On FACTORING QUADRATIC EXPRESSIONS -- if you must

I have a bad attitude towards this topic!
The algebra curriculum has students factor quadratic expressions for what purpose -- to quickly recognize the zeros of carefully crafted quadratic functions? to factor because the author spent some time crafting examples that factor nicely?
If one is forced to play the factoring game, then this is how I personally play it.
Enjoy(?)

[@amandaserenevy2291]

I think there are times when factoring can be useful. I have found times when working on problems in dynamical systems or number theory when being able to move backwards from a polynomial to its factored form can lead to a format that is illuminating in the context of the problem or which makes the result simpler to comprehend. Also, moving from a form in which adding is the outside operation to one where multiplying is the outside operation is useful when trying to simplify an expression or find zeroes. I know you mentioned that, and that many of the problems involving this idea in algebra and calculus and differential equations are contrived. I still think there is some value in the idea that expressions can be factored in various ways and that these different forms can tell us interesting things. For example, if we use function fitting techniques to find the quadratic function for the triangular numbers T(n) =1/2n^2+1/2n, then it can be helpful to see it is the same as n(n+1)/2, and maybe look for a geometric meaning in that if one did not occur to us before.

[@JamesTantonMath]

G'Day Amanda! I know ... my attitude is bad here. Of course, I am reacting to the standard way this is presented to students in an algebra 1 course, which often is "factoring for the sake of factoring." Yes, factoring is useful. e.g. Graphing a polynomial in in factored from (eg. p(x) = (x-18)^3 * (x-6)^100 * (x+78)^11 * (x+100) ) is totally cool and fun, versus graphing its non-factored form (ick!). I guess my true stance is -- of course, knowing how to factor things is helpful and can shed insight and light onto a problem at hand. But so much textbook work is without a problem at hand to shed light on to. I guess I am railing against teaching these things without a need in sight -- or without subtly creating a need first. (Teachers, if given the means and freedom to teach with innovation and flexibility and joy, naturally do this for kiddos. Departmental demands, an assessment culture, and pacing guides and such, on the other hand, hinder that freedom and flexibility.)

[@MrKrause]

To some extent, the way factoring is taught, tested and emphasized is one of my pet peeves, too. kzbin.infoZGynRfLYlEE But your bad attitude-and I appreciate your honesty about it--is keeping you from passing on the enlightenment, motivation and priorities that you usually pass on to math teachers. So I will try... First, there is no need for guesswork if students just write out the factors of ac to take advantage of the Rational Roots Theorem. It's that theorem, and not the kindness of authors, that makes factoring a reasonable thing to try, and limits the possibilities. Too many math teachers were good at solving them by guessing without writing down the factors themselves and leave the students who want more structure unsatisfied. Second, I think you'd agree that the structure of polynomials is an essential topic of algebra. Sure, we should spend more time creating polynomials with shapes and zeros we want using factors before reversing the process. But I think spending a couple of weeks showing how to undo that by factoring with quadratic examples--the easiest polynomials to factor--is a reasonable use of time. I agree it gets emphasized beyond that because it's easy to make multiple choice test questions about it; teaching factoring should glorify the Zero Product Property more than the SAT. Third, the patterns of the times tables through 12 are interesting on their own and here and in your work and in many textbooks are just convenient concrete placeholders to demonstrate multiplication-like operations. Factoring is painful for students who haven't learned their times tables by high school, but it's totally reasonable for later math teachers and books to expect you to be able to recognize those patterns on sight, and there is value in practicing times tables and, as you call them, "logic puzzles."

[@JamesTantonMath]

Thank you for this thoughtful response. I just made another video about factoring, it's power, along the lines of your second point -- something I did in the classroom to the horror of my colleagues, but kiddos could see and do just fine and get a sense of how factoring can really provide insight and structure. I think my problem is that the examples we use are too small and limited to properly utilize the power, and so it doesn't feel powerful. Here's the URL: kzbin.info/www/bejne/boHRaWN3h5WAn7M Thanks tremendously!

[@lornacy]

Factoring for the sake of factoring! It is kind of fun ...

[@FuchsiaRiv]

Thank you so much!

[@arraymac227]

Is there a way to solve "p+q=A; p×q=B" beyond just knowing the combinations? The structure you get looks like a perverse Pascal's Triangle, I have noticed.

[@JamesTantonMath]

This is the irony of matters here! To solve p+q = A , pq = B, write q = B/p and get p + B/p = A. This then is the same as p^2 - Ap + B = 0 and you are back to just solving a quadratic. It's annoyingly circular!

[@arraymac227]

@@JamesTantonMath It does look like a 'fixpoint' with that system.

[@carultch]

@@arraymac227 There is the simplified quadratic formula, m +/- sqrt(m^2 - p). The m stands for mean, and the p stands for product (nothing to do with your p). Another application of this formula, is a more direct way of finding the Eigenvalues of a 2x2 matrix. 3Blue1Brown has a video on that one. So with your symbols, m = A/2, and p = B. Solving for two numbers, q and r, such that q + r = A, and q*r = B: q = A/2 + sqrt(A^2/4 - B)

[@PigsOnBook]

very nice. and these videos, are related to factoring and square videos, Finger and Toe Multiplication -- kzbin.info/www/bejne/nJywcmdjltSKo7csi=xTKLziQiPAj2Fydv - factoring in 5 instead of x. Just in case, you're wondering what counting hands video you saw that was like this video, there is, Number Bases in Society -- kzbin.info/www/bejne/a3q1oZSFiqx4eZYsi=1eEBsuSdvHXCEydK - 12 digits using 3 joints on 4 fingers.