Bottoms Up Factoring, A Question of History

A few years ago(Dec 29, 2006) a lady named Nancy Kitt sent a question to the Teacher2Teacher Service at the Math Forum and asked about a factoring method called the Bottoms Up method:

One of my former students showed me the following method to factor
trinomials.
I want to know HOW and WHY this method works.
3x^2 + 14x + 8 Multiply AC, that is 3 x 8 = 24
Now look at B = 14. We are looking for two numbers
multiplied together to give 24 and added to give 14. The numbers will be
+12 and +2.

(x + 12)(x + 2)--- put the two factors 12 and 2 inside the parentheses,
but put x as the first term in both parentheses.

Now, since A was 3, divide the two factors 12 and 2 by 3

(x + 12/3) (x + 2/3)

12 will divide by 3 giving 4.

2 does not divide by 3. Therefore, multiply the x by 3, giving the final
factorization of (x + 4) (3x + 2).

(she followed this with a second example)...

This is the COOLEST method I've ever seen. However, I have NO CLUE HOW
or WHY it works!!!!!!

I want to use this method this semester, and I'd like to have an idea why
it works?????

I responded (helpfully, I hope) with a post to explain the substitution method

Well, the secret is that 8 = 24/3...

If you consider that the solutions of x^2 + bx +c = 0 are the same as the
solutions of 2x^2 + 2bx + 2c etc... then you are a step closer to
understanding the solution....

If we take 3x^2 + 14x + 8 = 0 and let x=u/3 (or u=3x) and substitute we get


(3(u/3)^2 + 14 (u/3) + 24/3) = 0 and now if we simplify the first term
we get

u^2/3 + 14 u/3 + 24/3 = 0

now if we multiply all terms by 3 we get

u^2 + 14u + 24... and solve to get the two solutions you had, u=12 and u=2,
but remember that we wanted x, not u, and x=u/3 thus the final solution...
(And then I added two other methods that are not well known or understood)

Then I posted a second note in case she might want some historical information...

Just a little addendum on the history of this method (I was writing up an
article on factoring and thought of your question). The substitution of Z=ax to
make a solution pliable dates back to the ancient Babylonian clay tablets
according to Boyer's History of Mathematics. They used it in order to make
a trinomial (ax)² + b(ax) =ac so that they could solve using their method
of completing the square. The idea of factoring had to wait a LONG time
until Thomas Harriot came up with it around 1600-1621 (he died in 1621 but
his method was not published until 1631, ten years after his death)..

By the way, I can not find any reference to "bottoms up" name for this...
can you help ME?

So several years later, I still wonder... does anyone have a clue how/why this term was applied, or any other detail about the history?

New Insights via Old Problems

While sending me some old documents recently in my quest to track down the first use of !n for the subfactorial symbol, Dave Renfro sent me an electronic copy of an old "Mathematical Questions from the Educational Times", 1880. With time on my hands I was looking through them when I came upon the one above. Not a remarkable problem, but it did remind me of a recent problem I came across that said something like "Prove that 2009 can be written as the difference of squares of two integers."
A no brainer, every odd number is the difference of two squares since (n+1)^2 -n^2 = 2n+1. But after reading the old problem, I began to wonder if 2009 could not be done as the difference of two squares in more than one way. As it turned out, it could... in fact, it could be expressed in that form in three ways... 1005² - 1004², and 147²-140², and again as 45²-4². Perhaps it would not have registered on me, but I noticed that the differences between the squares, 1, 7, and 41 were all factors of 2009.
So I tried 2007 = 3²(223) and sure enough it could be written as 1004²-1003², or as 336²-333², and also as 116²-107²; differences of 1, 3, and 9. The remaining factors, 223, or 669 (and of course 2007) were too big to be differences (later I realized they were sums, duh).

SO why??.. a moment's thought made me blush at my own ignorance... of course, algebra one... if a²-b² = n then factoring the difference of the squares gave us (a-b)(a+b)= n so a-b must be a factor of n... and suddenly I knew that 1999and 2011 could only be expressed as a difference of squares in one way; they are both prime. [Followup question for students, prove that neither of 1999 nor 2011 can be expressed as the sum of two squares... also show that 2009 CAN be expressed as the sum of squares, but in how many ways?]

Its a simple idea, so why did I have to struggle with a very old, very much harder problem to have it fall apart for me??? Now I'm going to go think about the even numbered years..... hmmm... but ONLY the leap years.

The Cheshire Math Curriculum

In a note to the Philosophical Magazine (January, 1846) by Professor J.R.Young of Belfast College, Working with the sum of infinite series he refers to the "evanescence" of x^inf when x < 1.. One of the things I love about old Math journals is the beauty of the language used. For those who may not know the word, it means to slowly disappear, to vanish.

I was reminded of this quote recently thinking about the evanescence of many of the ideas once common to the high school curriculum. I recently had a math teacher who teaches Alg I tell me that factoring was not in the curriculum for her students, and she would not be teaching it. I know many others for whom, although it is still in their curriculum, it will not be taught because they don't see a use for it. In a blog a teacher of introductory students in a community college stated, almost smugly, that they had never learned the quadratic formula. The apparent implication, "It is not needed because I don't use it and I'm the teacher."

I recently wrote some blogs about synthetic division, including a few things that some folks had said to their classes "can't be done" and received several notes reminding me that it was a waste of time, because they could just use long division.... BTW, guess what is being de-emphasized in elementary school these days...... evanescence...

Last year I noticed that Descartes Rule of Signs had essentially "evenesced" out of my Pre-calc sequence. The term and the idea are both absent from my new algebra II book, and appears only as the last problem (number 73) in an exercise set in the current Pre-calc/analysis book.

A brief treatment of vectors is still present, but seeming unsteady on legs as wobbly as a punch-drunk fighter. No vector equations (nor parametric if you see them as I do as almost interchangeable) in three space, although there is a unit on two space parametric equations to accomodate a few questions on projectile motion.

For most of the year I squeeze these in as extra items, one this week, one the next, so that by the time we get to this time of the year I can give a full couple of weeks to vectors and planes in three space and my best students will actually be able to write the equation of a line of intersection of two planes; or the foot of the altitude of a tetrahedron from its four coordinates.

It's not in the curriculum, and I have no answer when someone protests that the students will "NEVER have to use that." Maybe they are right..I know that long ago when I was selecting classes in North Side High School in Fort Worth, Texas, I could not imagine any time in my life when I would need to speak French... (guess where I am going back to on Spring Break because Christmas there was so wonderful).. and I can't imagine anyone sitting around doing math, punching buttons on their calculator or ??? and they suddenly think to themselves, "I wish I didn't know how to factor polynomials, and I can't wait to forget how to do synthetic division