
Understanding Algebra: Why do we factor equations? - joeyespo
http://betterexplained.com/articles/understanding-algebra-why-do-we-factor-equations/
======
kalid
Hi all! Author here. Didn't expect to see this on HN so early this morning :).

Lots of good feedback, I'll try to answer some of it collectively here.

HN readers who can spit out the quadratic formula in their sleep aren't the
real audience for this article :). It's someone googling "Why are we factoring
all the time?".

A high-level question needs a high-level answer before diving in. I'm not a
fan of "Because that's where the graph touches the x-axis" because it again
leads to... Why does touching the x-axis matter?

My intuition is that we have a system and we have something we want it to
become. The trick is to track the difference as its own system, write it as
interlocking parts, and break any of the parts.

"How do we break a chain?" => we break any of the links. Intuitively, that's
what we're enabling when we factor an equation into a series of
multiplications. I'm probably going to continue to reword the explanation, but
that is fundamentally _why_ we bother rewriting as a series of
multiplications.

On rigor and simplicity: guilty as charged. I believe you need to become
intuitively comfortable with an idea, even if slightly incorrect, before
getting into the nuance.

We tell kids a cat is a furry animal that has claws and a tail. Later, we
refine to say they're all descendants of a common ancestor. Later we say
there's this thing called DNA which holds genetic information, and all cats
have DNA in common.

See this article for more:

[http://betterexplained.com/articles/developing-your-
intuitio...](http://betterexplained.com/articles/developing-your-intuition-
for-math/)

I go into the 4 rigorous and common definitions of e, something which confused
me (and many, many engineering students) for _years_. But approaching with
intuition it all snaps together.

Again, that's my teaching style though! It's been very successful for me and
other students. Rigor is always available on Wikipedia and Mathworld if you
need it.

~~~
klochner
I applaud your effort to describe why we factor equations, I just don't think
you've done that here.

    
    
       A high-level question needs a high-level answer before
       diving in. I'm not a fan of "Because that's where the
       graph touches the x-axis" because it again leads to...
       Why does touching the x-axis matter?
    

Because that's where the the system "becomes what you want it to become", i.e,
where:

    
    
      x^2 + x - 6 = 0 
          x^2 + x = 6  
    

You're trying to describe factoring to someone without the basic fundamentals
to even need to do factoring.

By analogy, that's like saying "I want to explain why we need piston rods, but
I don't want to mention anything about engines."

Prerequisites are meaningful.

[Edit] I just verified that graphing is typically taught after factoring, so I
at least sympathize with the challenge you're facing.

~~~
kalid
Thanks for the feedback! Yep, graphing usually comes much later, so I'm trying
to find an explanation that works without it.

For the piston rods, I'd say something like "They help capture the power of an
explosion. And an engine is a really, really cool device to make this power
useful. Want to see how it works?" :)

------
psykotic
The bigger picture is that factorization is decomposition. Depending on the
domain, that can mean different things.

In geometry, factoring a polynomial in x and y decomposes its curve into a
union of subcurves. The curve of xy = 0 is the union of the curves of x = 0
(vertical line through origin) and y = 0 (horizontal line through origin), so
the curve of xy = 0 is a pair of crossing lines with a double point at the
origin. Similarly, a complete factorization of f(x) decomposes the
intersection of y = f(x) and y = 0 into a union of points (the linear
factors). But what about irreducible quadratics like x^2 + 1? From a
geometrical perspective, they're an algebraic nuisance that goes away with
complex numbers.

In linear time-invariant systems, a factorization of a transfer function
corresponds to a serial decomposition of the system because of the convolution
theorem. The most interesting case is when the system is recursive, so that
the transfer function is a general rational function (a ratio of polynomials)
and you have feedback loops in the block diagram. A parallel decomposition
corresponds to a partial fraction expansion, which is based on division and
factorization. These decompositions can also be mixed, so you can take a
transfer function that is a ratio of sextics, factor them into a pair of cubic
ratios (serial decomposition) and then break each of those halves into partial
fractions (parallel decomposition).

In calculus, the partial fraction expansion lets you integrate all rational
functions once you know how to integrate functions of the form x^n for any
integer n, positive or negative.

I could give more examples, in probability theory (characteristic functions
and moment-generating functions), in combinatorics (generating functions), in
the theory of linear differential equations (the Laplace transform
diagonalizes shift-invariant operators like differentiation, so differential
equations become algebraic equations), etc.

~~~
grandalf
Are there any interesting/useful engineering applications of the recursive
ones you mention?

~~~
psykotic
Tons. The whole theory of IIR filter design is predicated on understanding the
effects of the transfer's function poles and zeroes (generally complex
numbers) on the behavior of the filter. Pick up any book on signals and
systems like Oppenheim to get the details. Julius Smith's online books are
also very good albeit terse.

~~~
grandalf
Thanks! That is actually an area of interest. Thanks for the recs.

~~~
psykotic
Here's the relevant chapter in one of Julius Smith's books:

[https://ccrma.stanford.edu/~jos/filters/Transfer_Function_An...](https://ccrma.stanford.edu/~jos/filters/Transfer_Function_Analysis.html)

It might not make total sense if you don't have any background in this area,
but it's hopefully enough to give you an inkling.

~~~
grandalf
Thanks! I am actually learning the background theory at present so will
bookmark the link.

------
yequalsx
I've seen a number of criticisms of the article. Some things need to be kept
in mind.

Presently it is standard to introduce factoring before one knows how to graph
polynomial functions. It is possible to argue that we teach this before
factoring and I think a compelling case can be made for this but it isn't a
standard practice. Anyone who thinks of using x-intercepts to explain this
probably has had some College Algebra. The topic in the article is usually
covered in Beginning Algebra.

References to the discriminant in criticizing the article are not appropriate.
The discriminant is taught after teaching factoring.

The normal order of topics is

1\. factoring 2\. solving quadratic equations with factoring 3\. proving the
quadratic formula 4\. solving general quadratic equations

~~~
czzarr
Are you serious about college algebra? in France this is taught in middle
school

~~~
sp332
To be clear: yequalsx said _Anyone who thinks of using x-intercepts to explain
this probably has had some College Algebra. The topic in the article is
usually covered in Beginning Algebra._ So x-intercepts are taught in middle
school, but the idea of using them to explain this particular problem probably
wouldn't occur to someone with such little experience.

~~~
yequalsx
Indeed, college algebra students typically struggle when trying to tie in the
concepts of factoring, x-intercepts, remainder theorem, and zeros of a
polynomial. Lots of gnashing of teeth in this chapter.

------
lacker
This seems like a confusing explanation to me.

We factor equations because that makes it easy to figure out what x is. If you
did not know any algebra, and looked at an equation like

Ax^2 = Bx + C

it is not obvious what x can be. But if you factor it into

(x - D)(x - E) = 0

then if you know how zero multiplies, it is fairly clear that x must be D or
E.

~~~
jpeterson
Your explanation doesn't address where the zero came from in the second step.
In other words, you're presupposing that we already know what "factoring" is,
which is what this article is meant to explain.

------
wging
> _There’s formulas for more complex systems (with x^3, x^4, or x^5
> components) but they start to get a bit crazy._

This is just untrue! It can be proven that there's no general formula that can
work for x^5 without just defining away the problem by introducing a solution
to a polynomial not solvable by radicals.

~~~
kalid
True -- I got overzealous with x^5, some (not all) can be factored. I'll
reword to "some more complex systems.

~~~
psykotic
> I got overzealous with x^5, some (not all) can be factored

All polynomials can be factored but only some of the fifth or higher degree
are solvable by radicals--a crucial distinction.

Here's a fun puzzle that also serves as a gentle introduction to the central
idea of Galois theory: Show that any palindromic quintic is solvable by
radicals. A palindromic quintic is one of the form ax^5 + bx^4 + cx^3 + cx^2 +
bx + a. It might be useful to know that if you have a polynomial equation f(z)
= 0 and make the substitution z -> 1/z and multiply through by z^n, you get
the reversed polynomial. On the Riemann sphere, this inversion symmetry
reflects the northern hemisphere onto the southern hemisphere and vice versa,
interchanging the north and south pole, which are 0 and infinity in z
coordinates. Therefore a palindromic polynomial is one that "looks the same"
from either vantage point.

This idea of looking at the behavior of a polynomial simultaneously from the
vantage point of 0 and infinity is also the basis of how the fundamental
theorem of algebra is proved. The general model is 1 + z^n. Once you
understand that polynomial qualitatively near 0 and infinity, you can
understand every other polynomial by a simple perturbation analysis.

------
lotharbot
When solving high-school algebra problems, we try to reduce them to
arithmetic. When solving calculus problems, we try to reduce them to algebra.
When solving differential equations, we try to reduce them to integrals. In
essence, we use some new technique in order to take a problem we don't know
how to solve and _transform it_ into a problem we do know how to solve.

That's why we factor: because we can (sometimes) transform a higher-order
equation into a series of linear pieces, and solving a linear equation is
essentially arithmetic.

------
wcarey
Who's the target audience for that explanation? It seems like the initial
question can be solved much more easily by rewriting x^2 + x = 6 as (x)(x+1) =
6. Seeing that x can be two seems intuitive, and that x can be -3 a bit less
so.

I wonder whether talking about algebra as the manipulation of statements to
construct consistent sets is more productive than talking about error in
systems?

Instead of x hiding a value, we want to know what statements can we make about
x that are consistent with x^2 + x = 6.

~~~
pavel_lishin
> seems intuitive

You can't really write "well, it seems intuitive" into a logical proof, or
into an algorithm.

~~~
alanctgardner
But the point of this site is seemingly "understanding", not an algorithm -
see the title of the page ("right, not rote"). However, what they really seem
to be doing is breaking down what should be an intuitive process -
understanding equalities and how to work with them - into several smaller,
vague sub-processes which also aren't really easy to understand. I would argue
that explanations like this might hurt someone's chances of understanding
concepts like systems of equations later on.

~~~
pavel_lishin
I thought it was meant to give us an intuitive explanation for _why_ we
approach things via the "factoring algorithm" as a method of solving these
guys.

In any case, it's just a simple example case. x^2 + x = 6 can probably be
solved just by staring at it for a little bit, but once the equation gets more
complex, trying out a couple of numbers in the [-5,5] range is not gonna work
anymore.

------
alanctgardner
Looking at some other explanations from this site, it seems like it suffers
from the same things people complained about with Khan: a lack of rigour, and
oversimplification. They have some good visuals and metaphors, but ultimately
the 'plain' explanations of mathematical properties are a poor foundation for
someone to build upon when they get to higher concepts.

------
bsaul
I don't understand why adding error in "factoring the error" helps in any way
understand why we factor. It would have started with a factorized form with 0
on the right, saying that finding that a*b=0 is easy to solve because of the
way 0 and multiply work together.

~~~
maxerickson
(Apparently), people don't always understand that ax^2+bx=c is equivalent to
ax^2+bx-c=0.

So they don't see how factoring the latter reveals anything interesting about
the former.

~~~
kalid
Exactly! In the real world you are trying to get some system (x^2 + x) into
some desired state (2).

In math class, we usually "pre-optimize" these equations to (x^2 + x - 2 = 0)
and they look very different from the situations you have to set up on your
own. I want to make the connection explicit.

------
czzarr
thank god my math teacher didn't teach like that. This is the most confusing
explanation of factoring I have ever seen.

for the apparently numerous people here who have no idea how to solve this
equation generically, read this: <http://en.wikipedia.org/wiki/Discriminant>
or watch this:
[http://www.khanacademy.org/math/algebra/quadtratics/v/discri...](http://www.khanacademy.org/math/algebra/quadtratics/v/discriminant-
of-quadratic-equations)

------
louischatriot
I find this explanation overly complicated. For the general case a*x^2+bx+c,
the easiest way is to try to reduce it to the form A^2-B^2 which has the easy
factorisation (A+B)(A-B). So:

ax^2+bx+c = a[x^2+(b/a)x+(c/a)] = a[x^2+(b/a)x+(b/2a)^2-(b/2a)^2+c/a] (only
adding and removing the same thing) =a[(x+(b/2a))^2 - ((b^2/4a)-c/a)] (using
(A+B)^2=A^2+B^2+2AB)

Call D the term (b^2/4a)-c/a, if D is positive we can take its square root,
factor and have our two solutions.

I like more analytical solutions.

------
bazzargh
Wow, that's a terrible explanation. Try drawing a picture instead. The
factorization simply answers the question of where the graph touches the
x-axis.

------
tomrod
I think the explanation is thorough. Is it intended to be a full and complete
explanation, or an introduction? As an introduction it'd be very poor--it
would probably be too encyclopedic as an introduction.

------
eckyptang
Another thing to add to this: Factored equations can be computationally
cheaper to execute.

~~~
psykotic
Not if you use Horner's method. Evaluating a(x-p)(x-q) takes two additions and
two multiplications. Evaluating ax^2 + bx + c with Horner's method as (ax +
b)x + c takes two additions and two multiplications as well. Factoring a
polynomial to save on evaluation is as inefficient and roundabout as factoring
a pair of integers to help compute their greatest common divisor. You're
replacing an easy problem with a much harder one.

~~~
eckyptang
That's a fair point. Perhaps my example was badly chosen!

------
ten_fingers
The answer is simple, dirt simple: If you have to ask the question, then
DON'T!

That is, we factor algebraic expressions if and only if (iff) we have a good
reason to do so. If we don't have a good reason to factor, then there is no
need to bother.

Yes, in high school, 'factoring' is seen as an important algebraic
manipulation. It is. Then high school continues on and wants to factor
whenever possible and for no reason other than it is possible. This is dumb.

Also, commonly there is more than one way to factor. Then high school gets all
in a tizzy over which way is 'best'. Nonsense. Again, we factor for a reason
we have in mind, and of several possible ways to factor we select the one for
the reason we have in mind. Simple.

We factor when we have a reason to do so. Otherwise, f'get about it! High
school teachers: Understand that now?

My authority: I hold a Ph.D. in the applied math of stochastic optimal
control. I've taught math in college and graduate school. I've published peer-
reviewed original research in applied math and mathematical statistics.

~~~
majelix
The obvious counterpoint here is that we're trying to teach skills before
they're needed. After all, it would suck to have to rediscover Calculus on
your own in the middle of your Physics II exam just because you didn't see a
point to it at the time.

~~~
ten_fingers
Yes, and that's the way I learned in high school. At the time it appeared that
we factored to achieve some 'artistic' goals of making the algebraic
expressions 'look nice'. When later I concluded that we factored for some
serious purposes and that the artistic goals of look nice were silly, I
resented some of what I had been taught.

But the question on this thread is appropriate: "Why" do we factor? Sure, the
reason in some of high school is just to learn how to factor so that we will
be able to when we need to, say, working with integration by parts in
calculus. But likely this tread and the students want a reason more
substantial than just to learn for later. So, my answer was (say, beyond just
learning) to factor when there was a good reason and otherwise just f'get
about it, and basically that's the correct answer.

