I was having a conversation about math education in the USA. Would an excellent middle school student, high school student, or college freshman be able to answer this simple question:

Exercise 0 If $A(z) = a_0 + a_1 z + a_2 z^2$ and $B(z) = b_0 + b_1 z + b_2 z^2$, what are the coefficients $c_i$ in $C(z) = \sum_i c_i z^i = A(z) \times B(z)$? Only write the answer here, not the calculation (to do on scrap).
$\begin{aligned} \ & c_0 = \qquad\qquad c_3 =\\ \ & c_1 = \qquad\qquad c_4 =\\ \ & c_2 = \qquad\qquad \end{aligned}$

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    $\begingroup$ I don't think that age is the right parameter here. It is about mathematical maturity. At the very least, a student would need to understand the notation (so they probably need to have seen polynomials, e.g. in a high school algebra class, and sigma notation, which usually shows up in a high school or college precalc class (in the US)). But if you change the notation or scaffold it a bit, the actual content of the question would likely be doable by a high school freshman (or even a middle schooler). $\endgroup$
    – Xander Henderson
    Nov 9, 2022 at 14:43
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    $\begingroup$ It would also be interesting to know the context of this conversation. My immediate (probably overly defensive) reaction is that you (or someone else) was using a problem like this as an example of how terrible US math ed is (or how great European math ed is). But the "difficulty" of this problem is entirely due to the use of notation which may or may not be known to a particular student. $\endgroup$
    – Xander Henderson
    Nov 9, 2022 at 14:59
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    $\begingroup$ For a similar kind of example, most American high school students take trig, and would be expected to be able to evaluate $\sin(\pi/3)$. But would they know how to compute $\operatorname{haversin}(\pi/3)$? Should we bemoan the fact that students cannot answer a question which would have been considered simple and vital for navigation 100 years ago? This feels like trivia to me, not mathematics. :/ $\endgroup$
    – Xander Henderson
    Nov 9, 2022 at 15:00
  • $\begingroup$ @XanderHenderson I also don't know what a $\rm{haversin}$ is but I suspect that if you give me the definition, then I'll be able to figure it out at least with the precision required for navigation. The original question is self-contained, but yeah, for the school level I would just ask to "open parentheses in $(1+2z+3z^2)(4+5z+6z^2)$" and for the college level to "write a program that, given the coefficients of 2 polynomials, as arrays, outputs the coefficients of the product". Still, it all is an algorithmic stuff, so it doesn't really reflect much anyway. Is it what you meant by "trivia"? $\endgroup$
    – fedja
    Nov 9, 2022 at 22:33
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    $\begingroup$ "I also don't know what a haversin is but I suspect that if you give me the definition, then I'll be able to figure it out at least with the precision required for navigation." That is exactly my point. $\endgroup$
    – Xander Henderson
    Nov 9, 2022 at 22:34

2 Answers 2


I'm going to give an answer based on my feelings and memories, but I hope someone will give a more informed answer.

This question is testing

  1. Does the student know how to multiply polynomials with more than 2 terms? Many students just learn the FOIL mnemonic and are therefore confused by trinomials.

  2. Is the student comfortable with having many letters on the page, some of which are variables and some of which are coefficients?

  3. Does the student understand summation notation and the word "coefficient"?

Here are versions of the question which test subsets of these skills.

Just 1 Fill in the blanks: $$(1+2x+3x^2)(4+5x+6x^2) = \underline{\hspace{0.5 in}}+\underline{\hspace{0.5 in}} x +\underline{\hspace{0.5 in}} x^2 +\underline{\hspace{0.5 in}}x^3 + \underline{\hspace{0.5 in}} x^4.$$

Just 2 Fill in the blanks: $$(a_0+a_1 x)(b_0 + b_1 x) = \underline{\hspace{0.5 in}}+\underline{\hspace{0.5 in}} x +\underline{\hspace{0.5 in}} x^2.$$

1 and 2 Fill in the blanks: $$(a_0+a_1 x+a_2 x^2)(b_0 + b_1 x+b_2 x^2) = \underline{\hspace{0.5 in}}+\underline{\hspace{0.5 in}} x +\underline{\hspace{0.5 in}} x^2++\underline{\hspace{0.5 in}} x^3+\underline{\hspace{0.5 in}} x^4.$$

Now the question: At what ages can excellent students do this? I would love to see an answer which addressed where these skills lie in the common core curriculum. Being lame, I am going to give my subjective impression. Also, of course, there is the question of what is "excellent". I'm going to take this to mean the top few students in a typical school.

In my uninformed opinion, an excellent middle school student at a median middle school can do the "just 1" problem. MATHCOUNTS and AMC-8 are two competitions targeting students at this level, and this feels easy by their standards. However, if you notice, their problems are written in much more concrete language than your question, so I am not sure whether an excellent middle school student can do "just 1 and 2", let alone the original question.

I'm not sure much changes at high school? The level of the lower students moves up a lot: At middle school, IIRC, a large fraction of students take algebra but the median student doesn't whereas, by high school, they almost all have. But "the top math student from an unexceptional high school" is the sort of student I see a lot of teaching at U Michigan, and lots of them are still confused by skills 2 and 3.

Perhaps naively, I think that an excellent college freshman should be able to do all of this. Certainly, the top students coming into U Michigan would have no problem, but I would like to think that, at most universities, the top freshmen have all these skills.

At any of these ages, if you change "excellent" to "typical", the typical student struggles with the "just 1" version.


It would be no big deal for a strong US algebra 2 high school student, who is about half way through the course. Done their work on factoring and the like. Like around this time of year.

Stereotypically that is 11th grad (16-17ish), but many strong college prep students would take that course in 10th grade (15-16ish). They may or may not have memorized how to multiply trinomials, but anyone can do it the brute force way (multiply each thing in first by each thing in second and get 9 terms and then group them).

I don't think it's a fair question for kids younger than that. (Some might get it, but they really haven't practiced this sort of manipulation. Would be rare.)

For college kids, it should be doable, expecially if they are "excellent". I would not expect remedial students (who aren't even taking calculus first year) to handle it. But of course, they aren't excellent. They are remedial.

Of course we don't like in Lake Wobegone, so the average student is not an excellent one.


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