# Terminology: degree of coefficient?

It's clear that a polynomial has a "degree". For instance our $$x^2 + 2x + 3$$ is a of degree two.

Can we apply the "degree" terminology to the coefficients? Here we have the coefficient tuple $[1\ 2\ 3]$. Can we call $2$ the "degree one coefficient" by synecdoche (meaning it's the coefficient of the degree one term of the polynomial)?

If not, how else do we pinpoint a particular coefficient position? In particular, without depending on the order of the syntax?

Even if the formula is given non-canonically as $2x + 3 + x^2$, how can we fill in "the ___ coefficient" in a way which unambiguously refers to the $2$?

• I would say it is "the coefficient of the first degree term" (or the degree 1 term). I'm not sure that's what you want, because it doesn't fill in the blank. – Jim H Sep 6 '17 at 18:22
• @JimH It's too verbose. Doesn't the coefficient have a property that we can name by a simple noun? (If "degree" isn't it, what is its name?) – 509-249-3447 Sep 6 '17 at 18:48
• I guess you could say "degree 1 coefficient" but that has a bit of ambiguity. – kcrisman Sep 6 '17 at 19:10
• I would use the "leading coefficient" for the coefficient of the highest degree term, and the "constant term" for the... um... constant term. However, I am not sure that the intermediate coefficients occur often enough in computation to require anything less verbose than the "coefficient of the $n$-th degree term." – Xander Henderson Sep 7 '17 at 0:52
• @XanderHenderson Could you write that as an answer? – Tommi Brander Sep 7 '17 at 5:19

Personally, I would use "the leading coefficient" to refer to the coefficient of the highest degree term and "the constant term" to refer to the constant term. Aside from that, I am not sure that the intermediate coefficients occur often enough in computation to require anything less verbose that "the coefficient of the $n$-th degree term" or "the coefficient of $x^n$" (which only costs two extra syllables over "the $n$-the degree coefficient").

That being said, I can't see any harm in using "the $n$-th degree coefficient," as long as one is clear that this is non-standard terminology, which may momentarily confuse folk who have not been in your classroom.

• Nice answer. I feel "non-standard" terminology should be avoided at all costs. – Michael Sep 9 '17 at 14:44

In your example, I would call 2 the coefficient of the linear term, or the linear coefficient for short. Then 1 would be the quadratic coefficient. And 3 would be the constant.

In the following textbook, terms “$n$-th coefficient” and “$n$-th term” are used. A quote.

This ring is usually denoted by $R[[X]]$ and its elements are called formal series in the indeterminate $X$ with coefficients in $R$. If $f=(f_n)_{n\in\mathbb{N}}\in R[[X]]$, for every $n\in\mathbb{N}$, $f_n$ is called $n$-th coefficient and $f_n X^n$ the $n$-th term of the formal power series $f$. In particular $f_0 = f_0 X^0$ is called its constant term.

Source:

Menini, Claudia, and Freddy Van Oystaeyen. Abstract Algebra: A Comprehensive Treatment. Marcel Dekker, 2004.