Which of these is the correct ordering:

a polynomial multiplied by a monomial

a monomial multiplied by a polynomial

if what I want to achieve is something like 3x(x+5)?

Here are my initial thoughts:

When we use the phrase "divided by," order must stay the same, as in 9 divided by 3 = 9/3, or a polynomial divided by a monomial would imply polynomial/monomial. Thus the ordering of 9 multiplied by 3 should be 9(3), and monomial multiplied by polynomial is correct?

But, if I have a person hit by a truck, it is the truck that did the hitting, not the person. So 9 multiplied by 3 implies that the 3 is doing an action to the 9, thus 3(9), and polynomial multiplied by monomial implies that the monomial should be distributed to each term of the polynomial, so polynomial multiplied by monomial is correct?

Which is technically correct? Which phrase paints the picture the correct way in the mind of the typical student?


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    $\begingroup$ i dont think it really matters considering multiplication is commutative. for division it does matter because that is not a commutative operation. you may be over thinking this... $\endgroup$ – celeriko Jan 30 '17 at 15:49
  • $\begingroup$ True, but I know from teaching Algebra that asking students to write the expression, "What number is 6 less than x?" a lot of students write 6-x rather than x-6. Even though addition is commutative, accepting only x+6 for "What number is 6 more than x?" rather than 6+x helps students build the habit of logically thinking through what the phase is saying in both expressions, and thus having a higher chance of getting the former correct. I'd like to be consistent with my use of "multiplied by" as well. $\endgroup$ – Elem-Teach-w-Bach-n-Math-Ed Jan 30 '17 at 15:59
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    $\begingroup$ I think the reason that there is no convention is because it is commutative. I might focus on which operations have this property and which don't, rather than trying to teach a distinction of language which may or may not exist. Alternately: consider "eight times two." I would need to study the English language to determine which is intended, two eights, or eight twos, since both makes sense from what we have been given $\endgroup$ – David Steinberg Jan 30 '17 at 19:03
  • $\begingroup$ @DavidSteinberg I agree again with what you say here, but still have a few points of contention. First, I would say the "times" symbol simply replaces the word "times," thus although multiplication is commutative, 8 x 2 would be the proper way to write the statement symbolically. I quibble more over "6 more than x" meaning x+6 since convention exists for less than--I want to be consistent. Second, I want to use the correct terminology because I'm trying to emphasize a specific ordering over the other while talking about polynomial multiplication. Thus phrasing it the right way is important. $\endgroup$ – Elem-Teach-w-Bach-n-Math-Ed Jan 30 '17 at 20:28
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    $\begingroup$ +1 For the question which is of the sort (careful terminology) that I like. $\endgroup$ – Daniel R. Collins Jan 30 '17 at 21:35

a monomial multiplied by a polynomial

This is best. Maybe.

It is important for students to understand the commutative property of multiplication (and other binary operators) and become procedurally fluent in using it with numbers, variables, monomials, polynomials and all other forms of algebraic expressions. However, after reading your comments carefully, I see what you really mean and I think there is an answer for it.

When we write a monomial, we try to follow the convention of writing numbers before letters. For example, by convention we write $4x$ not $x4$. Both are mathematically correct. But we are human and some ways a clearer than other ways. Also, using conventions reduces interpretive effort, allowing us to get more done in less time.

$4x^2+4x$ factorises to $4x(x+1)$. We prefer $4x(x+1)$ over $(x+1)4x$. Therefore, by convention, not mathematical rule, we put monomials before polynomials in a factorised expression.

That’s fine for mathematical expressions, but how about the word expression “a monomial multiplied by a polynomial”? Conventions get a little muddy here. If conventions get muddy, it is probably because there isn’t really a need for a convention. You could argue that the order in which things should appear in a sentence is the order in which they appear in the mathematical expression. So, for $4x$ we would say “$4$ multiplied by $x$” and not “$x$ multiplied by $4$”.

Some will argue that because $x$ multiplied by $4$ means $4$ acts on $x$, the phrase “multiplied by” is not commutative. However, when used in the context of mathematics, the binary multiplication operator is commutaive and the product of $4$ acting on $x$ is the same as the product of $x$ acting on $4$.

So, do you strictly apply the language syntax order that matches the desired conventional order in the mathematical expression?

I suggest not.

I have recently been reading Steven Pinker (The Sense of Style) and what he emphasises in writing is “flow”. Consider the flow of your writing. Not just a touchy-feely “flow” but an actual cognitive flow. If students are trying to visualise mathematical expressions as you speak or write, then it would be choppy to say B before A if you then by convention write A before B, unless they are so proficient with the syntax of English language that they would view this as an unforgivable travesty.

If you intend to write the monomial before the polynomial in a mathematical expression (e.g. $3x(x+5)$) then you should say $3x$ multiplied by $x+5$, or a monomial multiplied by a polynomial.

For the sake of consistency (and to avoid confusion with your students, especially those for whom English is not their native language), use the word order that matches the expression order – from left to right!


Drawing an analogy to the multiplication of matrices (where order does matter):

If I read "$A$ multiplied by $B$" I would probably think of $AB$ first, but probably I would realize that multiplication of matrices is composition of linear maps, and I would than wonder if it could be $BA$.

Hence, my conclusion is, that I would not write "$A$ multiplied by $B$" at all, but rather

$A$ times $B$

for $AB$.

In more a general non-commutative context, e.g. group theory, there are specialized notions, e.g. $gh$ means to "multiply $h$ from the left with $g$" or "multiply $g$ from the right by $h$. Also an element $g$ may "act from the left" or "act from the right" which seems to indicate that neither the left nor the right position is canonically more passive or active than the other.

In an obviously commutative situation like to multiplication of polynomials I would find it disturbing to read an additional "left" or "right" when referring to the multiplication and would ask myself if I missed some subtlety.

  • $\begingroup$ Technically not an answer to the OP, but +1 for offering a word alternative that solves the root issue the OP presented. Makes my essay below look like overkill. $\endgroup$ – lukejanicke Feb 1 '17 at 9:20

Consider the article at the MAA website by Keith Devlin (Executive Director of the Human-Sciences and Technologies Advanced Research Institute at Stanford University and The Math Guy on NPR's Weekend Edition), "What Exactly is Multiplication?". In it, he writes:

For instance, the mathematician's concept of integer or real number multiplication is commutative: M x N = N x M. (That is one of the axioms.) The order of the numbers does not matter. Nor are there any units involved: the M and the N are pure numbers. But the non-abstract, real-world operation of multiplication is very definitely not commutative and units are a major issue. Three bags of four apples is not the same as four bags of three apples. And taking an elastic band of length 7.5 inches and stretching it by a factor of 3.8 is not the same as taking a band of length 3.8 inches and stretching it by a factor of 7.5...

With multiplication you have a multiplicand (written second) multiplied by a multiplier (written first). The unit for the multiplier has to be sets of the unit for the multiplicand. For example, if you have 3 bags each containing 5 apples, then you can multiply to give


So while Devlin makes a distinction and naming convention (that I've seen elsewhere) between different factors, unfortunately he doesn't clarify the natural-language ordering.

Let's look at a few mass-market elementary algebra books to see how the section titles are phrased for the simplification of something like $3x(x+5)$:

  • Martin-Gay, Prealgebra & Introductory Algebra (Ch. 10): "Multiplying Monomials by Polynomials"
  • Bittinger, Intermediate Algebra (Ch. 4): "Multiplying Monomials and Binomials"
  • Ratti & McWaters, Precalculus: a right triangle approach (Ch. P): "Multiplying a Monomial and a Polynomial".

So based on that short survey, it seems like your second formulation is more customary.

  • $\begingroup$ I'm surprised the first book chose "multiplying a monomial by a polynomial* -- that seems to imply, for me, that multiplication "acts on the right" by default and that the less complicated object, a monomial, is acted upon by a more complicated object. I find both of these preferences odd, particularly "acting on the right" because functions act exactly oppositely with the standard $f(x)$ notation. I always prefer to act on the left, even when multiplying, for students. $\endgroup$ – pjs36 Jan 31 '17 at 2:05
  • $\begingroup$ Although I realize my last comment, of course, is not always true. It seems that when dealing with graphing and transformations, we prefer that only multiplication acts on the left (for example, the parabola in vertex form $y = (x - h)^2 + k$). So I guess it's not so clear-cut! But I think, in general, multiplier-on-the-left is something we tend to do in practice. $\endgroup$ – pjs36 Jan 31 '17 at 3:22
  • $\begingroup$ @MichaelE2: I agree with your observation in the context of real numbers. I might guess that Devlin was momentarily thinking about the axioms that make something qualify as a field (a common misstep, from what I think I've seen). $\endgroup$ – Daniel R. Collins Jan 31 '17 at 7:51

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