It’s been some years since I’ve taken a math class, so I admit I did need to relearn a little bit, but factoring is actually really simple once you get the hang of it. It’s super important in pretty much any math class you take, so if you never understand it, life will be hard for you. So, if no teacher or tutor explained any of this in a way it made sense, allow me to give it a shot.
The baseline here is that factoring is taking out and simplifying complex equations. You can chop them up and organize them in a way that looks more orderly, so eventually you’ll organize “z³ – z² – 9z” into “(z – 3)(z + 3)(z – 1)”.
But lets start more simple. A factor is a whole number that, when multiplied with another number, makes a new one. So, 2 and 3 are factors of 6, because when you multiply them together, they equal 6. So if something multiplies into another thing, it is a factor of that thing. It’s worth noting that every number that can be multiplied into a larger number is a factor of it, so 1,2,3,4,6,8, and 12 are all factors of 24.
Which brings us to letters. “Letters don’t belong in math!” you proclaim, and you’d be right. Letters themselves hold almost no value in mathematics. But the letter itself is meaningless. “X” is a variable. The number can change because its “variable”: its a placeholder for a number we don’t know. It holds no linguistic value, because we aren’t referring to language here. We could just as easily use a drawing of a hippopotamus in place of the letter X. In math, letters aren’t letters at all, but convenient, understandable symbols. when we say A = L * W, it serves as both a sentence and a mathematical expression for “area equals length times width”. I could say “T = & * $” and get the same answer when I plug in the same numbers. We just use letters, or variables, because it’s easier to convey what each specific symbol means when we involve other people.
So, all variables mean in this context is “we don’t know what this number is”. The Area of that rectangle could be a lot of different things, but if you just say A, we know what you’re talking about even if we can’t quantify it with a number.
So, if we say “3x”, we’ll take that to mean “3 times whatever number X happens to be”. Without knowing what X is, we’ll have to leave the equation at that. We can treat “3x” as one number, though, because when it’s all said and done they would be mixed together.
So if we try to factor “4x + 8”, we are going to try to take out the common elements of both (or all, if there’s more than two) numbers. When we’re factoring, we want to simplify the equation as much as possible. So we can take out a 2 from both 4x and 8. We put that outside of some parenthesis, and we divide 2 from everything because that’s what we took out. This leaves us with 2(2x + 4).
But we’re not done. We can take out another 2 here, because they are still factors of both numbers inside the parenthesis. We’ll divide that from everything inside, and then multiply it to whatever is on the outside, leaving us with 4(x + 2). (If this doesn’t make any sense, sorry, it’s hard to describe numerical processes with words!)
If you’ll notice, we could have simply removed a four from both numbers in the beginning and saved ourselves the extra step, but we get the same answer in the end.
Now that we’ve got the basics, let’s throw in an extra step. Let’s bring in exponents. You probably know that anything squared (ex: 3²) is a number multiplied by itself. We can do that with variables, too! Let’s try factoring 6x² + 18x. We can easily take out the largest number, 6, and be left with 6(x² + 3x). You may think we’d be done, but we’re not. Once you start getting into more complex math, we’ll need to start factoring variables, too. If x² means x*x, then x² divided by x will just leave us with one x. So, from 6(x² + 3x), we can divide X from inside the parenthesis. x² divided by x will leave one x left, and dividing x by itself will just leave 1. Since 3*1 will just be 3, we can shorten this equation all the way down to 6x(x + 3).
“Hold on,” you say. “We’re still super far from being able to factor z³ – z² – 9z” into (z – 3)(z + 3)(z – 1).” And you’d of course be right, but in the interest of keeping things simple, we’ll stop here and continue on next week. Or, if you need to know how to do that now, here is the link to the website that helped refresh my memory.