Factoring (Distributive Property in Reverse)
We can work the distributive property in reverse—we just need to check our rear view mirror first for small children.
When we rewrite ab + ac as a(b + c), what we are actually doing is factoring. A factor in this case is one of two or more expressions multiplied together. Factoring an expression means breaking the expression down into bits we can multiply together to find the original expression. Why would we want to break something down and then multiply it back together to get something equivalent to what we started with in the first place? Oh, who knows. Those crazy mathematicians have a lot of time on their hands. It actually will come in handy, trust us.
The factoring of expressions is similar to factoring numbers. Click here for a refresher.
1. Factor the expression 3x2 - 27xy. Check to see that your answer is correct
Since each term of the expression has a 3x in it (okay, true, the number 27 doesn't have a 3 in it, but the value 27 does), we can factor out 3x to find that 3x(x - 9y). We can check that our answer is correct by using the distributive property to multiply out 3x(x - 9y), making sure we get the original expression 3x2 - 27xy. We do, and all of the Whos down in Whoville rejoice.
The value 3x in the example above is called a common factor, since it is a factor that both terms have in common. If these two ever find themselves at an uncomfortable office function, at least they'll have something to talk about. When we factor an expression, we want to pull out the greatest common factor. The greatest common factor is a factor such that, after we pull it out, we have no more factoring left to do. That would be great, because as much as we love factoring and would like nothing more than to keep on factoring from now until the dawn of the new year, it is almost our bedtime. Let's find ourselves a GCF and call this one a night.
2. Factor the expression 45x - 9y + 99z.
3 is a common factor, and therefore, we can factor the expression as 3(15x - 3y + 33z). However, we can factor this expression even further because all of the terms in parentheses still have a common factor, and 3 isn't the greatest common factor. Although it is still great, in its own way. Really, really great. As great as you can be without being the greatest. This is us desperately trying to save face.
Instead, let's be greedy and pull out 9. If we factor out 9 we find that 9(5x - y + 11z). The terms in parentheses have nothing else in common to factor out, and 9 was the greatest common factor. Not that that makes 9 superior or better than 3 in any way, it is that...well, it is that 3 is simply...oy. Insert foot into mouth.
To find the greatest common factor for an expression, look carefully at all of its terms. The number part of the greatest common factor will be the largest number that divides the number parts of all the terms. The variable part of a greatest common factor can be figured out one variable at a time. Much easier. Both to do and to explain.
For each variable, find the term with the fewest copies. Use that number of copies (power) of the variable. Finally, multiply together the number part and each variable part. This step will produce the greatest common factor.
Sometimes we have a choice of factorizations, depending on where we put the negative signs. We will show you now what we mean by that; grab a bunch of negative signs and follow us...
Factor the expression - 50x + 4y in two different ways.
First way: Factor out 2 to find that 2(- 25x + 2y).
Second way: Factor out - 2 to find that - 2(25x - 2y). You can double-check to see that this is a correct factorization by using the distributive property.
These factorizations are both correct. Neither one is more correct, so let's not get all in a tizzy. Which one you use is merely a matter of personal preference. Or maybe a matter of your teacher's preference, if your teacher asks you to do these problems a certain way. If they do, don't fight them on it. They are bigger than you. Or at least they were a few years ago.
Be Careful: Always check your answers to factorization problems. When you multiply factors together, you should find the original expression. This step is especially important when negative signs are involved, because they can be a tad tricky. In fact, you probably shouldn't trust them with your social security number. Especially if your social has any negatives in it.