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WHAT WORKS IN EDUCATION The George Lucas Educational Foundation

Recently, I applied to a fellowship with Math for America, a program dedicated to improving mathematics education in U.S. public schools by recruiting, training, and retaining highly qualified secondary school math teachers. In my quest to get the fellowship, I started to fiddle around with some math ideas that made me curious. One of those is the idea of factoring a polynomial, and specifically, how we teach it.

For one, I'm not a fan of FOIL (first-outside-inside-last) for a plethora of reasons. While I think it's handy to have an acronym that reminds students of a procedure, it only works in a very special case. In this case, FOIL works only for multiplying a binomial by another binomial. Does FOIL lead students toward understanding multiplication of all types of polynomials, or understanding why the distributive property works even with variables? I'm not so sure.

We do know that there are alternate ways of approaching multiplication of binomials, but I'd like to focus on using the geometric method of multiplication because, well, because I can.

Multiplication and Factoring Using Areas

Multiplying (x + 2) by (x + 3) can be represented like so (see Figure 1):

Figure 1

This makes the following operations look rather simple:

(x + 2)( x + 3)
2 + 2x + 3x + 6
x2 + 5x + 6

Using the area method for multiplying binomials also makes factoring an easy task. We can visualize the squares and rectangles in this shape while thinking to ourselves, "Which two numbers have a sum of 5 (second term) and a product of 6 (third term)?" If we look carefully, students have another method for understanding why we get the terms we do after multiplying the binomials shown. How does this relate to trinomials? Let's see.

Multiplication and Factoring of Cubes

Let's take the last example and multiply it by (x + 4). Like so (see Figure 2):

Figure 2

(x + 2)(x + 3)(x + 4)
(x2 + 5x + 6)(x + 4)
x3 + 9x2 + 26x + 24

Or geometrically (see Figure 2): This has awesome implications for finding both the surface area and volume of this figure. Since we already figured out the "face" of this cube earlier (x2+ 5x + 6), we're basically multiplying that face by the length of x and by the length of 4. This yields:

x(x2 + 5x + 6) + 4(x2 + 5x + 6)
. . .
x3 + 9x2 + 26x + 24

Factoring The Cube

Once we find the quadrinomial, the cube gives us a hint for finding the lengths that created the quadrinomial. One would only need to figure out which three numbers give us a sum of 9 (second term) and a product of 24 (last term). These numbers are 2, 3, and 4, so we'll get (x + 2)(x + 3)(x + 4).

Is this much better than using the cubic formula? Absolutely, especially for our students.

What about a quadrinomial like 2x3 - 11x2 + 12x + 9? We can try to determine all the real roots of this polynomial, or we can take a look at the second and last terms. The simplest combination for a product of 9 is multiplying 3, 3, and 1. The first term's coefficient, 2, makes getting a -11 tricky. We can't get -11 from the set of numbers without considering the first coefficient.

Yet, as we've seen with the other cubes (see Figure 3):

Figure 3

We will see that the term of 2x will multiply with any lengths that aren't associated with it. Thus, (2x - 3) + (2x - 3) will give us 6x + 6x, or 12x. Since we needed to get -11x, this means the remaining 1x is positive and the 12x is actually -12x

Therefore, our quadrinomial gets factored to (x - 3)( x - 3)(2x + 1) or (x - 3)2(2x + 1).

Word of Caution

I'm still exploring this for other cases (looking at x3 + 27, for example), and what imaginary numbers would look like using this model. These methods work great as a way to draw students in, but some special cases will probably require more space than I could lend it here.

Also, this was my rebellion against the cubic formula which, as many mathematicians know, makes little sense to introduce in the classroom.

Let me know what you think in the comments below.

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Michael Paul Goldenberg's picture
Michael Paul Goldenberg
mathematics coach

Having looked at the problems you linked to, @ccssi, I conclude that you've managed to prove. . . nothing at all. Any basic algebra book, after dealing with factoring trinomials, looks at additional factoring methods, including factoring by grouping. And yes, there are some nifty special cases that require a "combination of ingredients" (well, methods/techniques) to do the job. You must have felt all sort of excitement at finding three of those cases in order to torpedo the very idea of a model for factoring trinomials. But of course, you didn't succeed in doing so.

Try looking at James Tanton's brilliant (free) online course, complete with both video lectures and accompanying text with additional practice problems. Best thing I've ever seen for what many students never learn or do so only mechanically at best. And not a second of snark from Tanton. You could take a lesson just from his demeanor. My students love those videos, which I use in class in a particular way to give them a chance to really think about what he's doing and why it makes sense. When we then go through the traditional completing the square methods and finally explore the quadratic formula, they have an arsenal from which to choose if and when they need to factor/solve equations of the form ax^2 + bx + c = 0 without using graphing or CAS-equipped calculators. Leave it to you to change the game by bringing in an entirely different family of expressions, then whinging about how a model doesn't work on them. Funny, but I'm pretty sure students are asked far more frequently to deal with typical trinomials than with the sorts of things you linked to. But you knew that, didn't you?

http://gdaymath.com/courses/quadratics/

p.s.: Tanton's courses on the same site, Exploding Dots and the one on Combinatorics, are both brilliant as well. But YOU shouldn't bother with them. They're way too student-friendly for someone like you, I'm sure. Nothing to learn there for a know-it-all.

(1)
Michael Paul Goldenberg's picture
Michael Paul Goldenberg
mathematics coach

I guess I'm getting just a tad obsessed here. I hadn't realized until just now that @CCSSI Mathematics is the same person(s) behind Five Triangles Math. Seeing a link on the latter site to the former was a rather large hint, given that there are no other links offered other than internal ones. And of course, Five Triangles is also anonymous. My, but there seems to be quite a lot to hide going on.

At any rate, all three problems linked to are, of course, NOT trinomials. And they're not in one variable. Coincidence? Hardly. They're hand picked to NOT be relevant to the issues Jose is exploring and that James Tanton discusses in his free course and that countless students have to struggle with every year, often to ill effect (if you consider it an ill effect when something in mathematics that could be intriguing and beautiful - again, see those Tanton videos - is made tedious, pointless, and frustrating to the point where it becomes yet another hurdle that eliminates a sizeable number of, ahem, unfit students. Because clearly, if a kid can't factor quadratic equations given the classic treatment they are given in most US mathematics classrooms, s/he must be intellectually defective, fit only for a job a McDonalds, as long as the electronic cash registers are working. Otherwise, it's flipping burgers forever.

So here are the gems we were asked to consider:

First, x^2 - yz + xy - xz

Well, that's a little wicked. Of course, don't put the terms in a convenient order. Too easy. But factoring by grouping does seem to rear its head here. So let's try grouping them this way:
x^2 + xy | - yz - xz

The left group factors into x (x + y). The right factors into -z(x + y) and the distributive property gives us (x + y)(x - z).

Of course, if you have 3 variables rather than one, students are likely to freeze up, so this is a great problem to make a lot of them feel paralyzed and inferior. Always a goal for a certain breed of mathematics teacher.

2) a^2 - ac - b^2 + bc

Again, make sure not to put the terms in the most pliable order, because that would cut the frustration level a bit.

So we'll write instead a^2 - b^2 | + bc - ac

Neatly, we get the difference of two perfect squares on the left, suggesting

(a + b)(a - b). With a little thought, we can factor out a negative c on the right to get -c(a - b)
[It never hurts to think about what result MIGHT be desirable and then see if there's a way to get it]. So now we can use the distributive property to get (a-b)(a + b - c). Check to see that this multiplication of a binomial times a trinomial gives the original expression. Oh, so tricky-san is our Five Triangles host. Why, you'd almost think that these problems were created to "prove" that modeling something that is very different, namely quadratics in one variable, just won't yield to methods that of course we already KNOW they won't yield to. Hmm. What would the goal be in bringing them up in a conversation about something different?

Finally, we get this gem: x^2 - 2xy + y^2 - 1

This one is neatly designed to lead you down various primrose paths to failure.

For example, seeing that it is four terms, and given the previous two problems, we are likely to want to factor by grouping into pairs of binomials. And it would seem that a promising approach would be to group, say, x^2 - 1 | + y^2 - 2xy

But not only does this fail (you can factor both expressions, but not so you get a common factor), but so will other grouping into pairs (with grouping x^2 + y^2 obviously doomed before we start). Despair begins to set in, and our oh-so-clever problem-poser has once again made sure to list the terms in an unhelpful order.

However, we aren't going to be daunted. We try writing x^2 - 2xy + y^2| - 1
Since the left side is a perfect square trinomial, namely (x - y)^2, we have the difference of two perfect squares after all, but not as we first thought.

(x - y)^2 - 1 then factors as (x - y + 1)(x - y - 1). Bingo. But of course this is the product of two trinomials, not two binomials. We've gotten awfully far from ax^2 + bx + c = 0, haven't we? But then, that's the point. So it's not that the model being used to explore quadratic equations of the form most often dealt with in school algebra fails, but that it doesn't address problems that it was never claimed to explain. In a now archaic phrase that somehow still resonates, No Duh!

(2)
Michael Paul Goldenberg's picture
Michael Paul Goldenberg
mathematics coach

Can't speak for anyone else, but I prefer actually understanding how the results of the multiplication of the binomial and the trinomial show you precisely why the signs have to be that way. Examining the flow between factored and unfactored forms of the expressions are supposed to eventually sink in as to what's going on. If a mnemonic helps kids get through the exam, that's fine, if the goal is getting through an exam. But there's a downside. I like mnemonics best when they are used for arbitrary information, e.g., order of operations or the order of the cranial nerves or the names of the five Great Lakes. There's just no logical way to deduce the order or the names. So having aids to memory makes sense, if for nothing else than as a checklist to make sure you've got them all in the last case, and for the ordering in the second one. Order of operations does have a kind of logic to it, but it's still got an arbitrariness, too.

When there's something conceptual at play, however, I prefer to draw students' attention to that. Algebra is at least in part about structures, and I like to point back to how two-digit by two-digit multiplication can be broken down into (10a + b)(10c +d) followed by using the distributive property or the lattice method, or any reasonable approach that stresses the underlying structure, using specific numerical examples, and then returning to the algebraic setting. Doesn't work for every student (or elementary teacher, for that matter), but it's worth looking at. And I think we've got things like that going on with sums and differences of perfect cubes. I love that Jose is playing with this visually. Of course, as someone will surely want to say, that model breaks down when we go to one higher dimension, so obviously it's a complete waste of time to even consider. :^)

(2)
Bon Crowder's picture
Bon Crowder
Math Mom & Education Advocate

I saw the cubic model as a Montessori device in my new classroom (leftover from previous teacher). Thought it was interesting.

What I'm wondering is how we decide the "size" of the x length. And how things would look if we varied that - as it is a variable after all.

Also, I have a problem teaching factoring of 3rd degree polynomials this way without saying out loud and multiple times that there are VERY FEW of these that have nice factorizations. There are also ridiculously few quadratics with factorizations, but we have the quadratic formula to (hopefully) show that the "non-nice" factorizations can also be irrational or even imaginary.

Consider x^2 - 10x +34. It "factors" into (x-5-3i)(x-5+3i) and you'd only get that mess by setting the polynomial to zero, using the quadratic formula to find the zeros/roots and apply the idea that (x-c) is a factor iff c is a zero/root.

(1)
José Vilson's picture
José Vilson
Middle school math teacher and coach
Blogger 2014

Thank you for your comment. I might suggest that I would *have* to make the connection from the geometric to the abstract because it's hard to define a fourth dimension, unless we're talking about time or something. In any case, I guess if you MUST ... but personally, I think FOIL is a gateway drug whereas if we focus on the idea of distributive property, there's a better fluidity so people don't *have* to rely on first-outside-inside-last.

José Vilson's picture
José Vilson
Middle school math teacher and coach
Blogger 2014

I feel similarly about SOAP a) because I've never heard of it and b) because I'm not sure if that's going to help students progress upwards ...

José Vilson's picture
José Vilson
Middle school math teacher and coach
Blogger 2014

Thank you, Bon, and you raise legit questions here. Here's the thing, Bon. At some point, every "model" breaks down for special cases. That's why I'm playing with it right in front of you. At some point, you *must* transition into the abstract to deal with those pesky non-nice trinomials or any other polynomial. Really, it's a way to push our thinking just a bit to examine why we do what we do, or even what made us bring up polynomials at all.

Allison Weisel's picture
Allison Weisel
Math Teacher

A. Men. I was taught not to use FOIL and instead use the area/volume model to teach multiplication and factoring, and I never looked back. I have worked with lower achieving students and I say that this method not only teaches the concept, but allows them to think visually (and kinesthetically, if you use algebra tiles), where their number sense trips them up.

In my mind, you want models to be too cumbersome to serve at some point, to provide impetus for abstraction. Also, doesn't understanding of what is and what is not possible open up whole new worlds of ideas? Teaching with models does not forbid abstraction, or even impede it. Rather, it deepens connections so students know, understand, and remember concepts without (sometimes silly) acronyms.

(1)
Nikita Hertz's picture
Nikita Hertz
Passionate about Math & EdTech

I think this a really good way to explain factoring. Geometrical representation helps the student understand the concept much better.

Allison Weisel's picture
Allison Weisel
Math Teacher

A. Men. I was taught not to use FOIL and instead use the area/volume model to teach multiplication and factoring, and I never looked back. I have worked with lower achieving students and I say that this method not only teaches the concept, but allows them to think visually (and kinesthetically, if you use algebra tiles), where their number sense trips them up.

In my mind, you want models to be too cumbersome to serve at some point, to provide impetus for abstraction. Also, doesn't understanding of what is and what is not possible open up whole new worlds of ideas? Teaching with models does not forbid abstraction, or even impede it. Rather, it deepens connections so students know, understand, and remember concepts without (sometimes silly) acronyms.

(1)
Bon Crowder's picture
Bon Crowder
Math Mom & Education Advocate

I saw the cubic model as a Montessori device in my new classroom (leftover from previous teacher). Thought it was interesting.

What I'm wondering is how we decide the "size" of the x length. And how things would look if we varied that - as it is a variable after all.

Also, I have a problem teaching factoring of 3rd degree polynomials this way without saying out loud and multiple times that there are VERY FEW of these that have nice factorizations. There are also ridiculously few quadratics with factorizations, but we have the quadratic formula to (hopefully) show that the "non-nice" factorizations can also be irrational or even imaginary.

Consider x^2 - 10x +34. It "factors" into (x-5-3i)(x-5+3i) and you'd only get that mess by setting the polynomial to zero, using the quadratic formula to find the zeros/roots and apply the idea that (x-c) is a factor iff c is a zero/root.

(1)
Michael Paul Goldenberg's picture
Michael Paul Goldenberg
mathematics coach

Having looked at the problems you linked to, @ccssi, I conclude that you've managed to prove. . . nothing at all. Any basic algebra book, after dealing with factoring trinomials, looks at additional factoring methods, including factoring by grouping. And yes, there are some nifty special cases that require a "combination of ingredients" (well, methods/techniques) to do the job. You must have felt all sort of excitement at finding three of those cases in order to torpedo the very idea of a model for factoring trinomials. But of course, you didn't succeed in doing so.

Try looking at James Tanton's brilliant (free) online course, complete with both video lectures and accompanying text with additional practice problems. Best thing I've ever seen for what many students never learn or do so only mechanically at best. And not a second of snark from Tanton. You could take a lesson just from his demeanor. My students love those videos, which I use in class in a particular way to give them a chance to really think about what he's doing and why it makes sense. When we then go through the traditional completing the square methods and finally explore the quadratic formula, they have an arsenal from which to choose if and when they need to factor/solve equations of the form ax^2 + bx + c = 0 without using graphing or CAS-equipped calculators. Leave it to you to change the game by bringing in an entirely different family of expressions, then whinging about how a model doesn't work on them. Funny, but I'm pretty sure students are asked far more frequently to deal with typical trinomials than with the sorts of things you linked to. But you knew that, didn't you?

http://gdaymath.com/courses/quadratics/

p.s.: Tanton's courses on the same site, Exploding Dots and the one on Combinatorics, are both brilliant as well. But YOU shouldn't bother with them. They're way too student-friendly for someone like you, I'm sure. Nothing to learn there for a know-it-all.

(1)
Sue J's picture

(and what do you think of SOAP for difference/sum of cubes? Same, Opposite, Always Positive for getting the signs straight... )

(1)
Sue J's picture

Okay, you asked for comments :)

I tutor in community college, and it pleases me that while lots of the students say "do I FOIL that?" it's become shorthand for "apply the distributive property." (Usualy they'll say "Foil, distribute..." and I've heard our faculty tell 'em "yes, you learned FOIL but I don't like that; it's the distributive property.") They've sometimes said they were "FOIL'ing ... when it was a trinomial/binomial situation. I credit our faculty for taking them where they were and building, and suspect it's more effective than saying "don't use FOIL" which leads to students thinking math is this thing where you have to figure out which magic formula each teacher wants you to use and they change all the time.
Based on that, If I were teaching it for the first time, I think I would still teach FOIL for the sake of my kinesthetic learners -- but they'd better understand it as 'an example of applying the distributive property' and what it's doing mathematically, as well, not just a pencil dance.
The geometric model helps deal with basic, fundamental misconceptions about multiplying and adding that an awful lot of my students hold. Yes, I'd say it doesn't work as well with more complicated factoring -- but I'm not sure a creative student couldn't model even the 'more sophisticated' ones. In my experience, students can and will "over-rely" on the manipulatives and models... unless and until we show 'em the connections to the abstract and the symbols. However, they tend to rely on other fundamentally flawed models
I often start with just natural numbers with the pics and the problems ... show 'em more than once that (2 + 3) squared is not 4 + 9. The idea that multiplying can be expressed as multiple dimensions & directions is helpful.
The other thing I do with FOIL is, for my students who persist in thking (x+y)^2 is x^2 + y^2, to tell them how many times I've heard teachers threaten to set fire to student papers for doing that -- that half the class will get it wrong, and don't they want to be in the half that gets it right? -- that when they see (x+y)^2 they should think, "Oh, F......OIL!") and recognize that plus signs so often mean more work in algebra and get the job done.

(1)
Michael Paul Goldenberg's picture
Michael Paul Goldenberg
mathematics coach

Can't speak for anyone else, but I prefer actually understanding how the results of the multiplication of the binomial and the trinomial show you precisely why the signs have to be that way. Examining the flow between factored and unfactored forms of the expressions are supposed to eventually sink in as to what's going on. If a mnemonic helps kids get through the exam, that's fine, if the goal is getting through an exam. But there's a downside. I like mnemonics best when they are used for arbitrary information, e.g., order of operations or the order of the cranial nerves or the names of the five Great Lakes. There's just no logical way to deduce the order or the names. So having aids to memory makes sense, if for nothing else than as a checklist to make sure you've got them all in the last case, and for the ordering in the second one. Order of operations does have a kind of logic to it, but it's still got an arbitrariness, too.

When there's something conceptual at play, however, I prefer to draw students' attention to that. Algebra is at least in part about structures, and I like to point back to how two-digit by two-digit multiplication can be broken down into (10a + b)(10c +d) followed by using the distributive property or the lattice method, or any reasonable approach that stresses the underlying structure, using specific numerical examples, and then returning to the algebraic setting. Doesn't work for every student (or elementary teacher, for that matter), but it's worth looking at. And I think we've got things like that going on with sums and differences of perfect cubes. I love that Jose is playing with this visually. Of course, as someone will surely want to say, that model breaks down when we go to one higher dimension, so obviously it's a complete waste of time to even consider. :^)

(2)
Michael Paul Goldenberg's picture
Michael Paul Goldenberg
mathematics coach

I guess I'm getting just a tad obsessed here. I hadn't realized until just now that @CCSSI Mathematics is the same person(s) behind Five Triangles Math. Seeing a link on the latter site to the former was a rather large hint, given that there are no other links offered other than internal ones. And of course, Five Triangles is also anonymous. My, but there seems to be quite a lot to hide going on.

At any rate, all three problems linked to are, of course, NOT trinomials. And they're not in one variable. Coincidence? Hardly. They're hand picked to NOT be relevant to the issues Jose is exploring and that James Tanton discusses in his free course and that countless students have to struggle with every year, often to ill effect (if you consider it an ill effect when something in mathematics that could be intriguing and beautiful - again, see those Tanton videos - is made tedious, pointless, and frustrating to the point where it becomes yet another hurdle that eliminates a sizeable number of, ahem, unfit students. Because clearly, if a kid can't factor quadratic equations given the classic treatment they are given in most US mathematics classrooms, s/he must be intellectually defective, fit only for a job a McDonalds, as long as the electronic cash registers are working. Otherwise, it's flipping burgers forever.

So here are the gems we were asked to consider:

First, x^2 - yz + xy - xz

Well, that's a little wicked. Of course, don't put the terms in a convenient order. Too easy. But factoring by grouping does seem to rear its head here. So let's try grouping them this way:
x^2 + xy | - yz - xz

The left group factors into x (x + y). The right factors into -z(x + y) and the distributive property gives us (x + y)(x - z).

Of course, if you have 3 variables rather than one, students are likely to freeze up, so this is a great problem to make a lot of them feel paralyzed and inferior. Always a goal for a certain breed of mathematics teacher.

2) a^2 - ac - b^2 + bc

Again, make sure not to put the terms in the most pliable order, because that would cut the frustration level a bit.

So we'll write instead a^2 - b^2 | + bc - ac

Neatly, we get the difference of two perfect squares on the left, suggesting

(a + b)(a - b). With a little thought, we can factor out a negative c on the right to get -c(a - b)
[It never hurts to think about what result MIGHT be desirable and then see if there's a way to get it]. So now we can use the distributive property to get (a-b)(a + b - c). Check to see that this multiplication of a binomial times a trinomial gives the original expression. Oh, so tricky-san is our Five Triangles host. Why, you'd almost think that these problems were created to "prove" that modeling something that is very different, namely quadratics in one variable, just won't yield to methods that of course we already KNOW they won't yield to. Hmm. What would the goal be in bringing them up in a conversation about something different?

Finally, we get this gem: x^2 - 2xy + y^2 - 1

This one is neatly designed to lead you down various primrose paths to failure.

For example, seeing that it is four terms, and given the previous two problems, we are likely to want to factor by grouping into pairs of binomials. And it would seem that a promising approach would be to group, say, x^2 - 1 | + y^2 - 2xy

But not only does this fail (you can factor both expressions, but not so you get a common factor), but so will other grouping into pairs (with grouping x^2 + y^2 obviously doomed before we start). Despair begins to set in, and our oh-so-clever problem-poser has once again made sure to list the terms in an unhelpful order.

However, we aren't going to be daunted. We try writing x^2 - 2xy + y^2| - 1
Since the left side is a perfect square trinomial, namely (x - y)^2, we have the difference of two perfect squares after all, but not as we first thought.

(x - y)^2 - 1 then factors as (x - y + 1)(x - y - 1). Bingo. But of course this is the product of two trinomials, not two binomials. We've gotten awfully far from ax^2 + bx + c = 0, haven't we? But then, that's the point. So it's not that the model being used to explore quadratic equations of the form most often dealt with in school algebra fails, but that it doesn't address problems that it was never claimed to explain. In a now archaic phrase that somehow still resonates, No Duh!

(2)
Michael Paul Goldenberg's picture
Michael Paul Goldenberg
mathematics coach

@CCSSI Mathematics - If a model breaks down at some point, then, you want to dismiss that model at every point, right? And you figure that if you can make an analogy that is sufficiently demeaning, you've won. It's reasoning of this type that makes me increasingly inclined to dismiss you as any sort of useful resource regarding the teaching and learning of mathematics. I've seen Irving Kaplansky use models that break down at some point for pedagogical purposes. Since you won't reveal your identity, I can only speculate, but my guess is that you're not quite in Kaplansky's league mathematically. You're most definitely not in the league of many math teachers whose creativity and ingenuity in finding ways to help students understand key ideas through a wide range of analogies and models (none of them filled with snark), none of which are claimed to be bullet-proof.

The reason that wise teachers don't use FOIL is that it's a useless mnemonic that attempts to avoid having students understand the distributive property. I don't teach it, and if a student brings it up, I demonstrate its obvious flaws. And few students fail to get the point. But FOIL isn't a model at all. It's just a silly and unnecessary way to remember the steps in a procedure. It's not an explanation of that procedure. A solid understanding of the distributive property makes it obvious that no one need worry about FOIL, ever. And when I work with preservice and inservice teachers, I try to help them see that. Teachers, unfortunately, can be more stubbornly pig-headed than many students, but fortunately not all of them are.

What Jose is playing with (you do grasp the concept of intellectual play as a path towards new knowledge, understanding, insight, creativity, etc., don't you) is useful. Your attempt to deflate models for factorization is rather indicative of your mindset, one I'm glad that Jose doesn't have.

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