Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Perfect squares. The difference of squares is something of the form a ^2 -, b ^2 and that factors into a -, b * a + b. We could also think of that as A + b * a -, b if we wanted because of commutative property. So when we look at this, the first step is always to see if there's anything that they all have in common. X ^2 -, 9 doesn't have anything in common. So what times what gives me X ^2? And the answer is X&X. So that goes in the first location of our ordered pairs. Then what times what gives me? Nine. And one of them is going to be a minus and one's going to be a plus. If we thought about checking that by foiling it out, our first terms would give us X ^2. Our outer terms would give us a positive 3X our inner terms and -3 X our last terms and -9. So the positive 3X and -3 X would cancel, which indeed does check that we get X ^2 -, 9. We can always check these if we want. First step. Do they have anything in common? The answer's no. So what times? What gives me sixteen y ^2, 4 Y, and four Y? What times? What gives me 525? Sorry -5 and +5? If we check it, we will see that four y - 5 and four y + 5 do give us sixteen y ^2 - 25. First step. Anything in common? The answer is no. So 81 would be 9 * 9 A to the fourth. Well, a ^2 * a ^2 gives us A to the fourth, because remember, when we're multiplying variables, we add the exponents. B ^2 would just be b * b. So that whole thing is the first term of our parentheses. And then we're going to have one that's minus and one that's a + 49 seven times 7. And then C to the 8th, C to the fourth times C to the 4th would give us C to the 8th. So our final answer is going to be nine a ^2 B -, 7 C to the fourth, nine a ^2 B + 7 C to the 4th. We should maybe throw in a new one that has something in common before we start. Let's see, let's just do a kind of easy one. Let's do 4X squared minus 20. No, four X ^2 -, 36. First thing we do is we always factor out if there's anything in common. So we have X ^2 -, 9 left. So we'd get X -, 3 X plus three. That one's actually similar to this first one. If we had five X ^2 -, 80, let's say, we'd pull out the five first and we'd get an X ^2 -, 16 left, so we'd get an X -, 4 and an X + 4. So always make sure to check and see if there's something that can be pulled out first and then look for your difference of squares. We've got a few for you to try. And then we're going to look at perfect square trinomials. And this is a form that says a ^2 + 2 AB plus b ^2 or a ^2 -, 2 AB plus b ^2. And that's going to factor into A+B quantity squared or A -, b quantity squared. So this is a notation that I can do 2 formulas at once. So the big thing to look for here is this, this first one a perfect square and the last one a perfect square. And if the answer is yes, then do the two multiply together to give me the middle once I've actually factored it? So 49 Y squared would be 7 Y and seven Y +1 would be 1:00 and 1:00. But I needed it to multiply to give me a negative for the middle, multiply to give me a positive, but add to give me a negative. So there's a negative and a negative, so -7 Y and -7 Y gives me the -14 Y. So that indeed fits this formula. Here we're going to start by seeing is there anything that can factor out of everything? And the answer is a 2. So we get sixteen X ^2 + 20, four X + 9. Now to see if it's of the right form. 4X times 4X gives me the 16 X squared. I need it to multiply to give me a +9, but to add to give me a positive also. So +3 positive 3. When I look at the outside, I get 12X, the inside gives me 12X, so that gives me that 24X in the middle. If this didn't work, we would go back to a different method for trying to see if it's still factored here. Eight X ^2 -, 40 X plus 25. If we factor out of two, we get four X ^2 -, 20 X plus 25. So 2 * 2 X times 2X gives me the four X ^2, 5, and times 5 gives me a positive 25. But I need them to add to give me a negative. So it's going to be a -5 and a -5. When I check it, there's my -10 X another -10 X -20 X in the middle. So that indeed is the factorization. And then here's a few for you to try. Thank you and have a wonderful day.