1 00:00:05,000 --> 00:00:19,151 What are the possible directions for this problem? It could say simplify or 2 00:00:19,151 --> 00:00:20,240 multiply. 3 00:00:21,320 --> 00:00:26,120 Just multiply, which we need to use the distributive property. 4 00:00:31,780 --> 00:00:37,231 And of course we now need the distributive property of multiplication over the two 5 00:00:37,231 --> 00:00:37,620 additions. 6 00:00:37,620 --> 00:00:40,260 Okay, and there are no like terms. 7 00:00:41,140 --> 00:00:42,260 That would be it. 8 00:00:43,220 --> 00:00:44,600 You could be writing descending order. 9 00:00:48,240 --> 00:00:53,700 Okay, so today we're doing the opposite, or the reverse. 10 00:00:53,700 --> 00:01:03,400 First, we're going to start with this and we want to end with this. 11 00:01:06,220 --> 00:01:08,240 What's the reverse process called? 12 00:01:09,620 --> 00:01:10,960 It's called factoring. 13 00:01:10,960 --> 00:01:14,920 I'll start with the definition. There's actually two definitions of the word 14 00:01:14,920 --> 00:01:35,840 To factor means to write an expression as a product. 15 00:01:35,840 --> 00:01:40,840 Right, this is not a product, this is a sum. 16 00:01:42,460 --> 00:01:46,080 This polynomial, first off, what would this polynomial be called? 17 00:01:47,960 --> 00:01:49,740 A fourth degree trinomial. 18 00:01:49,740 --> 00:01:52,140 know if we're able to write it as a 19 00:01:53,200 --> 00:01:55,240 And we give you the number 24. 20 00:01:56,020 --> 00:01:57,480 And I said factor. 21 00:01:58,520 --> 00:01:59,720 What does factor mean? 22 00:02:03,280 --> 00:02:04,660 To write a product. 23 00:02:05,660 --> 00:02:07,640 So what are some possible answers here? 24 00:02:11,960 --> 00:02:13,440 1 times 24. 25 00:02:15,760 --> 00:02:17,100 2 times 12. 26 00:02:18,820 --> 00:02:21,480 3 times 8. 27 00:02:23,700 --> 00:02:26,360 4 times 6. 28 00:02:27,260 --> 00:02:29,040 6 times 4. 29 00:02:29,540 --> 00:02:30,720 Same sort of thing. 30 00:02:31,160 --> 00:02:32,620 Is that it? 31 00:02:35,520 --> 00:02:39,680 How about 2 times 2 times 6? 32 00:02:43,140 --> 00:02:47,180 How about 2 times 2 times 2 times 3? 33 00:02:54,220 --> 00:02:55,980 How about... 34 00:02:55,980 --> 00:02:56,880 What else do I got? 35 00:02:57,400 --> 00:03:02,380 2 times 3 times 4? 36 00:03:02,580 --> 00:03:03,480 Is that right? 37 00:03:07,700 --> 00:03:15,399 There's a lot of ways to write this as a product. These are often called factor 38 00:03:15,399 --> 00:03:16,149 pairs. 39 00:03:17,280 --> 00:03:25,468 Is there two numbers? Two different factors? These are often not. So, factor is kind 40 00:03:25,468 --> 00:03:26,560 of ambiguous. 41 00:03:27,860 --> 00:03:34,540 okay what it means to factor because there's lots of different possibilities okay 42 00:03:36,740 --> 00:03:55,360 This is called the prime factorization: 2 cubed times 3 squared. 43 00:04:03,840 --> 00:04:12,380 factored it into its prime factorization. The product of all its prime factors lead 44 00:04:12,380 --> 00:04:32,760 to the correct exponent. Okay. Why don't I put a number 1 in front? True. Well, what 45 00:04:32,760 --> 00:04:39,740 if it's one squared. Is that fine? No? Is it the same thing? What about one cube? 46 00:04:42,660 --> 00:04:49,580 They're all equal. But the reason we don't put a 1, because the number 1 is not 47 00:04:49,580 --> 00:04:56,380 prime, it's not a prime number. Why is the number 1 not a prime number? 48 00:05:05,760 --> 00:05:08,740 What is the definition of a prime number? 49 00:05:09,100 --> 00:05:10,000 Numbers prime yet? 50 00:05:16,220 --> 00:05:19,880 It's only divisible by one in itself? 51 00:05:21,000 --> 00:05:26,000 Or only be written as a product of one in itself? 52 00:05:26,300 --> 00:05:27,520 That's only half the definition. 53 00:05:28,960 --> 00:05:31,320 There's a second half of the definition. 54 00:05:32,820 --> 00:05:38,960 because the number one is only divisible by one in itself. The second half is a 55 00:05:39,780 --> 00:05:45,640 A number is prime if it is divisible only by one and itself. 56 00:05:45,640 --> 00:05:54,480 not a prime number. It's part of the definition. Okay so the reason one is not 57 00:05:54,480 --> 00:06:01,180 prime is because we define it not to be a prime number. It's in the definition. So 58 00:06:01,180 --> 00:06:11,960 why is one not prime? Because by definition. Okay. Which does make it, every 59 00:06:11,960 --> 00:06:19,180 number can be written uniquely as a factor of prime numbers to certain 60 00:06:19,180 --> 00:06:27,040 powers. Okay, that's an important thing in number theory. But so this is the only 61 00:06:27,040 --> 00:06:29,840 prime factorization of the number 24. 62 00:06:42,760 --> 00:06:44,540 To factor means to write as a product. 63 00:06:45,820 --> 00:06:48,200 Okay, so To factor means what? 64 00:06:50,880 --> 00:06:52,180 To write as a product. 65 00:06:53,660 --> 00:06:55,540 Okay, so what is a factor? 66 00:06:58,520 --> 00:07:15,782 Okay, one is a factor, 24 is a factor. In fact, what are the factors of 24? 1, 2, 3, 67 00:07:15,782 --> 00:07:20,960 4, 6, 8, 12, and 24. 68 00:07:24,560 --> 00:07:32,787 So what do all these numbers, these are all factors of 24. So what's it mean to be a 69 00:07:32,787 --> 00:07:33,537 factor? 70 00:07:37,660 --> 00:07:42,340 There's kind of two different definitions, but basically as far as numbers go, 71 00:07:42,340 --> 00:08:02,660 sorry this is the noun, a number is a factor of another number. 72 00:08:02,660 --> 00:08:09,080 divides evenly into it. 73 00:08:12,240 --> 00:08:15,080 We'll have to say if it divides evenly into it. 74 00:08:21,440 --> 00:08:35,928 Another definition is each, let me just call it a thing, in a product is called a 75 00:08:35,928 --> 00:08:36,780 factor. 76 00:08:46,440 --> 00:08:50,440 So these are a bunch of the, probably most are all the things. 77 00:08:50,940 --> 00:08:52,600 Each one of these things is a factor. 78 00:08:53,460 --> 00:08:55,580 A factor, a factor, a factor. 79 00:08:56,080 --> 00:09:00,340 Everything inside a product called a factor. 80 00:09:00,580 --> 00:09:03,300 Or if it divides evenly into it, it's a factor. 81 00:09:03,300 --> 00:09:31,140 Okay, once you find the prime factorization of 72 82 00:09:33,800 --> 00:09:37,920 I want each kind factor to the correct power. 83 00:09:37,920 --> 00:09:39,540 The prime factorization is 2 cubed times 3 squared. 84 00:09:41,340 --> 00:09:42,240 Is that right? 85 00:09:43,580 --> 00:09:45,420 These factor it, each of the prime factors. 86 00:09:45,420 --> 00:09:47,040 What are all the factors of 18x squared? 87 00:09:48,620 --> 00:09:49,680 Now that's the prime factor. 88 00:09:49,680 --> 00:10:06,151 What are all the factors of 18x squared? Now we got not just a number but 89 00:10:06,151 --> 00:10:07,180 polynomial. 90 00:10:07,180 --> 00:10:20,860 So we start with the numbers. There are factors of 18. Which are what? 1, 2, 3, 6, 91 00:10:20,860 --> 00:10:22,380 9, 18. 92 00:10:25,940 --> 00:10:37,160 Okay, is x a factor? Yeah, I could write this as 18x times x, it's a factor, or in 93 00:10:37,160 --> 00:10:47,100 other words 18x squared divided by x is 18x, it divides in it evenly. So x is a 94 00:10:47,100 --> 00:10:55,820 factor? Is x squared a factor? Yeah, right here we already got a product. 18 is a 95 00:10:55,820 --> 00:11:06,420 factor, x squared is a factor. What about x cubed? No. Okay, 18 x squared divided by 96 00:11:06,420 --> 00:11:11,960 x cubed is 18 over x, which does not divide any evenly. We end up with 97 00:11:11,960 --> 00:11:22,973 denominator. Okay so not x cubed. What other factors are there? Definitely 18x 98 00:11:22,973 --> 00:11:23,820 squared. 99 00:11:23,820 --> 00:11:51,040 is 2x a factor? Yeah 3x yeah 6x 9x 18x what about 2x squared yes 3x squared 6x 100 00:11:51,040 --> 00:12:01,280 squared, 9x squared, and 18x squared. Those are all the factors of 18x squared. 101 00:12:02,160 --> 00:12:16,940 Its prime factorization is 2 times 3 squared times x squared. 102 00:12:23,540 --> 00:12:31,565 Two times three squared times x squared. It's that simple. It's one of every prime 103 00:12:31,565 --> 00:12:32,100 factor. 104 00:12:32,100 --> 00:12:39,300 X is a prime factor to whatever power that's its prime factorization 105 00:12:51,100 --> 00:12:54,140 Find this prime factorization 106 00:12:54,140 --> 00:12:56,500 We get that. 107 00:12:58,340 --> 00:13:02,723 It's just the prime factorization of the coefficient times the rest of the 108 00:13:02,723 --> 00:13:03,473 variables. 109 00:13:09,060 --> 00:13:10,600 Slightly new topic. 110 00:13:13,500 --> 00:13:21,780 find the common sorry 111 00:13:37,480 --> 00:13:43,700 What are the common factors of 20 and 24? 112 00:13:43,700 --> 00:13:50,240 or let me list the factor to 20. 113 00:13:50,580 --> 00:13:54,340 One, two, five groups for. 114 00:14:14,620 --> 00:14:20,700 So, the common factors are... 115 00:14:36,040 --> 00:14:37,380 I forgot four here. 116 00:14:44,400 --> 00:15:04,440 The common factors are one, two, and four. The greatest common factor is four. 117 00:15:04,440 --> 00:15:08,700 largest of all the common factors. Okay. 118 00:15:13,700 --> 00:15:22,660 20 is 2 to the second times 5. 24 is 2 cubed times 3. 119 00:15:24,520 --> 00:15:26,700 Are those the two prime factorizations? 120 00:15:29,820 --> 00:15:33,720 Okay, and what acronym do we use for the greatest common factor? 121 00:15:35,220 --> 00:15:36,120 GCF. 122 00:15:36,920 --> 00:15:41,820 How can we use the prime factorizations to find the GCF? 123 00:15:41,820 --> 00:15:43,220 I 124 00:16:02,680 --> 00:16:09,520 How can we use the two prime factorizations to find the greatest common factor? 125 00:16:09,520 --> 00:16:12,260 Use every common prime factor to the lowest power. 126 00:16:20,900 --> 00:16:26,000 Use every common prime factor to the lowest power. 127 00:16:27,180 --> 00:16:30,160 Okay, and then how do I know? 128 00:16:30,620 --> 00:16:33,060 Is it 2 squared or 2 cubed or something else? 129 00:16:36,660 --> 00:16:41,360 Yeah, it's going to be the lowest of all the exponents. 130 00:16:41,360 --> 00:16:43,120 so it's going to be 2 squared. 131 00:16:46,760 --> 00:16:49,060 Okay, you could also think of this as 132 00:16:52,740 --> 00:16:57,820 2 times 3 to the 0 times 5 to the 1. 133 00:17:02,720 --> 00:17:07,942 Okay, so in fact it's one of every factor to the lowest power, so the two, between 134 00:17:07,942 --> 00:17:10,400 the two and the three is 2 squared. 135 00:17:11,660 --> 00:17:18,855 between 0 and 1 it's 0, which is just the number 1, and between 5 to the 1 and 5 to 136 00:17:18,855 --> 00:17:19,600 the 0, 137 00:17:19,600 --> 00:17:28,654 it's times 5 to the 0, which is just 2 squared. So it's one of every common prime 138 00:17:28,654 --> 00:17:29,660 factor to 139 00:17:30,480 --> 00:17:36,921 the lower of all the values, because 2 cubed does not divide evenly into this, but 2 140 00:17:36,921 --> 00:17:37,300 squared 141 00:17:37,300 --> 00:17:38,020 device in both. 142 00:17:38,020 --> 00:17:41,980 Now we've got some variables in there. 143 00:17:45,400 --> 00:17:48,580 So if we quickly find the prime factorization. 144 00:17:48,580 --> 00:17:51,520 It's the same thing but now we have variables in there as well. 145 00:17:52,100 --> 00:18:00,220 So between 2 cubed and 2 to the first, the smaller is 2 to the first. 146 00:18:00,860 --> 00:18:02,520 Between 3 squared and 3 squared. 147 00:18:03,700 --> 00:18:04,920 It's 3 squared. 148 00:18:07,440 --> 00:18:15,614 Between x cubed and x to the first we need a x to the first, y squared, y to the 149 00:18:15,614 --> 00:18:16,840 fourth, y squared, 150 00:18:17,900 --> 00:18:26,760 and there's no common Z so that'd be at 9 the 18 x y squared 151 00:18:52,340 --> 00:18:55,260 Okay, so what do we call this polynomial? 152 00:18:57,280 --> 00:18:58,860 Fourth degree trinomial. 153 00:18:59,300 --> 00:19:02,400 We're going to factor out the greatest common factor. 154 00:19:03,640 --> 00:19:08,220 So we look for the greatest common factor between three expressions or three 155 00:19:08,220 --> 00:19:09,120 monomials. 156 00:19:10,420 --> 00:19:13,160 We start the numbers 3, 12, and 18. 157 00:19:13,620 --> 00:19:16,546 We really don't even have to start factoring them. What's the largest number that 158 00:19:16,546 --> 00:19:17,800 goes into all three of them? 159 00:19:19,780 --> 00:19:22,560 Between x to the fourth, x cubed and x squared? 160 00:19:23,680 --> 00:19:25,500 x squared goes in all of them. 161 00:19:25,800 --> 00:19:29,060 That's the greatest common factor. You put parentheses. 162 00:19:33,040 --> 00:19:34,700 3x squared times what gives 3x to the fourth? x squared. 163 00:19:34,700 --> 00:19:38,540 gives you 3x to the fourth? x squared. 164 00:19:40,620 --> 00:19:44,360 3x squared times what gives you negative 12x cubed? 165 00:19:48,620 --> 00:19:51,500 3x squared times what gives you 18x squared? 166 00:19:51,920 --> 00:19:52,820 Six. 167 00:19:56,480 --> 00:19:57,480 And that's it. 168 00:19:59,680 --> 00:20:02,160 The result is 3x squared times the quantity x squared minus 4x plus 6. 169 00:20:02,440 --> 00:20:08,928 You find the greatest common factor between each of the terms, bring it on the 170 00:20:08,928 --> 00:20:09,360 outside, 171 00:20:09,360 --> 00:20:12,100 and then determine what we need to multiply to get that. 172 00:20:12,720 --> 00:20:15,820 In fact, that should be the results on our very first problem. 173 00:20:16,700 --> 00:20:17,900 Is that right there? 174 00:20:17,900 --> 00:20:19,420 Let's try another one. 175 00:21:10,360 --> 00:21:15,840 Between 4, 8, and 18, the numerical GCF is 2. 176 00:21:17,640 --> 00:21:23,780 Between x cubed, x squared and x to the 4th, the greatest common is x squared. 177 00:21:23,780 --> 00:21:27,060 y to the first, y to the fourth, y squared. 178 00:21:29,300 --> 00:21:31,760 y to the first 179 00:21:31,760 --> 00:21:33,680 is the smallest of the exponents. 180 00:21:38,280 --> 00:21:39,180 And then 181 00:21:39,660 --> 00:21:43,060 2x squared y times what gives us this? 182 00:21:46,700 --> 00:21:47,600 x 183 00:21:49,780 --> 00:21:52,380 2x squared y times what gives us this? 184 00:21:59,980 --> 00:22:02,880 2x squared y times what gives us this? 185 00:22:09,200 --> 00:22:13,160 We factored out the greatest common factor. 186 00:22:16,700 --> 00:22:21,725 I could factor out negative 2 instead, but that would not be the greatest common 187 00:22:21,725 --> 00:22:22,475 factor. 188 00:22:25,660 --> 00:22:28,400 But why is that not the greatest common factor? 189 00:22:28,400 --> 00:22:30,560 Because negative 2 is 190 00:22:31,420 --> 00:22:37,342 Less than positive. So we, when we're just factoring out the greatest common factor, 191 00:22:37,342 --> 00:22:39,880 we never factor out a negative. 192 00:22:40,160 --> 00:22:43,780 Sometimes we need to factor out a negative. That will be important when grouping. 193 00:22:44,420 --> 00:22:48,480 But for right now, the greatest is always a positive factor. 194 00:22:51,860 --> 00:22:53,300 And that's it. 195 00:22:53,300 --> 00:22:55,160 Okay, kind of a new topic. 196 00:23:13,940 --> 00:23:15,380 Okay, going back to multiplying where we 197 00:23:15,380 --> 00:23:22,051 We talked about this the last couple of days. What property do we use? The 198 00:23:22,051 --> 00:23:22,940 distributive property. 199 00:23:22,940 --> 00:23:29,180 properties. I'm gonna do it the long way, not our shortcut. I want you to write this 200 00:23:29,180 --> 00:23:34,500 out and you'll see why in a minute. I'm gonna first take this, distribute the 201 00:23:34,500 --> 00:23:57,300 whole thing, each of these. Then we use the distribute property again. Again. 202 00:23:57,300 --> 00:23:59,860 And then we typically write it in descending order. 203 00:24:23,140 --> 00:24:24,260 Okay, that's what we did the other day. 204 00:24:33,280 --> 00:24:35,140 Okay, next. 205 00:24:37,240 --> 00:24:41,820 I'm just going to say factor. 206 00:24:46,740 --> 00:24:48,560 We look at the four terms. 207 00:24:48,560 --> 00:24:55,140 What's the greatest common factor between those four terms? 208 00:24:57,140 --> 00:25:01,040 Yeah, one. So can we, we could factor out a one, we're not going to. 209 00:25:05,860 --> 00:25:08,020 What's it mean to factor? 210 00:25:10,560 --> 00:25:11,720 To factor means to write as a product. 211 00:25:11,720 --> 00:25:12,520 to write as a product. 212 00:25:12,520 --> 00:25:25,040 Besides multiplying by one, is it possible to write this polynomial as a product? 213 00:25:25,040 --> 00:25:38,780 We want to rewrite the four-term polynomial as a product of two binomials. 214 00:25:38,780 --> 00:25:48,480 We know it factors because we started with two binomials and multiplied them. 215 00:25:48,480 --> 00:26:08,760 started with it factored already. We multiplied it out. So how in the world 216 00:26:09,740 --> 00:26:11,620 The first group has a greatest common factor of 3x. 217 00:26:19,320 --> 00:26:20,960 The second group has a greatest common factor of negative 4. 218 00:26:22,300 --> 00:26:23,420 Negative four. 219 00:26:24,520 --> 00:26:31,100 and I factor it out. And then once I get it here, what do we have? We really only 220 00:26:31,100 --> 00:26:51,700 have one term, this plus this, but we have a common binomial factor. If we have 221 00:26:51,700 --> 00:26:54,640 for 3x minus 4. 222 00:26:57,480 --> 00:26:59,360 This is called factor by grouping. 223 00:27:11,340 --> 00:27:21,680 Can only be used when they're 224 00:27:21,680 --> 00:27:23,260 are four terms. 225 00:27:30,780 --> 00:27:35,000 Well, it could be six, but that gets complicated. 226 00:27:37,140 --> 00:27:40,862 It could be eight terms, but yeah, we're only we're only going to do it when there's 227 00:27:40,862 --> 00:27:41,612 four terms. 228 00:27:42,960 --> 00:27:46,820 The first step is always factor out the greatest common factor first. 229 00:27:47,860 --> 00:27:59,233 Side note, always factor the GCF first. But here the GCF is one, so we're not going 230 00:27:59,233 --> 00:28:01,240 to do that. 231 00:28:03,100 --> 00:28:12,220 Split the four terms into two groups: the first two terms plus the second two terms. 232 00:28:14,620 --> 00:28:19,200 And then within each group, we factor out the greatest common factor. 233 00:28:19,420 --> 00:28:26,840 So between 3x cubed and negative 4x squared, we factor an x squared out. 234 00:28:29,660 --> 00:28:31,280 3x minus 4. 235 00:28:34,260 --> 00:28:37,720 What's the greatest common factor between 6x and negative 8? 236 00:28:37,720 --> 00:28:39,500 2. 237 00:28:46,040 --> 00:28:50,000 Now we have two groups within each group. 238 00:28:50,220 --> 00:28:54,360 There's a binomial factor and 239 00:28:54,360 --> 00:28:59,280 it's a common binomial factor and so I 240 00:28:59,280 --> 00:29:00,360 factor that out. 241 00:29:12,020 --> 00:29:15,440 And when I factor it out of here, I'm left with the x squared. 242 00:29:16,920 --> 00:29:18,640 This plus is this plus. 243 00:29:19,860 --> 00:29:21,720 When I factor out of here, I'm left with the 2. 244 00:29:25,940 --> 00:29:32,580 So it's x squared plus 2 times 3x minus 4. 245 00:29:32,680 --> 00:29:33,400 And there it is. 246 00:29:33,400 --> 00:29:34,340 We factored that. 247 00:29:35,180 --> 00:29:42,973 a third degree or a cubic polynomial which has four terms into the product of two 248 00:29:42,973 --> 00:29:43,723 binomials. 249 00:29:47,140 --> 00:29:53,089 Okay, so we split up into two groups, factor out the GCF out of each group, and then 250 00:29:53,089 --> 00:29:53,420 check 251 00:29:53,420 --> 00:30:03,356 is there a common binomial factor, if so factor that out. Okay, why on this one are 252 00:30:03,356 --> 00:30:03,940 common 253 00:30:03,940 --> 00:30:09,560 binomial was 3x minus 4, but when we originally multiplied it, the common 254 00:30:09,560 --> 00:30:23,860 binomial was this. Why did that happen? It's because this polynomial with four 255 00:30:23,860 --> 00:30:31,680 terms was not written in descending order. If I took this and factor by 256 00:30:31,680 --> 00:30:36,320 grouping you will have a common x squared plus 2 with these two switched 257 00:30:37,580 --> 00:30:42,080 okay so but that was just side note let's try another one 258 00:30:42,080 --> 00:30:43,040 Actually, sorry. 259 00:30:43,220 --> 00:30:44,120 Let me change this. 260 00:31:00,680 --> 00:31:02,480 Now factor completely. First identify the GCF of all four terms. 261 00:31:09,240 --> 00:31:12,440 Okay, so the factor completely. 262 00:31:16,120 --> 00:31:18,040 The first thing we always do is what? 263 00:31:19,960 --> 00:31:22,620 Find the GCF and factor up the GCF. 264 00:31:22,760 --> 00:31:24,020 What is the GCF? 265 00:31:29,460 --> 00:31:30,380 Yeah, I think it's only 10. 266 00:31:31,920 --> 00:31:35,260 Factor out the common factor of 10. 267 00:31:49,180 --> 00:31:51,060 Now this has four terms. 268 00:31:51,200 --> 00:31:52,100 How are we going to do it? 269 00:31:53,920 --> 00:31:55,900 Factor by grouping. 270 00:31:55,900 --> 00:32:06,480 This 10 outside we just leave out here. Okay so I group into two groups this plus 271 00:32:06,480 --> 00:32:17,660 this. So between these two terms what's the greatest common factor? 3x squared. 272 00:32:28,380 --> 00:32:32,180 Between 12x and negative 16, what's the greatest common factor? 273 00:32:42,480 --> 00:32:47,340 And sure enough, we have a common binomial factor. 274 00:32:49,120 --> 00:32:53,920 Let's bring the factor that out. 275 00:33:06,060 --> 00:33:07,780 Let's put factor that out. 276 00:33:10,360 --> 00:33:31,720 and we're left with 3x squared plus 4. We usually put the coefficients of the 277 00:33:31,720 --> 00:33:35,820 monomial in front, but the order of the two binomials does not matter. 278 00:33:36,720 --> 00:33:37,620 That's fair. 279 00:34:16,400 --> 00:34:21,580 Find the GCF of the first group and the GCF of the second group. 280 00:34:33,500 --> 00:34:35,580 So the greatest common factor is? 281 00:34:36,980 --> 00:34:38,700 Two does not divide 15, so the four terms have no numerical GCF greater than one. 282 00:34:42,920 --> 00:34:45,020 So we don't factor out a one. 283 00:34:46,400 --> 00:34:47,360 We have four terms. 284 00:34:48,560 --> 00:34:51,180 Use factoring by grouping. 285 00:34:54,000 --> 00:34:55,800 First off, I can't write it like that. 286 00:34:56,140 --> 00:34:59,680 Why not? Because I changed that to multiplication. 287 00:35:02,700 --> 00:35:03,820 I can't 288 00:35:03,820 --> 00:35:06,620 do this. Why not? 289 00:35:08,820 --> 00:35:11,700 Okay, yet that negative 290 00:35:11,700 --> 00:35:15,360 is only with the 18. So, if there's a 291 00:35:15,360 --> 00:35:21,320 subtraction between the third term, we just have to put a little plus and then 292 00:35:21,320 --> 00:35:23,120 make sure we get the parentheses correct. 293 00:35:32,600 --> 00:35:35,760 And then we factor out the GCF out of here. 294 00:35:38,920 --> 00:35:40,740 Not just five. 295 00:35:42,960 --> 00:35:43,860 X squared. 296 00:36:04,860 --> 00:36:06,220 Factor out negative 6 so both groups contain the same binomial. 297 00:36:08,420 --> 00:36:09,320 It is very important. 298 00:36:10,600 --> 00:36:20,660 Factoring out positive 6 would produce the opposite binomial, so it would not match. 299 00:36:20,660 --> 00:36:39,240 Instead, factor out negative 6 so both groups contain 3x plus 8. 300 00:36:39,240 --> 00:36:49,640 Plus 8. Now we have a common binomial factor. 301 00:36:54,540 --> 00:37:02,305 Factoring by grouping works when four terms can be grouped to create a common 302 00:37:02,305 --> 00:37:02,860 binomial. 303 00:37:02,860 --> 00:37:07,004 because they have four terms doesn't mean it's going to work. Not all polynomials 304 00:37:07,004 --> 00:37:07,300 are 305 00:37:07,300 --> 00:37:14,497 able to be factored. Okay, but if it is able to factor, for example, if this were 306 00:37:14,497 --> 00:37:14,920 not, 307 00:37:14,920 --> 00:37:32,300 say this were 96 instead this would become well 16 and these would not be 308 00:37:32,300 --> 00:37:39,060 common so there'd be no way to factor that out so only sometimes it works it 309 00:37:39,060 --> 00:37:40,180 It has to work out nicely. 310 00:37:43,680 --> 00:37:47,220 Anyways, there are four. Let's do one more. If you get it right, 311 00:37:48,020 --> 00:37:48,960 we'll start homework. 312 00:37:48,960 --> 00:37:52,560 Try this final example. Start with the GCF. 313 00:37:52,560 --> 00:37:55,400 Factor out the GCF first. 314 00:37:55,400 --> 00:37:56,360 Check your result with a neighbor. 315 00:37:56,360 --> 00:37:57,460 We're seeing it got the same thing. 316 00:37:57,460 --> 00:38:03,922 On Thursday, we will learn to factor polynomials with three terms, called 317 00:38:03,922 --> 00:38:04,672 trinomials, 318 00:38:05,380 --> 00:38:09,920 and polynomials with two terms, called binomials.