1 00:00:05,000 --> 00:00:09,540 Alright, we're talking about polynomials. 2 00:00:14,280 --> 00:00:16,660 We're going to start out with the definition. 3 00:00:32,820 --> 00:00:42,176 A monomial is a number, a variable, or the product of numbers and variables raised 4 00:00:42,176 --> 00:00:42,800 to 5 00:00:42,800 --> 00:00:43,980 whole number powers. 6 00:00:43,980 --> 00:00:58,980 Let's see if this gets... 7 00:01:04,220 --> 00:01:13,239 A monomial is a number, a variable, or the product of numbers and variables raised 8 00:01:13,239 --> 00:01:13,840 to 9 00:01:13,840 --> 00:01:15,600 poll number. 10 00:01:50,730 --> 00:01:52,430 Uh, examples. 11 00:01:57,230 --> 00:01:59,610 Is three a monomial? 12 00:02:02,290 --> 00:02:03,070 Yes. 13 00:02:03,070 --> 00:02:03,750 Yes, why? 14 00:02:03,750 --> 00:02:23,326 Because three is a number. That's it. Simple. Is x a monomial? Yes. Because it's a 15 00:02:23,326 --> 00:02:24,550 variable. 16 00:02:24,550 --> 00:02:25,450 2. 17 00:02:28,730 --> 00:02:34,010 Is 4x squared a monomial? 18 00:02:34,470 --> 00:02:35,370 Yes. 19 00:02:35,730 --> 00:02:37,050 Yes, why? 20 00:02:37,490 --> 00:02:40,270 Product of numbers and variables raised to whole numbers. 21 00:02:41,230 --> 00:02:44,670 Product of numbers and variables raised to whole numbers. 22 00:02:45,130 --> 00:02:46,030 Powers. 23 00:02:47,370 --> 00:02:48,930 OK, not monomials. 24 00:03:04,110 --> 00:03:10,750 Is x to the negative 1 same thing as 1 over x? 25 00:03:12,550 --> 00:03:13,530 Is that a monolion? 26 00:03:14,630 --> 00:03:15,470 No. 27 00:03:15,470 --> 00:03:16,370 No, why not? 28 00:03:16,470 --> 00:03:17,370 It's nominal. 29 00:03:19,610 --> 00:03:20,510 Because? 30 00:03:22,670 --> 00:03:24,530 It's not raised to a whole number. 31 00:03:24,530 --> 00:03:25,430 number power. 32 00:03:27,110 --> 00:03:28,370 Negative 1 is not a whole number. 33 00:03:30,650 --> 00:03:31,950 OK, what are the whole numbers? 34 00:03:33,450 --> 00:03:34,350 Positive 2. 35 00:03:36,430 --> 00:03:41,690 So the 1, 2, 3, 4, and so on, the counting number 36 00:03:41,690 --> 00:03:43,290 is also known as the natural numbers. 37 00:03:45,350 --> 00:03:48,450 You add 0 to it, 0, 1, 2, 3, 4, and so on. 38 00:03:48,490 --> 00:03:50,230 Those are the whole number, set of whole numbers. 39 00:03:54,890 --> 00:04:05,130 the square root of X is that a monomial why not same thing as X to the 1 half 40 00:04:05,130 --> 00:04:21,590 which is not a whole number X squared plus 4x is that a monomial it's not 41 00:04:21,590 --> 00:04:22,490 Not because? 42 00:04:26,130 --> 00:04:28,230 It's not a product of numbers and variables. 43 00:04:28,390 --> 00:04:29,290 We have a sum here. 44 00:04:29,410 --> 00:04:30,410 This is not a monomial. 45 00:04:32,730 --> 00:04:33,630 Okay. 46 00:04:38,410 --> 00:04:39,810 What about the square root of 5? 47 00:04:39,950 --> 00:04:40,850 Is that a monomial? 48 00:04:44,490 --> 00:04:46,030 Same thing as 5 to what power? 49 00:04:47,730 --> 00:04:48,630 1 half power. 50 00:04:49,310 --> 00:04:50,490 Is it a monomial or no? 51 00:04:50,950 --> 00:04:51,850 No. 52 00:04:53,750 --> 00:04:55,270 Is the square root of 5 a number? 53 00:04:56,870 --> 00:04:57,770 Yeah. 54 00:04:57,810 --> 00:04:58,710 Yeah, it's a number. 55 00:04:58,850 --> 00:04:59,750 It's definitely a number. 56 00:05:01,230 --> 00:05:02,850 Okay, it is a monomial. 57 00:05:03,970 --> 00:05:07,210 Only the variables has to be raised to a whole number of powers. 58 00:05:08,590 --> 00:05:11,410 Oh, I noticed you put it in each example instead of five. 59 00:05:12,450 --> 00:05:16,170 And yes, I put it under the example, not the not a monomial. 60 00:05:16,550 --> 00:05:17,450 That's the other clue. 61 00:05:17,990 --> 00:05:20,270 OK, next definition. 62 00:05:22,930 --> 00:05:26,190 Actually, let's do one more, it's not... 63 00:05:27,930 --> 00:05:31,290 2 to the power of x. Is that a monomial? 64 00:05:31,790 --> 00:05:33,550 No. No, why not? 65 00:05:33,970 --> 00:05:36,690 Because the exponent's a monomial. 66 00:05:37,230 --> 00:05:40,850 Yeah, the variable's in the exponent. OK. 67 00:05:42,630 --> 00:05:45,290 We can't have any variables in the exponent. 68 00:05:45,690 --> 00:05:47,030 It's not a monomial. 69 00:05:47,750 --> 00:05:48,870 Next definition. 70 00:05:56,310 --> 00:06:04,090 A polynomial is a monomial for the sum of monomials. 71 00:06:51,610 --> 00:06:53,210 Is this a polynomial? 72 00:06:54,330 --> 00:06:55,810 Yes. Yes, why? 73 00:06:56,130 --> 00:06:57,030 Because it's a monomial. 74 00:07:06,510 --> 00:07:08,490 Is this a polynomial? 75 00:07:08,490 --> 00:07:13,910 Yes, yes, why it's the sum of three monomials 76 00:07:23,390 --> 00:07:26,070 Is this a polynomial 77 00:07:30,250 --> 00:07:33,270 For those who said no, why not 78 00:07:33,270 --> 00:07:34,170 It's not 79 00:07:36,910 --> 00:07:39,350 Is this the same thing 80 00:07:42,910 --> 00:07:49,750 Is negative 8x squared a monomial I was not I was trying to trick you 81 00:07:50,590 --> 00:07:52,970 But make sure we get the right definition 82 00:07:55,310 --> 00:07:59,190 This is a monomial sorry, this is a polynomial 83 00:07:59,260 --> 00:08:01,420 even though there's a subtraction, 84 00:08:01,560 --> 00:08:04,460 we always think that's just a plus and negative, that number. 85 00:08:05,660 --> 00:08:06,560 OK. 86 00:08:07,500 --> 00:08:08,500 Next definition. 87 00:08:12,560 --> 00:08:23,040 The number in a number being multiplied, not the exponent. 88 00:08:27,500 --> 00:08:40,520 Monomial is called the coefficient. Okay, the number, not the exponent. 89 00:09:02,460 --> 00:09:11,720 what's the coefficient here? 5. the coefficient here? 3. what is the 90 00:09:12,070 --> 00:09:40,710 here just it's very important it is negative 8 not 8 okay yeah hey Oh a term 91 00:10:35,020 --> 00:10:38,240 How many terms does this have? 92 00:10:40,360 --> 00:10:47,380 Because we think of this as 5x to the fourth plus negative 7x squared plus 8. 93 00:10:47,960 --> 00:10:52,260 It has three terms. One, two, three terms. 94 00:11:03,740 --> 00:11:17,300 A polynomial with two terms is called a binomial. 95 00:11:23,980 --> 00:11:26,560 A polynomial with two terms is called a binomial. 96 00:11:27,580 --> 00:11:31,760 A polynomial with one term is called a monomial. 97 00:11:31,760 --> 00:11:53,320 two-term binomial a polynomial with three terms is called a trinomial a 98 00:11:53,320 --> 00:12:10,960 polynomial with four terms could be called a quadrinal meal but we don't 99 00:12:11,360 --> 00:12:20,500 We just call it a called a polynomial with four terms. 100 00:12:20,980 --> 00:12:22,160 Oh wow, that's pretty elaborate. 101 00:12:28,360 --> 00:12:35,700 Once you get past three, four, and on, we call it a polynomial with n terms. 102 00:12:36,160 --> 00:12:36,800 What's that? 103 00:12:36,800 --> 00:12:37,880 What is that commutation line? 104 00:12:37,880 --> 00:12:42,440 That means repeat above a polynomial. 105 00:12:43,460 --> 00:12:46,600 A polynomial with four terms is called a polynomial with four terms. 106 00:12:47,160 --> 00:12:48,060 That's right. 107 00:12:48,260 --> 00:12:50,280 A polynomial with seven terms is called? 108 00:12:50,640 --> 00:12:51,740 A polynomial with seven terms. 109 00:12:51,760 --> 00:12:52,980 A polynomial with seven terms. 110 00:12:53,340 --> 00:12:55,340 So we've got a monomial, a binomial. 111 00:12:55,780 --> 00:12:57,740 Let me do a couple quick examples, right? 112 00:12:58,320 --> 00:13:06,900 For example, 3x cubed plus 4x is a binomial. 113 00:13:06,900 --> 00:13:15,580 Why? Because it has two terms. Is it also a polynomial? Yeah, definitely. Okay, 114 00:13:15,580 --> 00:13:16,820 three terms, 115 00:13:16,820 --> 00:13:21,360 X squared plus 2x minus 5 will be a trinomial 116 00:13:34,360 --> 00:13:36,760 This would be called a 117 00:13:36,760 --> 00:13:42,855 Polynomial with four terms so monomial binomial trinomial polynomial four terms 118 00:13:42,855 --> 00:13:46,180 polynomial five terms and so on 119 00:13:50,080 --> 00:14:12,520 okay next definition the degree of a polynomial sorry of a monomial the sum 120 00:14:12,520 --> 00:14:17,520 of the exponents. 121 00:14:27,400 --> 00:14:28,800 exponents 122 00:14:42,860 --> 00:14:45,660 So samples. 123 00:14:50,420 --> 00:14:53,800 What is the degree? 124 00:14:53,800 --> 00:14:55,120 6. 125 00:15:05,220 --> 00:15:08,500 What is the degree? 126 00:15:08,960 --> 00:15:10,320 2. 127 00:15:14,180 --> 00:15:18,200 What is the degree here? 128 00:15:18,200 --> 00:15:24,700 1. 1. Is the number 7 a monolu? 129 00:15:25,600 --> 00:15:28,320 Yeah. It is. What's the degree? 130 00:15:28,720 --> 00:15:34,740 1. I don't see any variables. This is degree 1. 131 00:15:35,480 --> 00:15:42,582 Oh, it's going to be 3. Ok, we could think of this as 7, same thing as 7x to the 0 132 00:15:42,582 --> 00:15:43,332 power. 133 00:15:43,420 --> 00:15:53,800 what is X and zero power one so what is the degree zero okay we're not going to 134 00:15:53,800 --> 00:16:00,240 deal with this a lot in this class but 7x squared y to the fourth what is the 135 00:16:00,240 --> 00:16:09,520 degree it's the sum of all the exponents of the variables so 2 plus 4 is 6 136 00:16:12,300 --> 00:16:28,900 Okay, every number has degree zero. We call this a constant. Okay, next definition. 137 00:16:31,520 --> 00:16:42,180 the degree of a polynomial not a monomial 138 00:16:58,900 --> 00:17:17,980 is the degree of the term with the largest degree. 139 00:17:40,200 --> 00:17:42,360 How many terms does this have? 140 00:17:43,380 --> 00:17:44,280 Three. 141 00:17:44,820 --> 00:17:46,880 What's the degree of this term? 142 00:17:47,280 --> 00:17:47,660 Two. 143 00:17:47,660 --> 00:17:48,420 One. 144 00:17:48,420 --> 00:17:49,320 One. 145 00:17:49,700 --> 00:17:50,600 Degree of this term? 146 00:17:50,960 --> 00:17:51,280 Two. 147 00:17:51,280 --> 00:17:51,800 Two. 148 00:17:51,800 --> 00:17:52,700 Degree of this term? 149 00:17:52,700 --> 00:17:53,240 Zero. 150 00:17:53,240 --> 00:17:54,140 Zero. 151 00:17:54,360 --> 00:17:56,640 The degree of the whole polynomial is? 152 00:17:56,680 --> 00:17:56,900 Two. 153 00:17:56,900 --> 00:17:57,800 Two. 154 00:17:59,720 --> 00:18:03,000 Because it's the degree of this term, which is two. 155 00:18:30,280 --> 00:18:32,420 What is the degree of this term? 156 00:18:33,840 --> 00:18:34,740 Degree of this term? 157 00:18:35,420 --> 00:18:36,460 Degree of this term? 158 00:18:36,560 --> 00:18:37,260 Four. 159 00:18:37,260 --> 00:18:38,180 Degree of this term? 160 00:18:38,260 --> 00:18:39,160 Five. 161 00:18:39,260 --> 00:18:43,240 The term with the largest is this, so the degree of the whole polynomial is? 162 00:18:43,340 --> 00:18:44,240 Seven. 163 00:18:45,680 --> 00:18:46,580 Seven. 164 00:18:50,940 --> 00:19:11,520 Another definition, the term with the largest degree is called the leading term. 165 00:19:20,100 --> 00:19:24,400 Even though it doesn't have to be in front, it is the leading term. 166 00:19:32,800 --> 00:19:35,940 What is the leading term in this? 167 00:19:35,940 --> 00:19:38,080 7x to the 7. 168 00:19:38,580 --> 00:19:40,160 Negative 7x squared. 169 00:19:45,960 --> 00:19:49,460 What is the leading term here? 170 00:19:49,900 --> 00:19:51,180 5x to the 7. 171 00:19:51,660 --> 00:19:53,000 5x to the 7. 172 00:19:56,380 --> 00:20:14,520 The coefficient of the leading term is called... anybody want to take a guess? 173 00:20:14,520 --> 00:20:15,700 Leading coefficient. 174 00:20:16,220 --> 00:20:17,120 Leading coefficients. 175 00:20:54,220 --> 00:20:56,460 What is the leading term? 176 00:20:58,120 --> 00:20:59,680 X cubed. 177 00:20:59,960 --> 00:21:02,240 What is the leading coefficient? 178 00:21:02,240 --> 00:21:03,140 1. 179 00:21:14,200 --> 00:21:18,620 What is the leading coefficient here? 180 00:21:21,240 --> 00:21:23,880 negative 7 181 00:21:38,540 --> 00:21:40,620 okay now we got some names 182 00:21:40,620 --> 00:21:44,420 of polynomials with different degrees 183 00:21:44,420 --> 00:22:11,980 so the degree the name and then I'll give examples the degree zero it's 184 00:22:11,980 --> 00:22:16,940 It's called a constant polynomial. 185 00:22:23,360 --> 00:22:30,980 Examples would be like 5 or negative pi. 186 00:22:32,780 --> 00:22:34,800 The constant doesn't change. 187 00:22:34,800 --> 00:22:39,800 degree one are called linear polynomials. 188 00:22:48,360 --> 00:22:53,360 Three x plus four, negative eight x. 189 00:23:03,320 --> 00:23:16,700 degree two are called quadratic polynomials. 190 00:23:23,400 --> 00:23:26,920 Examples would be x squared. 191 00:23:28,780 --> 00:23:34,140 Three x squared plus four x minus five. 192 00:23:37,440 --> 00:23:41,360 Two x squared minus eight x. 193 00:23:42,860 --> 00:23:44,860 Long as the degree is two, 194 00:23:45,080 --> 00:23:48,160 it's called a quadratic polynomial. 195 00:23:51,200 --> 00:23:58,829 what does quad usually mean? 4. that's not what it means here it has to do with the 196 00:23:58,829 --> 00:24:00,100 Latin word meaning 197 00:24:00,100 --> 00:24:10,900 square okay right because the exponent is second power which is a square right we 198 00:24:10,900 --> 00:24:13,780 even say x squared 199 00:24:13,780 --> 00:24:18,620 so don't confuse that degree three 200 00:24:28,060 --> 00:24:32,980 are called, you want to take a guess? 201 00:24:33,680 --> 00:24:37,120 Cubic polynomials. 202 00:24:42,320 --> 00:24:51,080 Examples are x cubed is a cubic polynomial. 4x cubed minus x squared. 203 00:25:03,060 --> 00:25:06,280 Anytime the degree three, the cubic polynomial. 204 00:25:09,240 --> 00:25:11,280 We have a couple of other names. 205 00:25:13,560 --> 00:25:17,960 degree four is called 206 00:25:19,920 --> 00:25:40,520 cork tick x to the fourth minus x cubed or so on. 207 00:25:40,520 --> 00:25:53,440 Often after cubic, we often just say polynomial of degree 4. 208 00:26:00,920 --> 00:26:07,040 5 is also known as quintic polynomial. 209 00:26:09,120 --> 00:26:20,960 But again, after 3, we usually just say a polynomial of degree 5. 210 00:26:23,920 --> 00:26:24,640 And we can keep going. 211 00:26:24,640 --> 00:26:46,800 n. Any degree n, we just say a polynomial of degree n. Like 5x to the 8th power is a 212 00:26:46,800 --> 00:26:48,220 n. Any degree n, we just say a polynomial of degree n. Like 5x to the 8th power is a 213 00:26:56,300 --> 00:27:02,420 Okay, the degree of the polynomial is the most important thing about the polynomial. 214 00:27:04,200 --> 00:27:09,904 Whether it has one term, two terms, three terms, and so on, that's a lot less 215 00:27:09,904 --> 00:27:10,654 important. 216 00:27:11,040 --> 00:27:15,540 But we will classify them by both degree and the name. 217 00:27:15,540 --> 00:27:27,380 Okay, so we're gonna classify by degree and name. 218 00:27:48,080 --> 00:27:50,920 What is the degree here? 219 00:27:51,540 --> 00:27:59,940 So the degree is 2, we call that a quadratic polynomial. 220 00:28:05,500 --> 00:28:11,120 Is it a monomial, binomial, trinomial? It's a trinomial. 221 00:28:11,120 --> 00:28:16,880 So when we call this a trinomial, you replace the word polynomial. 222 00:28:18,800 --> 00:28:23,520 So we would call this a quadratic trinomial. 223 00:28:26,160 --> 00:28:32,418 So any time you use the word monomial or binomial or trinomial, we just replace the 224 00:28:32,418 --> 00:28:33,200 word polynomial 225 00:28:33,200 --> 00:28:34,100 with that term. 226 00:28:41,040 --> 00:28:47,100 What would this be called? 227 00:28:49,820 --> 00:28:50,800 First off, what's the degree? 228 00:28:53,220 --> 00:28:54,600 What's the leading term? 229 00:28:56,020 --> 00:28:56,640 7x. 230 00:28:56,640 --> 00:28:59,180 The leading term would be 7x. 231 00:28:59,320 --> 00:29:01,420 The leading coefficient would be 7. 232 00:29:01,420 --> 00:29:08,120 OK, so since the degree is 1, it's called a linear polynomial. 233 00:29:11,660 --> 00:29:13,360 And then we replace the word polynomial 234 00:29:13,360 --> 00:29:16,740 with linear binomial. 235 00:29:28,900 --> 00:29:37,160 The number 8 would be of degree 0, that's x to the 0th power, 0. 236 00:29:37,160 --> 00:29:42,040 we call zero degree polynomials constant 237 00:29:49,680 --> 00:30:03,840 polynomial but we replace the word polynomial with monomial in fact all zero 238 00:30:03,840 --> 00:30:11,160 degree polynomials are constant and there's going to only be one term so all 239 00:30:11,160 --> 00:30:17,980 constant polynomials are constant monomials linear we could have either a 240 00:30:17,980 --> 00:30:22,460 constant binomial or a constant monomial okay 241 00:30:24,440 --> 00:30:26,800 Uh, let's do a few more. 242 00:30:27,720 --> 00:30:28,860 I think we're going to come up with this one. 243 00:30:31,020 --> 00:30:32,920 Except for the last apartment. 244 00:30:34,320 --> 00:30:35,260 There's a lot of x in there. 245 00:30:38,540 --> 00:30:39,900 There's a lot of x in there. 246 00:30:44,120 --> 00:30:45,720 Plus two, let's make it one. 247 00:30:48,520 --> 00:30:49,980 What is the degree here? 248 00:30:49,980 --> 00:31:04,819 6. 6. OK. So we just say a 6 degree polynomial. OK. Once we get past three terms, we 249 00:31:04,819 --> 00:31:05,600 stop 250 00:31:05,600 --> 00:31:16,072 calling it a monomial, binomial, trinomial. So we say 6 degree polynomial with four 251 00:31:16,072 --> 00:31:16,820 terms. 252 00:31:16,820 --> 00:31:17,720 Shh. 253 00:31:46,820 --> 00:31:49,300 What is the degree of this polynomial? 254 00:31:50,760 --> 00:31:56,420 Tenth degree. Okay, and there's only two terms we call it a tenth degree 255 00:31:59,700 --> 00:32:00,600 Binomial 256 00:32:07,420 --> 00:32:08,320 Okay 257 00:32:08,740 --> 00:32:11,380 Again, the degree is the most important thing 258 00:32:11,380 --> 00:32:13,240 however many terms 259 00:32:13,940 --> 00:32:19,647 Is less important and if you use a monomial binomial trying to use to replace the 260 00:32:19,647 --> 00:32:20,397 word polynomial 261 00:32:21,820 --> 00:32:25,280 Let's take we're gonna pause we still got a lot more to learn 262 00:32:26,340 --> 00:32:28,200 But we're gonna practice these 263 00:32:29,520 --> 00:32:30,920 Let me pause 264 00:32:35,260 --> 00:32:39,532 Now we're talking about adding and subtracting polynomials, but you've probably done 265 00:32:39,532 --> 00:32:40,600 in the past 266 00:32:41,220 --> 00:32:42,580 But let's just review 267 00:33:13,240 --> 00:33:21,680 So we're adding two quadratic polynomials to quadratic trinomials to be precise. 268 00:33:24,520 --> 00:33:26,080 Do we need these parentheses here? 269 00:33:27,320 --> 00:33:31,040 No, right, because this addition is the same as this addition, same as this. 270 00:33:42,040 --> 00:33:44,480 So, and how are we going to add these? 271 00:33:44,940 --> 00:33:46,060 We've done this in the past I believe. 272 00:33:47,040 --> 00:33:48,220 We're going to add like terms. 273 00:33:48,680 --> 00:33:49,980 We could rearrange it. 274 00:33:51,080 --> 00:33:52,420 You don't need to do this. 275 00:33:52,420 --> 00:33:54,740 we usually do this part in our head. 276 00:34:02,840 --> 00:34:05,340 Okay and what are like terms? 277 00:34:09,920 --> 00:34:18,202 Terms with the same degree, okay, are like terms. So they have to have the same 278 00:34:18,202 --> 00:34:18,720 variables 279 00:34:18,720 --> 00:34:24,076 raised to the same powers but these only have the X's in and how are we going to add 280 00:34:24,076 --> 00:34:24,826 like terms? 281 00:34:32,400 --> 00:34:40,248 1x squared plus 4x squared is 5x squared. You could factor out an x squared we get 1 282 00:34:40,248 --> 00:34:41,120 plus 4 283 00:34:41,120 --> 00:34:45,245 it's really because of the distributive property but again we do this part in our 284 00:34:45,245 --> 00:34:45,630 head. 285 00:34:45,630 --> 00:34:48,910 But same thing here, these are like terms. 286 00:34:55,930 --> 00:34:58,710 And we get, what is that, negative 5x. 287 00:34:59,070 --> 00:35:01,090 And the negative 5 to 6 is 1. 288 00:35:02,950 --> 00:35:05,130 And we usually don't write plus and negative, 289 00:35:05,250 --> 00:35:09,670 so we just rewrite as 5x squared minus 5x plus 1. 290 00:35:11,710 --> 00:35:11,810 OK. 291 00:35:11,810 --> 00:35:16,750 And typically we write it in, it's called descending order. 292 00:35:19,890 --> 00:35:22,570 Descending from the term with the largest degree 293 00:35:22,570 --> 00:35:24,310 down to the term with the smallest degree. 294 00:35:24,450 --> 00:35:27,990 So we've got a quadratic, a linear, a constant monomial. 295 00:35:28,850 --> 00:35:30,530 So we write it in descending order. 296 00:35:54,330 --> 00:35:55,170 I 297 00:35:55,170 --> 00:35:59,070 Will subtracting polynomials 298 00:36:01,430 --> 00:36:06,750 First up do I need these parentheses here, or can I just remove them? 299 00:36:08,390 --> 00:36:09,510 Definitely not. 300 00:36:10,490 --> 00:36:13,190 Because we're subtracting this entire thing. 301 00:36:13,690 --> 00:36:14,590 Not subtract. 302 00:36:14,670 --> 00:36:18,670 Without this, it would just be a minus 5x squared, which is not the case. 303 00:36:19,770 --> 00:36:21,570 So this is very important. 304 00:36:21,870 --> 00:36:28,970 We need to subtract, we add the opposite, and the opposite is we just distribute the 305 00:36:28,970 --> 00:36:32,930 negative is what I call it, even though technically it's not that. 306 00:36:44,690 --> 00:36:50,508 Okay, so you add the opposite and you just flip all the signs of the whatever is 307 00:36:50,508 --> 00:36:50,850 after 308 00:36:50,850 --> 00:36:57,356 this fraction or you just think of it as distributing the negative or the negative 309 00:36:57,356 --> 00:36:58,106 one. 310 00:36:59,310 --> 00:37:01,150 Does X cubed have any like terms? 311 00:37:01,150 --> 00:37:11,110 nope 4x negative 4x squared plus negative 5x squared is negative 9x 312 00:37:11,110 --> 00:37:27,730 squared x plus negative 6x is negative 5x and minus 7 that's it so we subtract 313 00:37:27,730 --> 00:37:32,930 distribute that negative or we add the opposite of a polynomial which is all 314 00:37:32,930 --> 00:37:42,010 the flip signs not too complicated right okay you've done that in the past okay 315 00:37:42,010 --> 00:37:45,750 so that should be a review now let's talk about multiplying polynomials 316 00:37:57,730 --> 00:37:59,570 I think I know where it is. 317 00:38:07,510 --> 00:38:11,730 So here we're multiplying a monomial times a trinomial. 318 00:38:13,050 --> 00:38:15,290 We're multiplying one quadratic times another quadratic. 319 00:38:17,150 --> 00:38:19,050 And what property do we have to use? 320 00:38:20,110 --> 00:38:23,330 The distributive property of multiplication over a distribution. 321 00:38:23,350 --> 00:38:27,070 Very good. The distributive property of multiplication, this multiplication. 322 00:38:27,730 --> 00:38:39,810 over addition we simply distribute just like we did with radicals the other day 323 00:38:39,810 --> 00:38:41,330 Two, let's draw. 324 00:38:47,110 --> 00:38:48,010 Three. 325 00:38:48,690 --> 00:38:49,590 Four. 326 00:38:50,590 --> 00:38:51,190 Six. 327 00:38:51,190 --> 00:38:51,990 Seven. 328 00:38:51,990 --> 00:38:52,710 Eight. 329 00:38:52,710 --> 00:38:53,610 Ten. 330 00:38:53,870 --> 00:38:54,710 Ten. 331 00:38:54,710 --> 00:38:57,850 And then we just use our rules of x on it that we've done. 332 00:38:58,930 --> 00:39:00,830 Three x squared times x squared. 333 00:39:01,910 --> 00:39:03,470 X to the two plus two. 334 00:39:04,030 --> 00:39:05,930 X to the fourth. 335 00:39:09,790 --> 00:39:11,390 minus 12. 336 00:39:13,510 --> 00:39:22,650 x squared times x is x to the 2 plus 1, which is x cubed. 337 00:39:25,270 --> 00:39:26,270 15x squared. 338 00:39:30,530 --> 00:39:32,630 It's just the rules of exponents. 339 00:39:34,170 --> 00:39:35,810 And we end up with this. 340 00:39:38,830 --> 00:39:40,270 What is the degree here? 341 00:39:41,570 --> 00:39:42,010 Four. 342 00:39:42,010 --> 00:39:42,910 Four. 343 00:39:43,510 --> 00:39:44,710 How many terms? 344 00:39:45,390 --> 00:39:46,010 Three. 345 00:39:46,010 --> 00:39:46,910 Do we call this a? 346 00:39:48,090 --> 00:39:49,510 Four degrees. 347 00:39:49,510 --> 00:39:51,270 He's trying to rattle me. 348 00:40:11,750 --> 00:40:13,570 Go ahead and distribute. 349 00:40:18,990 --> 00:40:20,390 you. 350 00:41:14,430 --> 00:41:19,250 42x to the fourth. 351 00:41:19,470 --> 00:41:23,390 What about that? 352 00:41:25,490 --> 00:41:27,510 Did we all get that? Yes, sir. Yeah. 353 00:41:28,110 --> 00:41:31,150 Multiplying is all about just using the distributive property. 354 00:41:32,390 --> 00:41:34,670 Because polynomials have multiple terms. 355 00:41:36,890 --> 00:41:40,030 Now, so that's multiplying monomials 356 00:41:40,030 --> 00:41:46,510 times any polynomial you just distribute the monomial to every term and then 357 00:41:46,510 --> 00:42:05,610 simplify now let's talk about a binomial times a binomial is a linear binomial 358 00:42:05,850 --> 00:42:10,050 The linear binomial what property we have to use 359 00:42:14,130 --> 00:42:17,850 The distributive property of multiplication over 360 00:42:18,370 --> 00:42:19,270 addition 361 00:42:19,450 --> 00:42:23,650 Just like we use radicals. We have to distribute the whole 362 00:42:24,930 --> 00:42:29,210 Binomial the whole polynomial to every term in the other 363 00:42:32,290 --> 00:42:34,750 We get 2x plus 3 364 00:42:34,750 --> 00:42:41,750 times x plus 2x plus 3 times negative 5. 365 00:42:50,410 --> 00:42:52,450 And now what do we have to do? 366 00:42:53,590 --> 00:42:56,350 Use the distributive property two more times. 367 00:42:59,670 --> 00:43:01,430 Just like we did with radicals. 368 00:43:28,070 --> 00:43:32,130 And then the last step. 369 00:43:32,130 --> 00:43:40,628 Find like terms. Find like terms. I only have two like terms here. So it's 2x 370 00:43:40,628 --> 00:43:41,690 squared 3 371 00:43:41,690 --> 00:44:02,870 minus 10 minus 7x. Okay is there a shortcut where we could just bypass this 372 00:44:02,870 --> 00:44:07,870 whole step. Why don't you talk to your neighbor, what's a shortcut? 373 00:44:07,870 --> 00:44:09,690 it. 374 00:44:25,550 --> 00:44:31,510 Uh, anybody got a shortcut? 375 00:44:37,490 --> 00:44:38,390 No shortcuts? 376 00:44:40,470 --> 00:44:44,962 Okay, let's just do the long cut from that one. That's fine. I like it. Let's try 377 00:44:44,962 --> 00:44:45,712 another one. 378 00:45:01,510 --> 00:45:02,410 I don't know. 379 00:45:02,450 --> 00:45:03,350 Can I spell it? 380 00:45:03,790 --> 00:45:04,690 Go. 381 00:45:05,490 --> 00:45:06,390 They can't. 382 00:45:06,430 --> 00:45:07,310 They can't. 383 00:45:07,310 --> 00:45:08,210 They can't. 384 00:45:08,270 --> 00:45:08,950 They can't. 385 00:45:08,950 --> 00:45:11,030 They can't. 386 00:45:11,030 --> 00:45:12,290 So what do we do? 387 00:45:15,430 --> 00:45:16,330 We distribute. 388 00:45:18,470 --> 00:45:19,370 OK. 389 00:45:19,970 --> 00:45:22,430 Does it matter if I distribute this here 390 00:45:22,430 --> 00:45:26,350 or if I distribute this here? 391 00:45:26,950 --> 00:45:27,850 It doesn't matter. 392 00:45:27,890 --> 00:45:28,950 It works out at the end the same. 393 00:45:32,010 --> 00:45:35,230 Let's just do that. I always go left to right. 394 00:45:57,870 --> 00:46:02,070 So we distribute the 4x cubed minus 3x squared. 395 00:46:03,330 --> 00:46:04,650 and now what do we gotta do? 396 00:46:06,350 --> 00:46:07,270 distribute again 397 00:46:10,370 --> 00:46:12,220 I wish there was an easier way 398 00:46:12,220 --> 00:46:14,230 In what order do we write our final answering? 399 00:46:14,850 --> 00:46:19,719 What's it called descending descending the term with the largest degree down to the 400 00:46:19,719 --> 00:46:21,110 term of a degree? 401 00:46:22,370 --> 00:46:23,990 So let's go back 402 00:46:25,950 --> 00:46:32,350 Is there a shortcut where we can skip this entire step, yes, what's the shortcut? 403 00:46:32,350 --> 00:46:34,250 aluminum foil method 404 00:46:34,250 --> 00:46:35,150 and 405 00:46:35,350 --> 00:46:36,310 well, five 406 00:46:37,210 --> 00:46:40,170 why are you guys talking about aluminum foil? 407 00:46:41,450 --> 00:46:43,310 I think like you multiply the front 408 00:46:43,310 --> 00:46:44,930 two terms together 409 00:46:44,930 --> 00:46:47,430 then you multiply the outside terms 410 00:46:47,430 --> 00:46:50,310 and then you multiply the inside terms 411 00:46:50,310 --> 00:46:52,150 and the last terms 412 00:46:52,150 --> 00:46:55,250 okay, I like that 413 00:46:55,250 --> 00:46:58,690 except what happens when we have 414 00:47:06,630 --> 00:47:09,130 Two terms times three terms. 415 00:47:19,010 --> 00:47:24,610 Okay so let's do a shortcut without any weird words like foil. 416 00:47:31,070 --> 00:47:34,170 Let's look at this. Where did these two terms come from? 417 00:47:39,610 --> 00:47:49,330 Was this times this. Okay. Which let me erase this. 418 00:47:54,950 --> 00:48:02,990 so it's this term times this term where did this the negative 3x squared and the 419 00:48:02,990 --> 00:48:18,410 2x squared come from this and then the 4x cubed and the 5x here and the negative 3x 420 00:48:18,410 --> 00:48:28,610 squared and the 5x there okay so we can skip this entire step by just taking 421 00:48:28,610 --> 00:48:38,370 each term and distribute it to the other terms in the other polynomial and then 422 00:48:38,370 --> 00:48:43,070 you go to the next term you distribute to each of the terms and the other one 423 00:48:43,070 --> 00:48:49,210 it's just a bunch of distributing we have a binomial times a binomial some 424 00:48:49,210 --> 00:48:57,610 people like the acronym foil okay first outside inside last but I don't mess with 425 00:48:57,610 --> 00:49:03,710 that because once you get two times three terms a binomial times a trinomial 426 00:49:04,750 --> 00:49:13,050 you just take the first term just distribute it to every term and then 427 00:49:13,050 --> 00:49:15,710 And you go to the next term, distribute to the next term. 428 00:49:21,370 --> 00:49:33,930 We got 2x times x squared plus 2x times 4x plus 2x times negative 5. 429 00:49:36,650 --> 00:49:37,770 Those should all be in purple. 430 00:49:43,550 --> 00:50:01,476 Oops. Let's try one more time. There we go. And then the second term times the all 431 00:50:01,476 --> 00:50:02,530 of 432 00:50:04,640 --> 00:50:13,420 3 times x squared plus 3 times 4x plus 3 times maybe 5. 433 00:50:14,060 --> 00:50:17,760 Eventually we even do this in our head and we start writing all 6 terms. 434 00:50:17,980 --> 00:50:22,400 So 2 terms times 3 terms, 2 times 3, we always get that number of terms. 435 00:50:22,400 --> 00:50:35,760 2x cubed plus 8x squared minus 10x 3x squared plus 12x minus 15. 436 00:50:37,080 --> 00:50:41,560 And once we do that, we just combine like terms. 437 00:50:43,860 --> 00:50:51,500 This guy likes this term 11x squared. 438 00:50:53,120 --> 00:50:54,680 That's two. 439 00:50:57,620 --> 00:50:58,520 Minus 15. 440 00:51:02,250 --> 00:51:04,000 I think it's a cubic polynomial. 441 00:51:07,660 --> 00:51:10,100 Cubic is correct, the degree is three, 442 00:51:11,100 --> 00:51:13,540 but the number of terms is four. 443 00:51:13,780 --> 00:51:17,460 So we spell it cubic polynomial with four terms. 444 00:51:19,380 --> 00:51:20,280 Okay. 445 00:51:22,040 --> 00:51:23,840 Try another one. 446 00:51:40,060 --> 00:51:42,200 Just take each term in any one of them. 447 00:51:42,520 --> 00:51:43,460 I always go left right. 448 00:51:44,820 --> 00:51:48,500 We distribute to every term in the other polynomial. 449 00:51:55,700 --> 00:52:00,880 Start distributing and then simplifying and then combining like turns. 450 00:52:43,560 --> 00:52:47,375 A trinomial times a trinomial will have to redistribute how many terms you're going 451 00:52:47,375 --> 00:52:48,120 to have. 452 00:52:48,120 --> 00:52:49,440 All I have is nine. 453 00:52:50,200 --> 00:52:50,580 Nine. 454 00:52:50,580 --> 00:52:51,940 Three times three is nine. 455 00:52:52,820 --> 00:52:54,620 So write out all nine terms. 456 00:52:55,860 --> 00:52:57,060 And then start simplifying. 457 00:53:23,400 --> 00:53:26,180 Once we get good at this, you can just multiply in your head. 458 00:53:27,720 --> 00:53:30,340 Expert times expert. 459 00:54:24,560 --> 00:54:25,360 And then... 460 00:54:25,360 --> 00:54:26,800 I'm always like... 461 00:54:26,800 --> 00:54:28,920 Still being wrong? 462 00:54:31,780 --> 00:54:32,680 Minus 7x... 463 00:54:33,020 --> 00:54:34,540 Plus 20, there we go. 464 00:54:34,940 --> 00:54:35,840 Oh my goodness. 465 00:54:36,700 --> 00:54:37,600 Wait, you didn't like it. 466 00:54:37,800 --> 00:54:39,160 You told me it was wrong. 467 00:54:41,180 --> 00:54:42,220 I said his was wrong. 468 00:54:42,860 --> 00:54:44,200 Why are you looking at the statement from the guy? 469 00:54:44,200 --> 00:54:45,500 Are you telling me to do it, bro? 470 00:54:46,880 --> 00:54:47,780 Wait a minute. 471 00:54:49,700 --> 00:54:51,960 Okay, what questions do we have on this? 472 00:54:53,140 --> 00:54:55,780 It's just a lot of distributing 473 00:54:56,500 --> 00:54:58,840 using the distributive property of addition, 474 00:55:00,100 --> 00:55:01,200 of multiplication over addition. 475 00:55:01,800 --> 00:55:02,840 Okay, we're gonna stop here. 476 00:55:03,400 --> 00:55:04,250 Yes.