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Understanding Algebra: Why do we factor equations?

Understanding Algebra: Why do we factor equations?

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.


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:

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{\text{Component A} \cdot \text{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:

x^2 + x - 6 &= (x + 3)(x -2) \\
&= \text{Component A} \cdot \text{Component B}

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:

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:

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

Algebra is pretty darn useful:

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!


A cheatsheet for the process:

algebra factoring process

Some more food for thought:

Happy math.

Other Posts In This Series

  1. Understanding Algebra: Why do we factor equations?
  2. A Quick Intuition For Parametric Equations
  3. Intuition for the Quadratic Formula
  4. Intuition for Slope-Intercept Form
  5. Intuition For Graphed Functions
  6. Intuition For Polynomials