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Welcome to Algebra I: Special Products

33 minutes

Hey, guys. Welcome to Algebra 1. Today's lesson focuses on finding special products. Remember how you multiplied binomials together because that will come in handy in this lesson. You ready to get started? Let's go. Okay. Let me take you back, just to jog your memory about multiplying binomials together. Now, we used FOIL to help us get through this. It's one strategy; you may know others. But it helps us multiply binomials together. I see this problem and I'd say, F, for FOIL is telling me to multiply the first two terms together. Let me move this up to get some space. Okay, so I'd say, all right, X times X, that's X squared. I'd keep working my way through FOIL. So, the O, those are my outer terms. So X times negative 6, that gives me negative 6X. So then, I'd come back up here, I'd write my minus 6X. Now, I'm at the I, so I'm at my inner terms. So 6 times X... that's 6X, so I have plus 6X. Then, the L, so I'm at my last terms. So 6 times negative 6, that's negative 36, right? So we'd bring that up here. After we're finished FOILing, we'll look at our terms and see if there were like terms to combine, and there are. Okay, let's get this work out of the way. Okay. Here, I can combine this negative 6X with this positive 6X. Negative 6X plus 6X, that's just 0. That's nothing, so these, when you combine them, just completely wipe out. And you're left with X squared minus 36. We used FOIL to work through that one. I'll do a few more. You'll notice a pattern whenever my factors are set up in this particular way. Look at these factors. Look at this next one, and you're going to start to notice something. All right, so this one, I'm multiplying X plus 4, times X minus 4, right? I'm still using FOIL. I won't show my work on the bottom while I'm FOILing through since you've already done FOILing and you probably already know the steps. So for the F, the first two terms, so X times X-- I know that's X squared. All right, so my O, outer terms. So X times negative 4, that's negative 4X, right? Still with me? Okay, the I, so my inner terms-- so 4 times X, so that's plus 4X. Then, my L, so my last terms, so 4 times negative 4, that's negative 16, right? Now you're at the point you can combine like terms. You have some in the middle here. Combine that negative 4X with that positive 4X. So negative 4X plus 4X, that's 0, just completely wiped out. Really, you're just left with X squared minus 16. Starting to notice anything yet? Let's do a couple more. So now I've got-- let's move this up a little bit-- X plus 7, times X minus 7. So I'll bring FOIL up here, keep me going. So my first terms: X times X, that's X squared. My outer terms, so X times negative 7, that's negative 7X, so minus 7X. My inner terms, 7 times X, so that's 7X, so plus 7X, okay? Then, the Ls are your last terms. 7 times negative 7, that's minus 49, okay. Remember, now we're at the point of combining your like terms. They're in the middle here. So, negative 7X plus 7X, that just wipes out, right? That's 0. You're left with X squared minus 49. Those wheels turning? See something that's going on here? Let's look at it all together. The first product, when you multiplied X plus 6, times X minus 6, our answer was X squared minus 36. Then, we multiplied X plus 4, times X minus 4; our answer was X squared minus 16. And when we multiplied X plus 7, times X minus 7, our answer was X squared minus 49. What's happening in this special product-- on the left, my factors-- they're a sum and a difference. For each, I have the sum of two terms and then their difference. The sum of two terms, then their difference; the sum, and then the difference. When we have a problem that's written in the form of M plus N, times M minus N-- to make it general, to use some variables-- but it's written in that format: a sum and then a difference, your answer is always in the same form. Look at our answers over here. So for the first one, let's identify the M and the N. In this one, our M is X... and our N is 6, right? Those are our M and our N. Our answer was X squared minus 6 squared, which is 36. Keep going with me. Over here, our M was X-- I'll write the X here-- and our N was 4, right? Those were our sum and our difference. We added and subtracted, and our answer was X squared minus 16, which is 4 squared, right? Let's look at this last one. In this case, our M was X. What was our N here? Our N was 7. And then our answer was X squared minus 49, which is 7 squared. The format of our answers is also always taking the same shape. It's always going to be whatever your M is, squared, minus whatever your N is, squared. That is the terms' sum and difference. Then, we call the answer "the difference of squares." Whatever those two terms are-- if written in this format, when finding your product-- the answer is this format. Square the first one, square the second one-- focusing on those terms-- and it's the difference of those two, okay? Let's do a couple to get the hang of it. So, find this product. It's in that format of a sum. I have the sum of X and 9, then the difference of X and 9. With this shortcut, you don't need to do FOIL. Recognizing that pattern, just use what you know. Because you know when things take the shape of M plus N, times M minus N, your answer is always M squared minus N squared. Let's figure out what the M and the N are. We can skip FOIL. In this case, I see that M is X, and the N is 9, right? Following this special products rule over here, I'll square these terms, and it's the difference of them. This answer would be X squared minus 81, which is 9 squared, right? Let's do another one. Okay, so here I see my first-- or my M is X; the N is 10 because it's in the format of X plus 10, X minus 10. I'll just focus on the M and the N. So, my M is X, so I need to square that. That's X squared. Then, the N is 10, so I need to square that. So 10 squared, that's 100. Then you're done. You could avoid FOIL completely with this. That's the trick of these special products. All right, so we're at the point where you can try some. I've got products on the left and some answers on the right. Press pause, take a few minutes, and see how you do with these. To compare answers with me, press play.

(female describer) Match. On the left, number 1: X plus 5, times x minus 5. Number 2: X plus 11, times X minus 11. Number 3: X plus 3, times X minus 3. On the right: X squared minus 9, X squared minus 25, and X squared minus 121.

(instructor) You ready to check? All right, so for number 1, I have X plus 5, times X minus 5. So, I think, okay, I see my M is X. That first term is X, the second one's 5. I'm looking for X squared minus 5 squared, and that is this one right here: X squared minus 25. Okay? Here, I see my first term is X, the second one's 11 for this sum and difference here. So I'm looking for X squared minus 11 squared. I scan, I see it's that last one. All right. The last one here... You could use process of elimination, but the math teacher in me won't let you. Let's think our way through it. That the first term is X. Ignoring my signs, I have X. The second one's 3. So I need X squared minus 3 squared, and that's 9. Here we go. That's how I knew that was that one. All right? Now, you could also, if you're not feeling comfortable about recognizing that pattern, just use FOIL for your answer, okay? I'll show you a different strategy. To use FOIL without recognizing those special products all the time, that's fine. Or use it to check your answer. We already knew this one, the answer should be X squared minus 25. To verify that, to check that, go ahead and FOIL it out to check. All right? Your first terms, X times X-- so I know that that's X squared, all right? The outer terms, so X times negative 5, that's negative 5X, right? Then your inner terms, 5 times X, so that's positive 5X. Then L, so your last terms, so 5 times negative 5. So that is minus 25. Look for where you can combine your like terms. We're noticing they're in the middle, generally, with these. I have negative 5X plus 5X, that just wipes out; that's 0. So you have X squared minus 25. That is another way to verify your answer. See if you're right before pressing play to see what I got. Here, if you wanted to check, you could make sure you were right. The answer was X squared minus 121. Let's just verify we get that when we FOIL this out, too. Let's go ahead and write FOIL. Okay, so the F, so X times X, that's X squared. The O, so the outer. X times negative 11, so that's negative 11X. Then my I's, or my inner terms-- 11 times X, so that's positive 11X. Then, my last terms, so 11 times negative 11, negative 121. You see those two like terms right in the middle, we can combine these. So negative 11X plus 11X wipes out; it's just 0. And you're left with X squared minus 121, okay? You checked it, you verified it, it was right. Let's check this last one. The answer was X squared minus 9, so let's just verify that with FOIL. All right, so our first terms-- so X times X, I know that that's X squared, right? Our outer terms, so X times negative 3, that's negative 3X, right? Inner terms, so 3 times X, that's 3X. Then L, so our last terms, 3 times negative 3, that's negative 9, so minus 9. You see you have your like terms, you can combine right in here. So negative 3X plus 3X wipes out and you're just left with X squared minus 9. So we verified our answer, we're good to go. All righty. There's another type-- or two more types of special products that I want to show you. Knowing what you do about exponents, we have the quantity (X plus 3) raised to the 2nd power. When you raise something to the 2nd power, that means you're multiplying it by itself. This X plus 3 raised to the 2nd power just means X plus 3 times X plus 3. That's what that means to do. To get this product, what do we do? We're going to FOIL, right. Okay. So our F, our first terms, so X times X, X squared. Outer terms, so X times 3, 3X. Inner terms, so 3 times X, is another 3X. Then, our last terms, so 3 times 3, that's 9. You have some like terms we can combine in the middle. So 3X plus 3X, that's 6X, right? So X squared plus 6X plus 9. So when we raise X plus 3 to the 2nd power, we get X squared plus 6X plus 9, okay? Let's do more. There's another pattern here to spot. So, X plus 5 squared, so you see that exponent-- what does it mean? If it's to the 2nd power then I'm multiplying X plus 5 by itself, all right? So X plus 5, times X plus 5, right? Then how do I handle this? I FOIL, right? Okay, so the F, my first terms, X times X; X squared. Now my outer terms, so X times 5, that's 5X. Now my inner terms, so 5 times X, that's 5X. Then, now my last terms, so 5 times 5, that's 25, okay? After you FOIL, what do you do next? Right, combine your like terms, and you see you have two in the center. You've got 5X plus 5X, that's 10X, right? So X squared plus 10X plus 25, okay? When we raise X plus 5 to the 2nd power, we end up with X squared plus 10X plus 25, okay? Starting to see something yet? Let's do a few more. We're switching up a bit by changing the sign, but I still think you'll notice what's going on here. So we see that we're raising that X minus 4 to the 2nd power. That means we're going to multiply it by itself. I've got my X minus 4, times X minus 4. FOIL it out, right? Okay, so your first terms, X times X, so that's X squared, okay? Then, your outer terms, so X times negative 4, negative 4X. Inner terms, so negative 4 times X, that's another negative 4X. Then, the last terms, so negative 4 times negative 4, that's a positive 16. Okay, so then, right, combining like terms, so negative 4X minus 4X, that'll give me negative 8X. o X squared minus 8X plus 16. Raising X minus 4 to the 2nd power gives me X squared minus 8X plus 16, okay? I'll do one more. Let's see if you've noticed. You've probably noticed by now, like we've been doing, that this means X minus 7, in this case, times X minus 7, right? Then, let's FOIL. Okay, so our first terms, X times X; X squared. Our outer terms, so X times negative 7; negative 7X. Then our inner terms, so negative 7 times X, that's another negative 7X. Then, negative 7 times negative 7, so that's 49, okay? Again, like terms to combine, so negative 7X minus 7X-- you combine that, you get X squared minus 14X right in there, plus 49. Okay, so we raised X minus 7 squared-- or raised X minus 7 to the 2nd power-- and got X squared minus 14X plus 49. Look at the answers all together and see if you spot it. Look at the ones that we were adding first. When we multiplied, when we wrote-- raised X plus 3 to the 2nd power, we got X squared plus 6X plus 9. When we raised X plus 5 to the 2nd power, we got X squared plus 10X plus 25. So thinking about what we noticed in that first special product, consider these terms separately: that you have an X, and then a 3, in this first one. You notice, if I write this again as M plus N squared here, because I'm squaring my two terms, my M and my N-- that sum. I consider the M first, consider this first term first. It's that first term squared for both of these. You end up squaring that first term for the answer. I'm going to write M squared here. Look at this middle term. Notice that as far as the coefficient is concerned, it's double that N, because 3 times 2 would give you 6, right? 2 times 5 would give you 10 in this one. That's how we're getting the number, but where did the X come from? Notice this; I'm going to write this first. You're actually doubling the product of your M and your N. Think about that for a second. I've got X and 3, right? X times 3, that's 3X, and if I double that, that's 6X, all right? Look a the second one: X times 5, that's 5X, and if I double that, that's 10X. That happens with all problems like this, every single time, okay? Now, let's look at that last term. I see here it's 9. Look back at our original problem. We see the number, the constant term was 3. What do we know about exponents and products? 3 squared is 9, okay; we know that to be true. Look at the next one; 5 squared is 25. So that last term is always N squared. Whatever the N was originally, it's always that one squared. And we call this the sum of squares. What we're doing right here, is called a perfect square trinomial--lot of words. Basically, you're squaring a binomial, right? And your answer, they call it a perfect square trinomial, because it ends up with three terms. And it always takes this format. So M plus N squared 2 always ends up in this format: M squared plus 2MN plus N squared. Use that shortcut when you recognize it, and remember you can always check your answer with FOIL. Let's look at when we had the difference here. So our answer here, if we think back, it was X squared minus 8X plus 16, right? Then, our answer for this one was X squared minus 14X plus 49. Because we just did that, the one, the product, that perfect square trinomial, you probably recognize this one a little bit faster, right? Because you see these are in the format M minus N squared, but essentially, the same thing's going on over here as it was before; it's just that the sign is different. Because our first term here is the M squared, right? The second term, it's still the product of these two terms. I'm still multiplying these together and doubling it. It's just that now it's always a minus. So this is always minus 2MN. Because X times 4, that's 4X, double that, that's 8X, but I was subtracting, so it's like you're treating this as a negative 4. Then, 7-- or X times negative 7-- so that's negative 7X, and then you're doubling that, so negative 14X, okay? The last terms here, if I look at this, all right, 4 squared, that's 16. 7 squared, that's 49. So it's still the end terms squared over here, okay? If you have a difference here, still a perfect square trinomial, you just have a slight sign change with this one. Use this shortcut if you have a problem. And you can always check your answer with FOIL, okay? All right, let's keep going. Look at this one. X plus 6 squared is equivalent to? We've got to pick the right answer. Use the shortcut and what we know about that special product. So, I see I'm adding here, right? I know this is in the form of M plus N squared, and knowing what we learned, I know my answer should take this format: M squared plus 2MN plus N squared, okay? I look through my answer choices and see which one takes that answer format. Then I'm good. So, let's see. To help me, I'm going to substitute over here. I'll say, all right, that M plus N squared, that's like X plus 6 squared, right? So M squared, that means I need to square my X, so X squared. Now, that middle term should be 2 times M, times N. Let's do M times N first. X times 6, that's 6X, and then I'm doubling that, so 12X, right? Then, that N squared should be 6 squared, so 6 squared, 36. Just scan your answer choices and find that one. And if you look, it's B. So you see how we can use that shortcut to get to the answer. Check it with FOIL, just to be sure. All right, this one. Okay, so similar idea, but this time we have subtraction. Okay, so X minus 3 squared is equivalent to? So I'm going to write my M minus N squared, and then I'll have M squared minus 2MN plus N squared because that was the pattern that we saw before. Let's figure out what's going on here. We've got our X minus 3 squared; I'll write that under it to keep straight. So, the first thing I need to is square the M. In this case, that's X, so my first term, that'd be X squared, right? Next, I need to multiply M times N together, double that, and it's a minus here, so write that down. M times N, that would end up giving me 3X, right? X times 3 and then I need to double that, so if I double 3X, that's 6X. Then, N squared, so that would be 3 squared, that's 9, okay? Now, just scan your answer choices and it's C. Okay. We're at the point where it's your turn to try. Use what you learned about special products and figure out the right answer. To check your answer, press play.

(female describer) X plus 9, squared, is equivalent to: Letter A: X squared plus 81X plus 18. Letter B: X squared plus 18X plus 81. Letter C: X squared minus 18X plus 81. You ready to check? Let's go. I see I'm adding here with my problem-- it's X plus 9, squared. My answer is going to take the format... let's see, M plus N squared-- I'll have M squared plus 2MN plus N squared. It helps to write that, to remember it, and then it just helps you keep things organized. You know what you're following as you move along. I have X plus 9, squared; that's what I'm working on here. If I'm following this pattern, then I need to square the first term, so that's X squared. I need to multiply X times 9, so that's 9X, and then double that, so that's 18X. I need to square the N, so in this case, that'd be 9 squared, which is 81. Okay, you've got your answer, now just scan the answer choices. It's B, okay? All right. To see another way-- because remember, we could always check with FOIL-- here you go. Remember when we started, this means X plus 9, times X plus 9. That's all that means. Okay, so if you FOIL, then I have X times X, so that's X squared. So my outer terms next, so X times 9, so that's 9X, right? My inner terms, so 9 times X, that's another 9X, okay? Then, your last terms, so 9 times 9, that's 81. Then, look and see where your like terms, like we've been noticing, they're in the center. So 9X plus 9X, that's 18X. So X squared plus 18X plus 81. You could always just throw back to FOIL to check your answer. I hope you got a handle on special products, and know we could still use FOIL just to verify that our answer's correct. See you back here soon. Bye!

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In this program, students will learn about special products. The term "product" refers to multiplying two amounts. In some instances, students will have to answer problems involving product of a sum, product of a difference, and product of a sum and a difference. Part of the "Welcome to Algebra I" series.

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Runtime: 33 minutes

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Episode 1
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Grade Level: 7 - 12
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