SM2 17.3.1 Factoring with Polynomial Identities

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[00:00:01]Hello and welcome. So we're going to be factoring  the following expression completely if possible.

[00:00:06]Now the first thing to check for with factoring  is a GCF which we don't have one here so moving on. The next thing would be to double check  your number of terms which usually tells you how to factor. Now here we have four terms  which generally says factoring by grouping which is a split down the middle. Look at your  first two and your last two terms for GCFs of the two. You can notice there's at least a GCF  of two there. They're both even numbers in this first one. You also have a GCF of w. Yay right.  So 2w is minimum there. There could be a larger one. Now of your last two terms however there  is nothing in common between those so bummer.

[00:00:44]Now factoring by grouping would generally say  keep rearranging the terms until it works. I'm just going to put a spoiler in here. Rearranging  isn't going to help us with this. All right. So then we'd say okay well how am I going to factor  this. A factor by grouping is not an option.

[00:01:00]Well take a look at some of the numbers we have in  here. We have 144, 25, and 121, right? Also look at the variables attached to these. Everything  here is a perfect square. The perfect square of your first term is 12w. If I were to take that and  square it you get 144w squared. Same thing here. I have 5n is the number you would have to — oops not  the n — you would have to square in order to get 25n squared. Same thing here. 11b would be your  number to square in order to get 121b squared.

[00:01:38]There's a lot of perfect squares here. So then  I'd say okay let's look at some of our polynomial identities that work with perfect squares.  There's a couple of them. You have your perfect square trinomials, right? So you have a plus b  squared equals a squared plus 2ab plus b squared is going to be that one. Now that plus minus  is indicating if it's a plus it will be a plus there in our equation. If it's a minus it'll be  a minus. Nothing too crazy. And then our other perfect square identity is going to deal with our  difference of squares, right? If you have a minus b and a plus b multiplied it's called a difference  of squares. There that's going to be your a squared minus b squared. Okay. So when I'm looking  at that I'd say all right well when I'm first looking at this we have a lot more than just two  things, right? We have four things but I'm going to break this down. I'm going to say okay what  if I just look at the first three things on their own. I look at just these. Do I have my perfect  square trinomial? And the answer, yes. Because if I'm looking at this I'd say okay so perfect square  which is 12w squared. Perfect square which is our 5n. Right. And then we have our middle term which  is the product of our first two perfect squares.

[00:03:07]So 12w times 5n and then that has to be doubled.  And we're going to say is that the same as what we have there? Well 12 times 5 is 60 times 2 is 120  and the w n multiply just w n. Boo yah, ding ding, right they match. Okay. The sign I'm not going  to look at too deeply because honestly the plus minus is not going to make a difference. It will  just tell you what's in the parentheses. We're just seeing if we get the same numeric value as  the middle term there when we multiply. So it matches. Yay. That tells you that perfect square  trinomial is what we're going to be doing here. So that means we have our first perfect square 12w,  our second perfect square 5n, and the sign in the middle is a minus so we'll just use minus here, squared. Minus our 121b squared. Our second term, right? Our minus 121b squared. That hasn't  changed. It hasn't done anything. It's just tacked on behind. All right. So then I say great. Now we  have first term minus second term, right? We have factored the first three into one thing together.  So I'd say great now what do I do because my goal with factoring, right, is just parentheses being  multiplied. No other addition or subtraction, right? So this continues to factor. And then you  might be wondering how? Well keep in mind you have two terms. You have two terms that are squared  meaning that you have a difference of squares here, right? We have two squares with subtraction  between them. Your first square is the (12w minus 5n). Your second square is the 121b. All right. Which means — oh my apologies almost made a mistake there — that is 11b. So our  second square is 11b. Okay. Same thing. Your first perfect square 12w minus 5n. Your second perfect  square 11b. And then from there the identity says that when you have those two perfect squares  being subtracted then you can write them out separately with addition and subtraction. It feels  a little weird because you have two terms inside your first perfect square but that is actually a  totally valid identity because this is the same thing twice, right? It's (12w minus 5n) squared.  Expressions can be perfect squares as well. So there's your first perfect square altogether. Now  would I put in those orange parentheses? No. My true answer would actually honestly just be that (12w - 5n + 11b) and (12 - 5n... whoops keep forgetting my variables today.  Uh minus or plus 11b. Essentially this thing. I'm having a harder time writing it  smaller. There you go. Thanks for watching.