1 00:00:00,000 --> 00:00:00,600 2 00:00:00,600 --> 00:00:02,770 And now I want to do a bunch of examples dealing with 3 00:00:02,770 --> 00:00:07,160 probably the two most typical types of polynomial 4 00:00:07,160 --> 00:00:10,550 multiplication that you'll see, definitely, in algebra. 5 00:00:10,550 --> 00:00:13,640 And the first is just squaring a binomial. 6 00:00:13,640 --> 00:00:20,600 So if I have x plus 9 squared, I know that your temptation is 7 00:00:20,600 --> 00:00:24,630 going to say, oh, isn't that x squared plus 9 squared? 8 00:00:24,629 --> 00:00:26,259 And I'll say, no, it isn't. 9 00:00:26,260 --> 00:00:28,260 You have to resist every temptation on the 10 00:00:28,260 --> 00:00:29,660 planet to do this. 11 00:00:29,660 --> 00:00:32,280 It is not x squared plus 9 squared. 12 00:00:32,280 --> 00:00:40,609 Remember, x plus 9 squared, this is equal to x plus 9, 13 00:00:40,609 --> 00:00:43,630 times x plus 9. 14 00:00:43,630 --> 00:00:47,940 This is a multiplication of this binomial times itself. 15 00:00:47,939 --> 00:00:49,349 You always need to remember that. 16 00:00:49,350 --> 00:00:52,899 It's very tempting to think that it's just x squared plus 17 00:00:52,899 --> 00:00:55,759 9 squared, but no, you have to expand it out. 18 00:00:55,759 --> 00:00:59,159 And now that we've expanded it out, we can use some of the 19 00:00:59,159 --> 00:01:02,250 skills we learned in the last video to actually multiply it. 20 00:01:02,250 --> 00:01:04,370 And just to show you that we can do it in the way that we 21 00:01:04,370 --> 00:01:09,910 multiplied the trinomial last time, let's multiply x plus 9, 22 00:01:09,909 --> 00:01:15,759 times x plus a magenta 9. 23 00:01:15,760 --> 00:01:18,850 And I'm doing it this way just to show you when I'm 24 00:01:18,849 --> 00:01:20,879 multiplying by this 9 versus this x. 25 00:01:20,879 --> 00:01:21,739 But let's just do it. 26 00:01:21,739 --> 00:01:23,969 So we go 9 times 9 is 81. 27 00:01:23,969 --> 00:01:25,799 Put it in the constants' place. 28 00:01:25,799 --> 00:01:29,629 9 times x is 9x. 29 00:01:29,629 --> 00:01:32,829 Then we have-- go switch to this x term-- we have a yellow 30 00:01:32,829 --> 00:01:36,239 x. x times 9x is 9x. 31 00:01:36,239 --> 00:01:38,021 Put it in the first degree space. 32 00:01:38,022 --> 00:01:40,960 x times x is x squared. 33 00:01:40,959 --> 00:01:43,669 34 00:01:43,670 --> 00:01:45,700 And then we add everything up. 35 00:01:45,700 --> 00:01:52,520 And we get x squared plus 18x plus 81. 36 00:01:52,519 --> 00:01:57,649 So this is equal to x squared plus 18x plus 81. 37 00:01:57,650 --> 00:02:00,560 Now you might see a little bit of a pattern here, and I'll 38 00:02:00,560 --> 00:02:02,965 actually make the pattern explicit in a second. 39 00:02:02,965 --> 00:02:06,010 But when you square a binomial, what happened? 40 00:02:06,010 --> 00:02:07,750 You have x squared. 41 00:02:07,750 --> 00:02:11,189 You have this x times this x, gives you x squared. 42 00:02:11,189 --> 00:02:13,919 You have the 9 times the 9, which is 81. 43 00:02:13,919 --> 00:02:16,949 And then you have this term here which is 18x. 44 00:02:16,949 --> 00:02:18,909 How did we get that 18x? 45 00:02:18,909 --> 00:02:23,210 Well, we multiplied this x times 9 to get 9x, and then we 46 00:02:23,210 --> 00:02:26,510 multiplied this 9 times x to get another 9x. 47 00:02:26,509 --> 00:02:29,959 And then we added the two right here to get 18x. 48 00:02:29,960 --> 00:02:33,250 So in general, whenever you have a squared binomial-- let 49 00:02:33,250 --> 00:02:33,889 me do it this way. 50 00:02:33,889 --> 00:02:38,250 I'll do it in very general terms. Let's say we have a 51 00:02:38,250 --> 00:02:39,629 plus b squared. 52 00:02:39,629 --> 00:02:42,340 53 00:02:42,340 --> 00:02:44,740 Let me multiply it this way again, just to give you the 54 00:02:44,740 --> 00:02:45,629 hang of it. 55 00:02:45,629 --> 00:02:53,250 This is equal to a plus b, times a plus-- I'll do a green 56 00:02:53,250 --> 00:02:54,770 b right there. 57 00:02:54,770 --> 00:02:58,439 So we have to b times b is b squared. 58 00:02:58,439 --> 00:03:00,409 Let's just assume that this is a constant term. 59 00:03:00,409 --> 00:03:04,509 I'll put it in the b squared right there. 60 00:03:04,509 --> 00:03:05,859 I'm assuming this is constant. 61 00:03:05,860 --> 00:03:07,060 So this would be a constant, this would be 62 00:03:07,060 --> 00:03:08,659 analogous to our 81. 63 00:03:08,659 --> 00:03:11,639 a is a variable that we-- actually let me change that up 64 00:03:11,639 --> 00:03:13,039 even better. 65 00:03:13,039 --> 00:03:21,269 Let me make this into x plus b squared, and we're assuming b 66 00:03:21,270 --> 00:03:22,750 is a constant. 67 00:03:22,750 --> 00:03:30,889 So it would be x plus b, times x plus a green b, right there. 68 00:03:30,889 --> 00:03:34,699 So assuming b's a constant, b times b is b squared. 69 00:03:34,699 --> 00:03:38,060 b times x is bx. 70 00:03:38,060 --> 00:03:40,090 And then we'll do the magenta x. 71 00:03:40,090 --> 00:03:43,140 x times b is bx. 72 00:03:43,139 --> 00:03:46,899 And then x times x is x squared. 73 00:03:46,900 --> 00:03:50,330 So when you add everything, you're left with x squared 74 00:03:50,330 --> 00:03:56,840 plus 2bx, plus b squared. 75 00:03:56,840 --> 00:04:02,009 So what you see is, the end product, what you have when 76 00:04:02,009 --> 00:04:06,340 you have x plus b squared, is x squared, plus 2 times the 77 00:04:06,340 --> 00:04:10,490 product of x and b, plus b squared. 78 00:04:10,490 --> 00:04:13,750 So given that pattern, let's do a bunch more of these. 79 00:04:13,750 --> 00:04:17,910 80 00:04:17,910 --> 00:04:19,709 And I'm going to do it the fast way. 81 00:04:19,709 --> 00:04:23,850 So 3x minus 7 squared. 82 00:04:23,850 --> 00:04:25,980 Let's just remember what I told you. 83 00:04:25,980 --> 00:04:29,340 Just don't remember it, in the back of your mind, you should 84 00:04:29,339 --> 00:04:30,779 know why it makes sense. 85 00:04:30,779 --> 00:04:33,349 If I were to multiply this out, do the distributive 86 00:04:33,350 --> 00:04:36,740 property twice, you know you'll get the same answer. 87 00:04:36,740 --> 00:04:44,730 So this is going to be equal to 3x squared, plus 2 times 88 00:04:44,730 --> 00:04:48,990 3x, times negative 7. 89 00:04:48,990 --> 00:04:49,280 Right? 90 00:04:49,279 --> 00:04:53,629 We know that it's 2 times each the product of these terms, 91 00:04:53,629 --> 00:04:57,699 plus negative 7 squared. 92 00:04:57,699 --> 00:05:01,199 And if we use our product rules here, 3x squared is the 93 00:05:01,199 --> 00:05:03,949 same thing as 9x squared. 94 00:05:03,949 --> 00:05:06,439 This right here, you're going to have a 2 times a 3, which 95 00:05:06,439 --> 00:05:11,980 is 6, times a negative 7, which is negative 42x. 96 00:05:11,980 --> 00:05:16,670 And then a negative 7 squared is plus 49. 97 00:05:16,670 --> 00:05:17,930 That was the fast way. 98 00:05:17,930 --> 00:05:20,600 And just to make sure that I'm not doing something bizarre, 99 00:05:20,600 --> 00:05:22,950 let me do it the slow way for you. 100 00:05:22,949 --> 00:05:32,319 3x minus 7, times 3x minus 7. 101 00:05:32,319 --> 00:05:36,939 Negative 7 times negative 7 is positive 49. 102 00:05:36,939 --> 00:05:42,310 Negative 7 times 3x is negative 21x. 103 00:05:42,310 --> 00:05:47,220 3x times negative 7 is negative 21x. 104 00:05:47,220 --> 00:05:52,280 3x times 3x is 9 x squared. 105 00:05:52,279 --> 00:05:54,059 Scroll to the left a little bit. 106 00:05:54,060 --> 00:05:55,050 Add everything. 107 00:05:55,050 --> 00:06:02,060 You're left with 9x squared, minus 42x, plus 49. 108 00:06:02,060 --> 00:06:05,050 So we did indeed get the same answer. 109 00:06:05,050 --> 00:06:07,579 Let's do one more, and we'll do it the fast way. 110 00:06:07,579 --> 00:06:12,129 So if we have 8x minus 3-- actually, let me do one which 111 00:06:12,129 --> 00:06:13,920 has more variables in it. 112 00:06:13,920 --> 00:06:19,370 Let's say we had 4x squared plus y squared, and we wanted 113 00:06:19,370 --> 00:06:21,360 to square that. 114 00:06:21,360 --> 00:06:22,720 Well, same idea. 115 00:06:22,720 --> 00:06:27,820 This is going to be equal to this term squared, 4x squared, 116 00:06:27,819 --> 00:06:32,839 squared, plus 2 times the product of both terms, 2 times 117 00:06:32,839 --> 00:06:39,209 4x squared times y squared, plus y 118 00:06:39,209 --> 00:06:42,709 squared, this term, squared. 119 00:06:42,709 --> 00:06:44,799 And what's this going to be equal to? 120 00:06:44,800 --> 00:06:48,430 This is going to be equal to 16-- right, 4 squared is 16-- 121 00:06:48,430 --> 00:06:53,319 x squared, squared, that's 2 times 2, so it's x to the 122 00:06:53,319 --> 00:06:54,730 fourth power. 123 00:06:54,730 --> 00:06:58,330 And then plus, 2 times 4 times 1, that's 124 00:06:58,329 --> 00:07:01,819 8x squared y squared. 125 00:07:01,819 --> 00:07:07,300 And then y squared, squared, is y to the fourth. 126 00:07:07,300 --> 00:07:10,389 Now, we've been dealing with squaring a binomial. 127 00:07:10,389 --> 00:07:13,139 The next example I want to show you is when I take the 128 00:07:13,139 --> 00:07:15,699 product of a sum and a difference. 129 00:07:15,699 --> 00:07:18,349 And this one actually comes out pretty neat. 130 00:07:18,350 --> 00:07:20,160 So I'm going to do a very general one for you. 131 00:07:20,160 --> 00:07:28,160 Let's just do a plus b, times a minus b. 132 00:07:28,160 --> 00:07:30,400 So what's this going to be equal to? 133 00:07:30,399 --> 00:07:36,659 This is going to be equal to a times a-- let me make these 134 00:07:36,660 --> 00:07:41,260 actually in different colors-- so a minus b, just like that. 135 00:07:41,259 --> 00:07:45,899 So it's going to be this green a times this magenta a, a 136 00:07:45,899 --> 00:07:55,379 times a, plus, or maybe I should say minus, the green a 137 00:07:55,379 --> 00:07:56,670 times this b. 138 00:07:56,670 --> 00:07:59,560 I got the minus from right there. 139 00:07:59,560 --> 00:08:04,370 And then we're going to have the green b, so plus the green 140 00:08:04,370 --> 00:08:07,230 b times the magenta a. 141 00:08:07,230 --> 00:08:10,920 I'm just multiplying every term by every term. 142 00:08:10,920 --> 00:08:17,250 And then finally minus the green b-- that's where the 143 00:08:17,250 --> 00:08:19,050 minus is coming from-- minus the green b 144 00:08:19,050 --> 00:08:22,370 times the magenta b. 145 00:08:22,370 --> 00:08:23,590 And what is this going to be equal to? 146 00:08:23,589 --> 00:08:27,219 This is going to be equal to a squared, and then 147 00:08:27,220 --> 00:08:29,020 this is minus ab. 148 00:08:29,019 --> 00:08:32,840 This could be rewritten as plus ab, and then we have 149 00:08:32,840 --> 00:08:35,038 minus b squared. 150 00:08:35,038 --> 00:08:39,129 These right here cancel out, minus ab plus ab, so you're 151 00:08:39,129 --> 00:08:42,600 just left with a squared minus b squared. 152 00:08:42,600 --> 00:08:44,430 Which is a really neat result because it 153 00:08:44,429 --> 00:08:46,120 really simplifies things. 154 00:08:46,120 --> 00:08:49,190 So let's use that notion to do some multiplication. 155 00:08:49,190 --> 00:08:55,470 So if we say 2x minus 1, times 2x plus 1. 156 00:08:55,470 --> 00:08:56,710 Well, these are the same thing. 157 00:08:56,710 --> 00:09:01,810 The 2x plus 1, you could view this as, if you like, a plus 158 00:09:01,809 --> 00:09:06,750 b, and the 2x minus 1, you can view it as a minus b, where 159 00:09:06,750 --> 00:09:10,830 this is a, and that b is 1. 160 00:09:10,830 --> 00:09:13,440 This is b. 161 00:09:13,440 --> 00:09:15,010 That is a. 162 00:09:15,009 --> 00:09:18,659 Just using this pattern that we figured out just now. 163 00:09:18,659 --> 00:09:19,679 So what is this going to be equal to? 164 00:09:19,679 --> 00:09:24,819 It's going to be a squared, it's going to be 2x squared, 165 00:09:24,820 --> 00:09:29,490 minus b squared, minus 1 squared. 166 00:09:29,490 --> 00:09:32,789 2x squared is 4x squared. 167 00:09:32,789 --> 00:09:36,179 1 squared is just 1, so minus 1. 168 00:09:36,179 --> 00:09:39,209 So it's going to be 4x squared minus 1. 169 00:09:39,210 --> 00:09:41,560 Let's do one more of these, just to really 170 00:09:41,559 --> 00:09:42,889 hit the point home. 171 00:09:42,889 --> 00:09:45,590 172 00:09:45,590 --> 00:09:47,290 I'll just focus on multiplication right now. 173 00:09:47,289 --> 00:09:52,039 If I have 5a minus 2b, and I'm multiplying that 174 00:09:52,039 --> 00:09:54,909 times 5a plus 2b. 175 00:09:54,909 --> 00:09:58,639 And remember, this only applies when I have at a 176 00:09:58,639 --> 00:10:00,750 product of a sum and a difference. 177 00:10:00,750 --> 00:10:02,409 That's the only time that I can use this. 178 00:10:02,409 --> 00:10:03,850 And I've shown you why. 179 00:10:03,850 --> 00:10:06,320 And if you're ever in doubt, just multiply it out. 180 00:10:06,320 --> 00:10:07,879 It'll take you a little bit longer. 181 00:10:07,879 --> 00:10:09,629 And you'll see the terms canceling out. 182 00:10:09,629 --> 00:10:13,070 You can't do this for just any binomial multiplication. 183 00:10:13,070 --> 00:10:14,590 You saw that earlier in the video, when we were 184 00:10:14,590 --> 00:10:17,080 multiplying, when we were taking squares. 185 00:10:17,080 --> 00:10:19,560 So this is going to be, using the pattern, it's going to be 186 00:10:19,559 --> 00:10:28,309 5a squared minus 2b squared, which is equal to 25 a squared 187 00:10:28,309 --> 00:10:31,399 minus 4b squared. 188 00:10:31,399 --> 00:10:34,329 And, well, I'll leave it there, and I'll see you in the 189 00:10:34,330 --> 00:10:35,900 next video.