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Factoring - Why use it?

Whenever math gives you an easy topic, take it and run with it! I am sometimes surprised by how many students struggle (really struggle) with factoring. That said, I always struggled with quadratics when I was in high school. And forgetting to move negative signs and decimals when simplifying (whoops!). Fortunately, I've gotten past that. My point is that even tutors and teachers start somewhere, and common mistakes are called 'common mistakes,' for a reason. Each of us has an area where we are just not as strong. No shame in that!
 
Factoring a number is nothing more than pulling apart its pieces. Glorified division, if you will. The difference with factoring is that you are dividing a given number by one of its smaller parts.
 
For example:
 
Let's use the number 20. We know from multiplication that we can multiply every number by 1, and come out with the same number. Our first two factors of 20 are 1 and 20.
 
We also know that since 20 is an even number, we can divide it evenly by 2. Two times 10 equals 20. Those are our second factors of 20.
 
We can also divide 20 by 4 evenly. We end up with 5. 5 x 4 = 20.
 
We know we're done factoring when we reach consecutive (numbers that are right after the other) numbers. Four and five are consecutive numbers.
 
Our factors of 20 are 1 x 20, 2 x 10, and 4 x 5. And since mathematicians like things neat and orderly, we re-arrange these numbers from smallest to largest: 1, 2, 4, 5, 10, 20. Voila!
 
 
 
But what about 3? or 6? I mean, theoretically, we can divide 20 by lots of different numbers.
 
 
 
Well, yes and no. We CAN divide 20 by lots of different numbers, but they won't come out as a whole number. We'll end up with a whole bunch of decimals. (Or fractions - whichever you prefer.) And factors MUST be whole numbers, which means NO DECIMALS OR FRACTIONS.
 
 
'Ok,' you might be saying, 'so, what? Why do I need to know this?'
 
 
Here's where factoring gets fun.
 
Let's say you're taking a math test with no calculators allowed. Fun, yeah, I know. Stay with me,  we're getting there! You're working on a proportions problem and come up with some big weird number like 176/24. You're running out of time on your test and REALLY don't want to divide that bad boy out to simplify, but you don't want to get the problem wrong simply from not simplifying OR make some stupid mistake that will cause you to get the answer wrong anyway. What now?
 
FACTOR, FACTOR, FACTOR! Factoring to the rescue!
 
 
We have two even numbers, which means that they're both divisible by 2 evenly.
 
You end up with:
 
2 x 88
2 x 12
 
Ok, good start. Let's keep going and divide by 2 again.
 
2 x 2 x 44
2 x 2 x 6
 
Now, do we HAVE to divide by the same number on both? Heck no, we're factoring! Your largest number, 44, has factors of 4 and 11. And, we know that 2 x 2 is 4 (make sure you keep track of all those 2's). When we have the whole thing factored all the way out, we end up with:
 
2 x 2 x 2 x 2 x 11
2 x 2 x 2 x 3
 
So... how about all those 2's? Since we're working with a fraction, we can actually cancel some of those 2's out, since we know that dividing any number by itself equals 1. (2 รท 2 = 1)
 
2 x 2 x 2 x 2 x 11
2 x 2 x 2 x 3
 
We come out with 22/3, or 7.33. That's a lot easier to divide out, now isn't it?
 
 
Obviously, you'll need some practice before using this as a time-saver on a test. This method does require knowledge on your part on how to factor as well as how to divide and multiply. It's a fabulous way to check your long division.
 
Eventually, you will use this technique in more advanced math - like quadratics and square roots.  And as a final note, you do NOT necessarily need to factor all the way down to prime numbers, as long as you're sure your math is correct.