not attached to variables in an expression.

3. From Precalculus by Paul Sisson (2nd ed):

   The terms of an algebraic expression are those parts joined by addition (or subtraction), while the factors of a term are the individual parts of the term that are joined by multiplication (or division). The coefficient of a term is the constant factor of the term, while the remaining part of the term is the variable factor. [p. 11]

   Polynomials are a special class of algebraic expressions. Each term in a polynomial consists only of a number multiplied by variables (if it is multiplied by anything at all) raised to positive integer exponents.

   The number in any such term is called the coefficient of the term... [p. 39]

4. From Advanced Algebra: an Introduction by Thomas Hungerford (2nd ed):

   Theorem 4.1: If $R$ is a ring, then there exists a ring $P$ that contains an element $x$ that is not in $R$ and has these properties:

   ...

   (iii) Every element of $P$ can be written in the form $$ a_0 + a_1 x + a_2 x^2 + \dotsb + a_n x^n, $$ for some $n\ge 0$ and $a_i \in R$.

   ...

   The elements of the ring $P$ in Theorem 4.1 are called polynomials with coefficients in $R$, the elements $a_i$ are called coefficients, and the special element $x$ is called an indeterminate. [pp. 81–2]

Other texts which I have on my shelves provide similar definitions. Based on this very quick survey of texts, I would expect that most sources treat the constant term as a coefficient—Hungerford and Wikipedia are explicit about this; Sisson is not quite explicit, but seems to imply this interpretation; and a basic English dictionary is ambiguous, but doesn't rule out this usage.

As such, I would say that almost any mathematician would not look askance at the text

Mom writes it all on the whiteboard, and I stare at it all. $$ 2x^2 + 3x + 4 \qquad\qquad ax^2 + bx + c$$ "So the $2$, $3$, and $4$ are the … co-efficients? And the $a$, $b$, and $c$ mean you could have any numbers there?"

This appears to be perfectly cromulent, and should be perfectly understood by any mathematically sophisticated reader (and is certainly correct usage in the context of a mother talking to a child in a YA novel).

---

Would a fictional character call the constant term a "coefficient"?

Why not?

Beyond the fact that (as noted above) the constant term is a coefficient in most contexts, fictional characters are allowed to say things which are wrong (even if this isn't wrong), or which elide details in order to simplify exposition or improve pacing.

Moreover, most people are not as precise when speaking, so even if a term is not quite right, I would imagine that many mathematicians would allow for a little bit of imprecision when speaking in order to make a larger point. So, even if one should not refer to the constant term as a "coefficient" (thought this appears to be perfectly fine), it is fair to call it a coefficient in the less precise world of spoken mathematics.

Answer 2

It is 100% accurate to call the constant term a coefficient. By definition, the constant term is just the coefficient of the $x^0$ term. A polynomial of degree $n$ (other than the 0 polynomial, undefined degree) is DEFINED to be

$$\sum_{k=0}^na_kx^k $$ where the $a_k$ terms are the coefficients, $a_n\neq 0$ It's flat out wrong to say the constant term isn't a coefficient.