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Whats algebra about? When learning about variables (x, y, z), they seem to hide a number:

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

t seems that arithmetic still works, even when we dont have the exact numbers up front. Later on, we might arrange

hese hidden numbers in complex ways:

Whoa a bit harder to solve, but its possible. Today lets figure out how factoring works and why its 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

esult is 6.

^2, x and 6 are all numbers, but now were keeping track of how theyre 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 lets fight with a trick of our own: we can make a differentsystem to track the error in our original

ne (this is mind-bending, so hang on).

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

will track the difference between the original system and the desired state. When are we happiest? When theres no

ifference:

Ah! thats why were so interested in setting polynomials to zero! If we have a system and the desired state, we can make

new equation to track the difference and try make it zero. (This is deeper than just subtract 6 from both sides

were trying to describe the error!)

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

Factor That Mamma Jamma

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

maller systems.

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

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eepee:

\

Thats a 2-d example, with two sticks).

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

s a series of multiplications:

weve put the sticks in a teepee. If Component A or Component B becomes 0, the structure collapses, and we get 0 as a

esult.

Neat! Thatis why factoring rocks: we re-arrange our error-system into a fragile teepee, so we can breakit. Well find what

bliterates our errors and puts our system in the ideal state.

Remember: Were breaking the error in the system, not the system itself.

Onto The Factoring

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

f 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) + -3 = 6

When x = 2, the error collapses, and we get 2 + 2 = 6

Putting It All Together

ve wondered about the real purpose of factoring for a long, long time. In algebra class, equations are conveniently set to

ero, and were not sure why. Heres 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. Were 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 i

a fair game?

he idea of matching a system to its desired state is just one interpretation of why factoring is useful. If you have more,

d like to hear them!

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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 i

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, shouldnt there be 4 individual x components being

multiplied? If there were 3, you could never get to x^4, and if there were 5, youd 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. Theres 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?). Dont forget, w

thought systems like x^2 + 1 were non-zeroable until imaginary numbers came along.