1 00:00:08,870 --> 00:00:10,740 - This is section 8.7, 2 00:00:10,740 --> 00:00:12,883 and it's on the difference of squares. 3 00:00:21,240 --> 00:00:22,910 So this is a special kind 4 00:00:22,910 --> 00:00:25,820 of a factoring trick that we can use, 5 00:00:25,820 --> 00:00:28,160 and it's for factoring exactly this, 6 00:00:28,160 --> 00:00:31,360 we're factoring the difference of squares. 7 00:00:31,360 --> 00:00:35,900 So we're gonna have A squared minus B squared, 8 00:00:35,900 --> 00:00:37,380 and we wanna factor it, 9 00:00:37,380 --> 00:00:40,960 and remember factoring means write as a product. 10 00:00:40,960 --> 00:00:45,160 So what we wanna do is somehow write this like that, right? 11 00:00:47,750 --> 00:00:49,300 That would be factored. 12 00:00:49,300 --> 00:00:52,620 And we'll use an area model to kind of figure out, 13 00:00:52,620 --> 00:00:54,460 okay, well, how can we factor it? 14 00:00:54,460 --> 00:00:55,750 So in this area model, 15 00:00:55,750 --> 00:00:58,740 I wanna think of this as a length, and this is a width. 16 00:00:58,740 --> 00:01:00,910 So I want an area that's a rectangle, 17 00:01:00,910 --> 00:01:02,410 so that I can just say, oh, this is the length, 18 00:01:02,410 --> 00:01:03,590 and that's the width, 19 00:01:03,590 --> 00:01:06,560 and the whole area will be the product. 20 00:01:06,560 --> 00:01:07,750 And so what I'll do is I'll think, 21 00:01:07,750 --> 00:01:10,630 okay, so over here I need to interpret this as an area. 22 00:01:10,630 --> 00:01:15,040 This is A squared is a square, right, an A by A square, 23 00:01:15,040 --> 00:01:17,110 and I'm gonna cut a B by B square out of it. 24 00:01:17,110 --> 00:01:18,283 So I'm gonna put here, 25 00:01:19,610 --> 00:01:21,393 here's my A by A square. 26 00:01:28,510 --> 00:01:29,910 And then down in the corner, 27 00:01:32,390 --> 00:01:35,973 I cut this little B by B square out. 28 00:01:40,812 --> 00:01:42,710 That square's a little wonky there, 29 00:01:42,710 --> 00:01:46,423 but what I want is this blue area here, right? 30 00:01:48,030 --> 00:01:50,500 That would be the big area, 31 00:01:50,500 --> 00:01:52,680 which is A squared minus the little area, 32 00:01:52,680 --> 00:01:53,513 which is B squared. 33 00:01:53,513 --> 00:01:56,813 So the blue area there is A squared minus B squared, 34 00:01:57,660 --> 00:01:59,610 and I want to rewrite it as a product. 35 00:01:59,610 --> 00:02:01,310 So I wanna rearrange the blue area 36 00:02:01,310 --> 00:02:03,850 so that it's a rectangle. 37 00:02:03,850 --> 00:02:05,340 And what I'm gonna do is this, 38 00:02:05,340 --> 00:02:08,596 I'm gonna imagine that this is made out of wood 39 00:02:08,596 --> 00:02:10,620 or out of paper or something, 40 00:02:10,620 --> 00:02:12,220 and I'm gonna cut it right here. 41 00:02:16,480 --> 00:02:19,150 So now I'll take this little piece here, 42 00:02:19,150 --> 00:02:21,970 and what I wanna do is put it over there, 43 00:02:21,970 --> 00:02:23,360 but I'm not sure if it'll fit. 44 00:02:23,360 --> 00:02:27,910 So what is the length of that dotted line there? 45 00:02:27,910 --> 00:02:29,580 - [Student] Isn't it A? 46 00:02:29,580 --> 00:02:34,090 - Well, let's see, so A is here. 47 00:02:34,090 --> 00:02:36,769 So A is bigger. 48 00:02:36,769 --> 00:02:37,990 - Oh, you took away B. - Right. 49 00:02:37,990 --> 00:02:38,850 And we took that B away, 50 00:02:38,850 --> 00:02:42,010 so this dotted line is A minus B, right? 51 00:02:42,010 --> 00:02:44,180 The whole length minus that bit. 52 00:02:44,180 --> 00:02:46,508 And then this is the same, right? 53 00:02:46,508 --> 00:02:47,341 - Oh yeah. 54 00:02:47,341 --> 00:02:51,080 - 'Cause it would be the whole thing, which is A minus B. 55 00:02:51,080 --> 00:02:51,913 - [Student] Right, right. 56 00:02:51,913 --> 00:02:52,809 - So we're okay. 57 00:02:52,809 --> 00:02:54,650 So what we can do is this bit down here, this rectangle, 58 00:02:54,650 --> 00:02:58,270 and just put it up there and it'll match. 59 00:02:58,270 --> 00:03:00,300 So it's gonna now look like this. 60 00:03:04,853 --> 00:03:06,883 This blue area will be gone, 61 00:03:11,740 --> 00:03:13,340 and we will have put it up here. 62 00:03:17,880 --> 00:03:19,070 So of course it's just been moved, 63 00:03:19,070 --> 00:03:22,563 so that blue area is still A squared minus B squared, 64 00:03:23,700 --> 00:03:25,493 but now it's a rectangle. So we just have to think, 65 00:03:25,493 --> 00:03:27,540 well, what are the dimensions of that rectangle? 66 00:03:27,540 --> 00:03:28,840 So this one we have, right? 67 00:03:28,840 --> 00:03:32,803 This we just figured out is A minus B. 68 00:03:35,740 --> 00:03:38,920 And what about this long dimension? 69 00:03:38,920 --> 00:03:41,320 - [Student] Well, you added something to that A. 70 00:03:42,340 --> 00:03:45,023 - Yep, what did we add to A? 71 00:03:46,370 --> 00:03:48,060 - [Student] Did you add B? 72 00:03:48,060 --> 00:03:49,833 - Yeah, right, because it was this, 73 00:03:51,500 --> 00:03:52,333 this whole thing that moved up. 74 00:03:52,333 --> 00:03:55,100 So this B is right down here now. 75 00:03:55,100 --> 00:03:57,293 So this is A plus B. 76 00:04:00,920 --> 00:04:03,310 So again, the area of that rectangle 77 00:04:03,310 --> 00:04:04,470 is just length times width, 78 00:04:04,470 --> 00:04:07,090 so it's A minus B times A plus B, 79 00:04:07,090 --> 00:04:09,440 but it's also the original blue area, 80 00:04:09,440 --> 00:04:12,580 which was A squared minus B square, just rearranged, 81 00:04:12,580 --> 00:04:13,880 and we factored it, right? 82 00:04:14,810 --> 00:04:16,160 So now let's use this. 83 00:04:16,160 --> 00:04:17,460 So what we're gonna do is. 84 00:04:23,880 --> 00:04:25,960 We come across some polynomial, 85 00:04:25,960 --> 00:04:29,963 like for example, X squared minus 4, 86 00:04:30,800 --> 00:04:33,160 and we can say, aha, that's gonna be, 87 00:04:33,160 --> 00:04:35,146 that's the difference of two squares. 88 00:04:35,146 --> 00:04:37,107 So that's X plus 2 times X minus 2. 89 00:04:41,680 --> 00:04:43,180 - How did you know it was two? 90 00:04:44,170 --> 00:04:45,160 - Yes, good question. 91 00:04:45,160 --> 00:04:46,740 So we have to spot this and say, 92 00:04:46,740 --> 00:04:49,130 okay, so it's difference of two squares. 93 00:04:49,130 --> 00:04:52,283 This is X squared and this is 2 squared. 94 00:04:53,346 --> 00:04:54,720 - [Student] So I could rewrite it 95 00:04:54,720 --> 00:04:56,500 as X squared minus 2 squared? 96 00:04:56,500 --> 00:04:57,650 - Yeah, that would be smart actually. 97 00:04:57,650 --> 00:05:00,910 So you could write here, exactly, 98 00:05:00,910 --> 00:05:04,150 you could say this is X squared minus 2 squared, 99 00:05:04,150 --> 00:05:05,240 and then you can see, 100 00:05:05,240 --> 00:05:09,360 oh, look at this A is just X, and B is 2, 101 00:05:09,360 --> 00:05:11,140 and then it's gonna fit our template. 102 00:05:11,140 --> 00:05:12,933 So I guess I could have written it like. 103 00:05:21,920 --> 00:05:24,610 - [Student] I missed what you said the directions were. 104 00:05:24,610 --> 00:05:26,190 So if you give me this problem, 105 00:05:26,190 --> 00:05:28,150 it's going to say, 106 00:05:28,150 --> 00:05:30,130 it's just gonna have X squared minus 4, 107 00:05:30,130 --> 00:05:32,360 or is it gonna tell me to do something? 108 00:05:32,360 --> 00:05:34,510 - This is a great question, it's gonna say, 109 00:05:38,750 --> 00:05:39,583 just like that. 110 00:05:41,290 --> 00:05:42,370 - [Student] Oh, okay. 111 00:05:42,370 --> 00:05:45,520 - Factor, and remember factor here is like, 112 00:05:45,520 --> 00:05:48,250 we had like factor the noun, 113 00:05:48,250 --> 00:05:51,583 like this is a factor, and we had factor the verb. 114 00:05:53,260 --> 00:05:54,330 This is factor the verb, 115 00:05:54,330 --> 00:05:56,450 so it just means write it as a product. 116 00:05:56,450 --> 00:05:59,940 - [Student] Okay, thank you. 117 00:05:59,940 --> 00:06:03,370 - Yeah, so that's it. 118 00:06:03,370 --> 00:06:05,020 That's difference of two squares.