 This video is called a mash-up of problems. The reason I call it this is all five of them are all a little bit different but it's important to be able to go back and forth between different kinds of problems and know what to do. This one is called simplify. So that lets you know it's not a factoring problem. You're not going to take out a GCF, you're not going to try to make it simpler, you're actually going to be doing some multiplication. It is also important to remember in a problem like this what's going on with the exponent of 2. There are parentheses and outside of the parentheses is the exponent of 2. That tells you that everything inside the parentheses belongs to the exponent of 2, not just the 4. So if you're going to expand this before you started you would take everything in the parentheses and multiply it by everything in the parentheses. Then hopefully you recognize it's a binomial times a binomial and we have to do FOIL. So I'll multiply my firsts, X times X is X squared, my outside negative 4X, my inside negative 4X and the lasts. Negative 4 times negative 4 is a positive 16. We have like terms to combine, negative 4X times negative 4X or negative 4X plus a negative 4X, excuse me, is a negative 8X plus 16. So this would be our answer for simplify the quantity of X minus 4 squared. Our next problem is a factoring problem. So now you know you are going to be doing things like looking for GCF, making it simpler, breaking it down, dividing things out. So when I look here I ask can I take out a GCF? I don't know if we can because this term has a 1, a negative 10 and a 16 so the biggest number I could divide out is a 1 so it wouldn't change anything. Well this term has an X and so does the second term but the third term does not so I can't factor out an X. But the third term it doesn't have an X but it does have a Y. So does the second term but the first term does not so a Y cannot come out either. So it's time to factor as it is. Make your parentheses, let's break up the X squared to X and X, the Y squared to Y and Y and now we're going to spend a little bit of time thinking what adds to give us negative 10 and multiplies to give us 16. So let's make our list of factors of 16. I have 16 and 1, 8 and 2 and 4 and 4. Well any of those add up to a negative 10? They sure do. A negative 8 plus a negative 2 is negative 10. A negative 8 times a negative 2 is a positive 16. So we have a negative 8 and a negative 2. Let's try our next problem. This is a factoring problem and this is also called the difference of squares. The difference comes from the fact that this is a subtracting problem, there's a subtraction sign and the squares refers to the 49. You have a perfect square. So if you ever have a problem that looks like this with a negative, that's a difference and a perfect square it's called a difference of squares problem. The reason it's special is it almost allows us a little bit of a shortcut. Let's explore. This problem is really saying that they want x squared plus 0xw minus 49w squared. So it's asking what multiplies to give you a negative 49 and adds to give you a 0. Well when you do your factors of 49 you have 49 and 1 and you have 7 and 7. Well the only way to add to give you a 0 would be a plus 7 minus 7. So when you break this up the x squared becomes x and x, the w squared becomes w and w. You'll use a plus 7 and a minus 7. So this is powerful. When you've got a difference of squares you simply look at the perfect square. Well what's a square out of 49? It's 7 because 7 times 7 gives you 49. So use the perfect square, one of them, one of the 7's plus, one of the 7's minus and you will land at your answer. A second difference of squares problem. I know it's a difference of squares because I have a subtraction sign. I have two terms and the number behind my subtraction sign is a perfect square. So remember it's really asking you well x squared plus 0xy minus 121y squared. So it's really asking you what multiplies to give you a negative 121 and adds to give you a 0. Well you can kind of skip making your factoring list here because you recognize that you can use the perfect square. So square root of 121 is 11. So you're simply going to break up your x's, break up your y's and since the square root of 121 is 11 you'll do a plus 11 and a minus 11. Alright, let's move on to the problem. This one says factoring. There's two terms. I know it's not a difference of squares for a couple reasons. One, there's an adding sign here and two, 39 is not a perfect square. So to factor it, the first thing you're going to want to ask yourself is can I take out a GCF? And you certainly can. 13ab to the third could be 13ab, bb and b. 39 is 13ab, bb, bb, b. And I'm sorry, 13 and 3. So then, let's see, what does it have in common so I can take out a GCF? So I've got a 13 in common so I can take out a 13 for my GCF. I've got an a in common so I can take out an a for my GCF. How many pairs of b's do I have? I have one pair of b's, a second pair of b's, a third pair of b's. So my GCF is 13ab to the third but now the problem isn't asking me for the GCF it's asking me to factor. So I just did the first step and took out a GCF. Now to complete my factoring I need my parentheses and remember I started with two terms so in the parentheses I have to represent two terms. So let's think about this. 13 times what gets me back to the 13? A 1. I've got an a. How many a's do I need here to get back to just an a? None of them. I've got b to the third here so how many b's do I need here to get back to b to the third? None. So I really just have a 1 here because notice my GCF of 13ab to the third was the entire first term. I used the entire 13ab to the third. I used everything in that first list but keep in mind be very careful you still need this placeholder of a 1 because you need two slots and placeholders because you started with two terms. So let's finish up. 13 times what gets me back to 39? A plus 3. If I have an a I have an a. How many a's do I need here to get me back to a squared? You just need one. And then for the b's I have b to the third. How many b's do I need here to get back to b to the fourth? Just one. So my final answer is 13ab to the third times the quantity of 1 plus 3ab.