Understanding Algebra: Why do we factor equations?

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What’s algebra about? When learning about variables (x, y, z), they seem to “hide” a number:

\displaystyle{x + 3 = 5}

What number could be hiding inside of x? 2, in this case.

It seems that arithmetic still works, even when we don’t have the exact numbers up front. Later on, we might arrange these “hidden numbers” in complex ways:

\displaystyle{x^2 + x = 6}

Whoa — a bit harder to solve, but it’s possible. Today let’s figure out how factoring works and why it’s useful.

Polynomials

When we write a polynomial like “x^2 + x = 6″, we can think at a higher level.

We have an unknown number, x, which interacts with itself (x * x = x^2). We add in the original number (+ x) and the result is 6.

x^2, x and 6 are all “numbers”, but now we’re keeping track of how they’re made:

  • x^2 is a component interacting with itself
  • x is a component on its own
  • 6 is the desired state we want the entire system to become

After the interactions are finished, we should get 6. What number could be hiding inside of x to make this true?

Hrm — this is tricky. So let’s fight with a trick of our own: we can make a different system to track the error in our original one (this is mind-bending, so hang on).

Our original system is x^2 + x. The desired state is 6. A new system:

\displaystyle{x^2 + x - 6}

will track the difference between the original system and the desired state. When are we happiest? When there’s no difference:

\displaystyle{x^2 + x - 6 = 0}

Ah! that’s why we’re so interested in setting polynomials to zero! If we have a system and the desired state, we can make a new equation to track the difference — and try make it zero. (This is deeper than just “subtract 6 from both sides” — we’re trying to describe the error!)

But… how do we actually get the error to zero? It’s still a jumble of components: x^2, x and 6 are flying everywhere.

Factor That Mamma Jamma

Factoring the rescue. My intuition: factoring lets us re-arrange a complex system (x^2 + x – 6) as a bunch of linked, smaller systems.

Imagine taking a pile of sticks (our messy, disorganized system) and standing them up so they support each other, like a teepee:

/\

(That’s a 2-d example, with two sticks).

Remove any stick and the entire structure collapses. If we can rewrite our system:

\displaystyle{x^2 + x - 6 = 0}

as a series of multiplications:

\displaystyle{Component \ A \cdot Component \ B = 0}

we’ve put the sticks in a “teepee”. If Component A or Component B becomes 0, the structure collapses, and we get 0 as a result.

Neat! That is why factoring rocks: we re-arrange our error-system into a fragile teepee, so we can break it. We’ll find what obliterates our errors and puts our system in the ideal state.

Remember: We’re breaking the error in the system, not the system itself.

Onto The Factoring

Learning to “factor an equation” is the process of arranging your teepee. In this case:


\begin{align*}
x^2 + x - 6 &= (x + 3)(x -2) \\
&= Component \ A \cdot Component \ B
\end{align*}

If x = -3 then Component A falls down. If x = 2, Component B falls down. Either value causes the error to collapse, which means our original system (x^2 + x, the one we almost forgot about!) meets our requirements:

  • When x = -3, the error collapses, and we get (-3)2 + -3 = 6
  • When x = 2, the error collapses, and we get 22 + 2 = 6

Putting It All Together

I’ve wondered about the real purpose of factoring for a long, long time. In algebra class, equations are conveniently set to zero, and we’re not sure why. Here’s what happens in the real world:

  • Define the model: Write how your system behaves (x^2 + x)
  • Define the desired state: What should it equal? (6)
  • Define the error: The error is its own system: Error = actual – desired (i.e., x^2 + x – 6)
  • Factor the error: Rewrite the error as interlocking components: (x + 3)(x – 2)
  • Reduce the error to zero: Zero out one component or the other (x = -3, or x = 2).

When error = 0, our system must be in the desired state. We’re done!

Algebra is pretty darn useful:

  • Our system is a trajectory, the “desired state” is the target. What trajectory hits the target?
  • Our system is our widget sales, the “desired state” is our revenue target. What amount of earnings hits the goal?
  • Our system is the probability of our game winning, the “desired state” is a 50-50 (fair) outcome. What settings make it a fair game?

The idea of “matching a system to its desired state” is just one interpretation of why factoring is useful. If you have more, I’d like to hear them!

Appendix

A cheatsheet for the process:

Some more food for thought:

  • Multiplication is often seen as AND. Component A must be there AND Component B must be there. If either condition is false, the system breaks.

  • The Fundamental Theorem of Algebra proves you have as many “components” as the highest polynomial. If your highest term is x^4, then you can factor into 4 interlocked components (discussion for another day). But this should make sense: if you rewrite an “x^4 system” into multiplications, shouldn’t there be 4 individual “x components” being multiplied? If there were 3, you could never get to x^4, and if there were 5, you’d overshoot and get an x^5 term.

  • Do you have a real-world system in a “teepee” arrangement, where a single failing component collapses the entire structure?

  • The quadratic formula can “autobreak” any system with x^2, x and constant components. There’s formulas for complex systems (with x^3, x^4, or even some x^5 components) but they start to get a bit crazy.

  • Is there any way to prevent a system from having these weak points? (Unfactorable? Non-zeroable?). Don’t forget, we thought systems like x^2 + 1 were “non-zeroable” until imaginary numbers came along.

Happy math.

Other Posts In This Series

  1. Rethinking Arithmetic: A Visual Guide
  2. Techniques for Adding the Numbers 1 to 100
  3. Understand Ratios with "Oomph" and "Often"
  4. Mental Math Shortcuts
  5. How to Develop a Sense of Scale
  6. Easy Permutations and Combinations
  7. How To Understand Combinations Using Multiplication
  8. Surprising Patterns in the Square Numbers (1, 4, 9, 16…)
  9. Navigate a Grid Using Combinations And Permutations
  10. A Quick Intuition For Parametric Equations
  11. Understanding Algebra: Why do we factor equations?
  12. How To Learn Trigonometry Intuitively
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28 Comments

  1. Must be that it’s 25 year ago that i was in school but you lost me at
    Onto The Factoring
    How do you get from \displaystyle{x^2 + x-6} to \displaystyle{(x+3)(x-2)}

    I like the breaking the system though

  2. Reverse engineering
    \displaystyle{(x + 3)(x - 2) = x^2 - 2x + 3x - 6 = x^2 + x - 6}

    But can’t figure out the intuition on the reverse the splitting of the x?

    Ehhh \displaystyle{2 * 3 = 6} nope still no light

  3. Hi Franc, great question! For polynomials with only a squared term, a common factoring trick is to see what numbers could multiply together to get 6.

    As you reverse-engineered, the last two numbers (x – a)(x – b) need to multiply to get 6:

    \displaystyle{a * b = 6}

    In this case, we have choices 1 & 6, 2 & 3, 3 & 2, or 6 & 1. At this point it’s trial and error to find two factors with a difference of 1 (i.e., leaving us with a single “x”). In this case it’s 2 and 3. Then we have a little work to figure out the signs [3 should be positive because we want positive 1x left over].

    Of course, this is a lot of tedium. The quadratic equation gives a formula to “autofactor” any quadratic equation [i.e., anything with an x^2 term].

    As we get to higher powers [x^3, x^4] it becomes really hard to compute by hand, and we usually rely on computer systems to factor the equations.

  4. I understand the math, but this article’s use of the word “error” is confusing to me. What does it mean in this context? What’s erroneous about the equations?

  5. I have 4 years of a math major behind me and 6 years of a science PhD, and I don’t understand what you mean by “error”.

    Why not just say, you want to have zero on one side because of the special way that zero works with multiplication?

  6. @James, yonemoto: Thanks, I need to clarify: the error is the difference between the current state and the desired state.

    For example, if x = 1, we have

    current state = 1^2 + 1 = 2
    desired state = 6
    error = 2 – 6 = -4

    i.e., if we attempt to use x = 1, our error will be -4 [we will be 4 less than our target value].

    If we try to use x = 4, our error is 16 + 1 – 6 = 9. We overshot our goal.

    The point of factoring is to easily figure out where our error will be zero, which means we’re exactly at the desired state. Thanks for the comments, I’ll clarify!

  7. This is great. I totally get that when you use the analogy of the “error”, what your referring to is that multiplication that results in a zero is a unique mathematical anomaly. You’re targeting the “Zero Product Property” as a stable point of reference in a sea of numbers.

    I think what’s more difficult about factoring though, is intuitively figuring out the whole “reverse FOIL” thing. For me, it’s always been this thing where I just sit and debunk the magic combination of values that both multiply to the coefficient, and also add to the constant. Usually I descend into brute force, and while that’s fine for destroying my entire weekend, and I browse through the infinite permutations that confront me and taunt me in my homework, this literally kills me on exams with time limits.

    Any shortcuts to that?

  8. @Sim (you like Wayne’s world too? :)): Great question. I think the reverse FOIL thing comes down to pattern recognition. Realistically, we usually do manual factorization on nice, premade homework problems that have nice clean solutions. In the real world we let computers factorize for us.

    After doing enough problems, you start to see that factorizations work out like this:

    \displaystyle{(x + a)(x + b) = x^2 + ax + bx + ab}

    Notice that the a*b term has no “x’s” involved. So going backwards, if you see a regular number by itself (like -15, in x^2 + 2x – 15), you start wondering what “a” and “b” could result in -15 (remember, a and b can be negative too):

    * -1 and 15
    * -15 and 1
    * -3 and 5
    * -5 and 3

    The next step is to find the combination that has a *difference of 2* (since we need a 2x term left over). In this case, it’s +5 and -3. So we get

    \displaystyle{(x + 5)(x - 3) = x^2 + 5x - 3x - 15 = x^2 + 2x -15}

    That’s the process that goes into my head. It is brute force to some extent, and we can use the quadratic formula to blast through any equation automatically.

    Honestly, most homework problems will be set up nicely so you can recognize the factors pretty quickly [what a and b have the difference we need?].

  9. This is great. Thank you very much for the perspective. Please continue to post more insight in math!

  10. I like the “error” analogy very much. For years I always wonder why there is a error
    function in a PDE. After your “system” analogy , I just remember that PDF used to discribe the control system, and the goal is minimum the error function. Though I don’t know whether there is someway to “factor” a PDE system .

  11. As usual a very thoughtful post. I’ll spend the week thinking about how this relates to my own sense that the reason to find each “component” of an equation is to figure out relative maxima and minima. Figuring out the “roots” of an equation is not anywhere near as interesting as figuring out the roots of its derivative. Is there an equation to track the error in my thinking?

  12. @Chris: Thanks!

    @hdhoang: Great find, thank you. Now to figure out how it works :).

    @gjing: Nice, glad it worked. I don’t have much experience with PDEs but I think that analogy could be extended. “Factoring” is just a process used to set up something which can be zero’d… there could be other ways.

    @mark: Hah, I wish I had an mental error function! (Hello stock market picks). Exactly, finding the roots of the derivative can give us better insights. And again, we factor the derivative to find when “the changes are zero” (i.e., any required component of the entire change process becomes zero).

  13. I went back to college almost a year ago and despite hating math when I was younger, I am loving it now. I’m nearly finished with Calculus 1, and while I got all A’s in algebra, I never really understood why one would set a polynomial equation to zero. Obviously I knew that the math worked out this way, but I’ve never had a professor explain it as you just did (or even try to explain it at all for that matter). I really enjoyed your error analogy. It instantly clicked for me, like a light was switched on. I won’t be able to stop thinking about it for a little while.

  14. Hi Robert, happy the analogy clicked! I’m having a blast going back and trying to find metaphors for all the mathematical truths I memorized but didn’t internalize. Appreciate the comment.

  15. Im a junior high student and the teacher is teaching factoring, Im good at the simple factoring but Im stuck in this question:
    (2 x^2+x-6)/(x^3-3 x^2+2)(x^2+4 x-5)/(4 x^2-6 x)
    I would really appreciate it if you could do it step by step and explain it slowly
    Thx
    Best Regards
    Robbie

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  17. @ Robbie

    Are you sure you wrote the question properly? There are two “/” signs in there, making it a bit confusing as to whether you’re dividing the denominator or numerator.

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  20. I tend towards very visual solutions when solving algebraic equations. Systems of equations makes sense to me because the process of solving boils down to finding out where two lines intersect, if they intersect at all.

    y(1) = 6
    y(2) = \displaystyle{x^2 + x = x * (x + 1)}

    I know the parabola of y(2) has roots of 0 and -1. I also know that the parabola is increasing at equal rates about the line x=-0.5. Those two points where the parabola meets y=6? The ‘negative point’ of the parabola (left of x=-1) is a touch more negative than the ‘positive point’ (right of x=0). At least, if you’re drawing everything to scale. x = (-3, 2)? Sure, I’ll buy that answer. It fits the prior estimate.

    Why do I do this? Because as the functions get more varied and complicated in these equations (\displaystyle{2/x = 5x + 3}), I can still graph (\displaystyle{y = 5x(x + .6)}; \displaystyle{y = 2}) on the back of an envelope and get a good idea of what the solution set should look like.

    I admit this does have the drawback of making this equation 2D, and some people will wonder if they need to use the y coordinate. I call that a learning opportunity.

  21. I hope this is related. Lately I was troubled about this very question: why is that the zeros of one parabola \displaystyle{y=x^2+x-6} happen to be the very solution we need for \displaystyle{x^2+x=6}? Then is dawned on me that \displaystyle{y=x^2+x-6} is really the composite function when we take the difference of \displaystyle{y=x^2+x} and \displaystyle{y=6}. And the very places where this new function end up being zero is where the former two functions had exactly the same value. ie.: the very places where the former two functions intersected! This might be completely obvious to most people, but for me it was a new little insight.

  22. You are super intelligent. I have been studying, and waiting for an “aha moment” for a long time, and it never came. I am glad that I stumbled onto your site, as it helps me visualize WHAT I am studying. I mean, it’s fine to memorize the unit circle, but what do all those values mean? I love seeing the practical uses for math – it is around us everywhere, in everyday life! My mom said I would never need higher math, but every aspect of life is structured around mathematical principles.

    By the way, I think you are very cute.

  23. How does this answer the question – why do we factor …
    Answer: it doesn’t, maybe it does if you already know the answer.
    Tripe.

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