6
$\begingroup$

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 $2$?

$\endgroup$
  • 1
    $\begingroup$ 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. $\endgroup$ – Jim H Sep 6 '17 at 18:22
  • $\begingroup$ @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?) $\endgroup$ – 509-249-3447 Sep 6 '17 at 18:48
  • $\begingroup$ I guess you could say "degree 1 coefficient" but that has a bit of ambiguity. $\endgroup$ – kcrisman Sep 6 '17 at 19:10
  • 2
    $\begingroup$ 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." $\endgroup$ – Xander Henderson Sep 7 '17 at 0:52
  • $\begingroup$ @XanderHenderson Could you write that as an answer? $\endgroup$ – Tommi Brander Sep 7 '17 at 5:19
7
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Nice answer. I feel "non-standard" terminology should be avoided at all costs. $\endgroup$ – Michael Sep 9 '17 at 14:44
2
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.