Thursday 7 December 2023

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
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?

A few days later I got a comment:  "
Blogger Nate said...

As a matter of fact, I just learned this method today from a few of my students who refer to it as 'slide and divide' although I'm not sure if that will help you to trace the history." After a little searching I came across a solution to calling it "Bottoms Up".  The name relates to the nearly final state where after, in our example, you get to (x + 12/3) (x + 2/3), You can simplify the left side fraction to a whole number, (x+4) but the right side requires the "magic, bottoms up, so we take the 3 in the denominator and write it in front of the leading term on the top, ie (3x+2)... BLAH!!!!

I still have not found any historical references to this method and the creation of either name.  If you have information or sources, please share.

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