WEBVTT 00:00:05.000 --> 00:00:09.540 Alright, we're talking about polynomials. 00:00:14.280 --> 00:00:16.660 We're going to start out with the definition. 00:00:32.820 --> 00:00:42.176 A monomial is a number, a variable, or the product of numbers and variables raised 00:00:42.176 --> 00:00:42.800 to 00:00:42.800 --> 00:00:43.980 whole number powers. 00:00:43.980 --> 00:00:58.980 Let's see if this gets... 00:01:04.220 --> 00:01:13.239 A monomial is a number, a variable, or the product of numbers and variables raised 00:01:13.239 --> 00:01:13.840 to 00:01:13.840 --> 00:01:15.600 poll number. 00:01:50.730 --> 00:01:52.430 Uh, examples. 00:01:57.230 --> 00:01:59.610 Is three a monomial? 00:02:02.290 --> 00:02:03.070 Yes. 00:02:03.070 --> 00:02:03.750 Yes, why? 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 00:02:23.326 --> 00:02:24.550 variable. 00:02:24.550 --> 00:02:25.450 2. 00:02:28.730 --> 00:02:34.010 Is 4x squared a monomial? 00:02:34.470 --> 00:02:35.370 Yes. 00:02:35.730 --> 00:02:37.050 Yes, why? 00:02:37.490 --> 00:02:40.270 Product of numbers and variables raised to whole numbers. 00:02:41.230 --> 00:02:44.670 Product of numbers and variables raised to whole numbers. 00:02:45.130 --> 00:02:46.030 Powers. 00:02:47.370 --> 00:02:48.930 OK, not monomials. 00:03:04.110 --> 00:03:10.750 Is x to the negative 1 same thing as 1 over x? 00:03:12.550 --> 00:03:13.530 Is that a monolion? 00:03:14.630 --> 00:03:15.470 No. 00:03:15.470 --> 00:03:16.370 No, why not? 00:03:16.470 --> 00:03:17.370 It's nominal. 00:03:19.610 --> 00:03:20.510 Because? 00:03:22.670 --> 00:03:24.530 It's not raised to a whole number. 00:03:24.530 --> 00:03:25.430 number power. 00:03:27.110 --> 00:03:28.370 Negative 1 is not a whole number. 00:03:30.650 --> 00:03:31.950 OK, what are the whole numbers? 00:03:33.450 --> 00:03:34.350 Positive 2. 00:03:36.430 --> 00:03:41.690 So the 1, 2, 3, 4, and so on, the counting number 00:03:41.690 --> 00:03:43.290 is also known as the natural numbers. 00:03:45.350 --> 00:03:48.450 You add 0 to it, 0, 1, 2, 3, 4, and so on. 00:03:48.490 --> 00:03:50.230 Those are the whole number, set of whole numbers. 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 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 00:04:21.590 --> 00:04:22.490 Not because? 00:04:26.130 --> 00:04:28.230 It's not a product of numbers and variables. 00:04:28.390 --> 00:04:29.290 We have a sum here. 00:04:29.410 --> 00:04:30.410 This is not a monomial. 00:04:32.730 --> 00:04:33.630 Okay. 00:04:38.410 --> 00:04:39.810 What about the square root of 5? 00:04:39.950 --> 00:04:40.850 Is that a monomial? 00:04:44.490 --> 00:04:46.030 Same thing as 5 to what power? 00:04:47.730 --> 00:04:48.630 1 half power. 00:04:49.310 --> 00:04:50.490 Is it a monomial or no? 00:04:50.950 --> 00:04:51.850 No. 00:04:53.750 --> 00:04:55.270 Is the square root of 5 a number? 00:04:56.870 --> 00:04:57.770 Yeah. 00:04:57.810 --> 00:04:58.710 Yeah, it's a number. 00:04:58.850 --> 00:04:59.750 It's definitely a number. 00:05:01.230 --> 00:05:02.850 Okay, it is a monomial. 00:05:03.970 --> 00:05:07.210 Only the variables has to be raised to a whole number of powers. 00:05:08.590 --> 00:05:11.410 Oh, I noticed you put it in each example instead of five. 00:05:12.450 --> 00:05:16.170 And yes, I put it under the example, not the not a monomial. 00:05:16.550 --> 00:05:17.450 That's the other clue. 00:05:17.990 --> 00:05:20.270 OK, next definition. 00:05:22.930 --> 00:05:26.190 Actually, let's do one more, it's not... 00:05:27.930 --> 00:05:31.290 2 to the power of x. Is that a monomial? 00:05:31.790 --> 00:05:33.550 No. No, why not? 00:05:33.970 --> 00:05:36.690 Because the exponent's a monomial. 00:05:37.230 --> 00:05:40.850 Yeah, the variable's in the exponent. OK. 00:05:42.630 --> 00:05:45.290 We can't have any variables in the exponent. 00:05:45.690 --> 00:05:47.030 It's not a monomial. 00:05:47.750 --> 00:05:48.870 Next definition. 00:05:56.310 --> 00:06:04.090 A polynomial is a monomial for the sum of monomials. 00:06:51.610 --> 00:06:53.210 Is this a polynomial? 00:06:54.330 --> 00:06:55.810 Yes. Yes, why? 00:06:56.130 --> 00:06:57.030 Because it's a monomial. 00:07:06.510 --> 00:07:08.490 Is this a polynomial? 00:07:08.490 --> 00:07:13.910 Yes, yes, why it's the sum of three monomials 00:07:23.390 --> 00:07:26.070 Is this a polynomial 00:07:30.250 --> 00:07:33.270 For those who said no, why not 00:07:33.270 --> 00:07:34.170 It's not 00:07:36.910 --> 00:07:39.350 Is this the same thing 00:07:42.910 --> 00:07:49.750 Is negative 8x squared a monomial I was not I was trying to trick you 00:07:50.590 --> 00:07:52.970 But make sure we get the right definition 00:07:55.310 --> 00:07:59.190 This is a monomial sorry, this is a polynomial 00:07:59.260 --> 00:08:01.420 even though there's a subtraction, 00:08:01.560 --> 00:08:04.460 we always think that's just a plus and negative, that number. 00:08:05.660 --> 00:08:06.560 OK. 00:08:07.500 --> 00:08:08.500 Next definition. 00:08:12.560 --> 00:08:23.040 The number in a number being multiplied, not the exponent. 00:08:27.500 --> 00:08:40.520 Monomial is called the coefficient. Okay, the number, not the exponent. 00:09:02.460 --> 00:09:11.720 what's the coefficient here? 5. the coefficient here? 3. what is the 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 00:10:35.020 --> 00:10:38.240 How many terms does this have? 00:10:40.360 --> 00:10:47.380 Because we think of this as 5x to the fourth plus negative 7x squared plus 8. 00:10:47.960 --> 00:10:52.260 It has three terms. One, two, three terms. 00:11:03.740 --> 00:11:17.300 A polynomial with two terms is called a binomial. 00:11:23.980 --> 00:11:26.560 A polynomial with two terms is called a binomial. 00:11:27.580 --> 00:11:31.760 A polynomial with one term is called a monomial. 00:11:31.760 --> 00:11:53.320 two-term binomial a polynomial with three terms is called a trinomial a 00:11:53.320 --> 00:12:10.960 polynomial with four terms could be called a quadrinal meal but we don't 00:12:11.360 --> 00:12:20.500 We just call it a called a polynomial with four terms. 00:12:20.980 --> 00:12:22.160 Oh wow, that's pretty elaborate. 00:12:28.360 --> 00:12:35.700 Once you get past three, four, and on, we call it a polynomial with n terms. 00:12:36.160 --> 00:12:36.800 What's that? 00:12:36.800 --> 00:12:37.880 What is that commutation line? 00:12:37.880 --> 00:12:42.440 That means repeat above a polynomial. 00:12:43.460 --> 00:12:46.600 A polynomial with four terms is called a polynomial with four terms. 00:12:47.160 --> 00:12:48.060 That's right. 00:12:48.260 --> 00:12:50.280 A polynomial with seven terms is called? 00:12:50.640 --> 00:12:51.740 A polynomial with seven terms. 00:12:51.760 --> 00:12:52.980 A polynomial with seven terms. 00:12:53.340 --> 00:12:55.340 So we've got a monomial, a binomial. 00:12:55.780 --> 00:12:57.740 Let me do a couple quick examples, right? 00:12:58.320 --> 00:13:06.900 For example, 3x cubed plus 4x is a binomial. 00:13:06.900 --> 00:13:15.580 Why? Because it has two terms. Is it also a polynomial? Yeah, definitely. Okay, 00:13:15.580 --> 00:13:16.820 three terms, 00:13:16.820 --> 00:13:21.360 X squared plus 2x minus 5 will be a trinomial 00:13:34.360 --> 00:13:36.760 This would be called a 00:13:36.760 --> 00:13:42.855 Polynomial with four terms so monomial binomial trinomial polynomial four terms 00:13:42.855 --> 00:13:46.180 polynomial five terms and so on 00:13:50.080 --> 00:14:12.520 okay next definition the degree of a polynomial sorry of a monomial the sum 00:14:12.520 --> 00:14:17.520 of the exponents. 00:14:27.400 --> 00:14:28.800 exponents 00:14:42.860 --> 00:14:45.660 So samples. 00:14:50.420 --> 00:14:53.800 What is the degree? 00:14:53.800 --> 00:14:55.120 6. 00:15:05.220 --> 00:15:08.500 What is the degree? 00:15:08.960 --> 00:15:10.320 2. 00:15:14.180 --> 00:15:18.200 What is the degree here? 00:15:18.200 --> 00:15:24.700 1. 1. Is the number 7 a monolu? 00:15:25.600 --> 00:15:28.320 Yeah. It is. What's the degree? 00:15:28.720 --> 00:15:34.740 1. I don't see any variables. This is degree 1. 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 00:15:42.582 --> 00:15:43.332 power. 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 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 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 00:16:12.300 --> 00:16:28.900 Okay, every number has degree zero. We call this a constant. Okay, next definition. 00:16:31.520 --> 00:16:42.180 the degree of a polynomial not a monomial 00:16:58.900 --> 00:17:17.980 is the degree of the term with the largest degree. 00:17:40.200 --> 00:17:42.360 How many terms does this have? 00:17:43.380 --> 00:17:44.280 Three. 00:17:44.820 --> 00:17:46.880 What's the degree of this term? 00:17:47.280 --> 00:17:47.660 Two. 00:17:47.660 --> 00:17:48.420 One. 00:17:48.420 --> 00:17:49.320 One. 00:17:49.700 --> 00:17:50.600 Degree of this term? 00:17:50.960 --> 00:17:51.280 Two. 00:17:51.280 --> 00:17:51.800 Two. 00:17:51.800 --> 00:17:52.700 Degree of this term? 00:17:52.700 --> 00:17:53.240 Zero. 00:17:53.240 --> 00:17:54.140 Zero. 00:17:54.360 --> 00:17:56.640 The degree of the whole polynomial is? 00:17:56.680 --> 00:17:56.900 Two. 00:17:56.900 --> 00:17:57.800 Two. 00:17:59.720 --> 00:18:03.000 Because it's the degree of this term, which is two. 00:18:30.280 --> 00:18:32.420 What is the degree of this term? 00:18:33.840 --> 00:18:34.740 Degree of this term? 00:18:35.420 --> 00:18:36.460 Degree of this term? 00:18:36.560 --> 00:18:37.260 Four. 00:18:37.260 --> 00:18:38.180 Degree of this term? 00:18:38.260 --> 00:18:39.160 Five. 00:18:39.260 --> 00:18:43.240 The term with the largest is this, so the degree of the whole polynomial is? 00:18:43.340 --> 00:18:44.240 Seven. 00:18:45.680 --> 00:18:46.580 Seven. 00:18:50.940 --> 00:19:11.520 Another definition, the term with the largest degree is called the leading term. 00:19:20.100 --> 00:19:24.400 Even though it doesn't have to be in front, it is the leading term. 00:19:32.800 --> 00:19:35.940 What is the leading term in this? 00:19:35.940 --> 00:19:38.080 7x to the 7. 00:19:38.580 --> 00:19:40.160 Negative 7x squared. 00:19:45.960 --> 00:19:49.460 What is the leading term here? 00:19:49.900 --> 00:19:51.180 5x to the 7. 00:19:51.660 --> 00:19:53.000 5x to the 7. 00:19:56.380 --> 00:20:14.520 The coefficient of the leading term is called... anybody want to take a guess? 00:20:14.520 --> 00:20:15.700 Leading coefficient. 00:20:16.220 --> 00:20:17.120 Leading coefficients. 00:20:54.220 --> 00:20:56.460 What is the leading term? 00:20:58.120 --> 00:20:59.680 X cubed. 00:20:59.960 --> 00:21:02.240 What is the leading coefficient? 00:21:02.240 --> 00:21:03.140 1. 00:21:14.200 --> 00:21:18.620 What is the leading coefficient here? 00:21:21.240 --> 00:21:23.880 negative 7 00:21:38.540 --> 00:21:40.620 okay now we got some names 00:21:40.620 --> 00:21:44.420 of polynomials with different degrees 00:21:44.420 --> 00:22:11.980 so the degree the name and then I'll give examples the degree zero it's 00:22:11.980 --> 00:22:16.940 It's called a constant polynomial. 00:22:23.360 --> 00:22:30.980 Examples would be like 5 or negative pi. 00:22:32.780 --> 00:22:34.800 The constant doesn't change. 00:22:34.800 --> 00:22:39.800 degree one are called linear polynomials. 00:22:48.360 --> 00:22:53.360 Three x plus four, negative eight x. 00:23:03.320 --> 00:23:16.700 degree two are called quadratic polynomials. 00:23:23.400 --> 00:23:26.920 Examples would be x squared. 00:23:28.780 --> 00:23:34.140 Three x squared plus four x minus five. 00:23:37.440 --> 00:23:41.360 Two x squared minus eight x. 00:23:42.860 --> 00:23:44.860 Long as the degree is two, 00:23:45.080 --> 00:23:48.160 it's called a quadratic polynomial. 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 00:23:58.829 --> 00:24:00.100 Latin word meaning 00:24:00.100 --> 00:24:10.900 square okay right because the exponent is second power which is a square right we 00:24:10.900 --> 00:24:13.780 even say x squared 00:24:13.780 --> 00:24:18.620 so don't confuse that degree three 00:24:28.060 --> 00:24:32.980 are called, you want to take a guess? 00:24:33.680 --> 00:24:37.120 Cubic polynomials. 00:24:42.320 --> 00:24:51.080 Examples are x cubed is a cubic polynomial. 4x cubed minus x squared. 00:25:03.060 --> 00:25:06.280 Anytime the degree three, the cubic polynomial. 00:25:09.240 --> 00:25:11.280 We have a couple of other names. 00:25:13.560 --> 00:25:17.960 degree four is called 00:25:19.920 --> 00:25:40.520 cork tick x to the fourth minus x cubed or so on. 00:25:40.520 --> 00:25:53.440 Often after cubic, we often just say polynomial of degree 4. 00:26:00.920 --> 00:26:07.040 5 is also known as quintic polynomial. 00:26:09.120 --> 00:26:20.960 But again, after 3, we usually just say a polynomial of degree 5. 00:26:23.920 --> 00:26:24.640 And we can keep going. 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 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 00:26:56.300 --> 00:27:02.420 Okay, the degree of the polynomial is the most important thing about the polynomial. 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 00:27:09.904 --> 00:27:10.654 important. 00:27:11.040 --> 00:27:15.540 But we will classify them by both degree and the name. 00:27:15.540 --> 00:27:27.380 Okay, so we're gonna classify by degree and name. 00:27:48.080 --> 00:27:50.920 What is the degree here? 00:27:51.540 --> 00:27:59.940 So the degree is 2, we call that a quadratic polynomial. 00:28:05.500 --> 00:28:11.120 Is it a monomial, binomial, trinomial? It's a trinomial. 00:28:11.120 --> 00:28:16.880 So when we call this a trinomial, you replace the word polynomial. 00:28:18.800 --> 00:28:23.520 So we would call this a quadratic trinomial. 00:28:26.160 --> 00:28:32.418 So any time you use the word monomial or binomial or trinomial, we just replace the 00:28:32.418 --> 00:28:33.200 word polynomial 00:28:33.200 --> 00:28:34.100 with that term. 00:28:41.040 --> 00:28:47.100 What would this be called? 00:28:49.820 --> 00:28:50.800 First off, what's the degree? 00:28:53.220 --> 00:28:54.600 What's the leading term? 00:28:56.020 --> 00:28:56.640 7x. 00:28:56.640 --> 00:28:59.180 The leading term would be 7x. 00:28:59.320 --> 00:29:01.420 The leading coefficient would be 7. 00:29:01.420 --> 00:29:08.120 OK, so since the degree is 1, it's called a linear polynomial. 00:29:11.660 --> 00:29:13.360 And then we replace the word polynomial 00:29:13.360 --> 00:29:16.740 with linear binomial. 00:29:28.900 --> 00:29:37.160 The number 8 would be of degree 0, that's x to the 0th power, 0. 00:29:37.160 --> 00:29:42.040 we call zero degree polynomials constant 00:29:49.680 --> 00:30:03.840 polynomial but we replace the word polynomial with monomial in fact all zero 00:30:03.840 --> 00:30:11.160 degree polynomials are constant and there's going to only be one term so all 00:30:11.160 --> 00:30:17.980 constant polynomials are constant monomials linear we could have either a 00:30:17.980 --> 00:30:22.460 constant binomial or a constant monomial okay 00:30:24.440 --> 00:30:26.800 Uh, let's do a few more. 00:30:27.720 --> 00:30:28.860 I think we're going to come up with this one. 00:30:31.020 --> 00:30:32.920 Except for the last apartment. 00:30:34.320 --> 00:30:35.260 There's a lot of x in there. 00:30:38.540 --> 00:30:39.900 There's a lot of x in there. 00:30:44.120 --> 00:30:45.720 Plus two, let's make it one. 00:30:48.520 --> 00:30:49.980 What is the degree here? 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 00:31:04.819 --> 00:31:05.600 stop 00:31:05.600 --> 00:31:16.072 calling it a monomial, binomial, trinomial. So we say 6 degree polynomial with four 00:31:16.072 --> 00:31:16.820 terms. 00:31:16.820 --> 00:31:17.720 Shh. 00:31:46.820 --> 00:31:49.300 What is the degree of this polynomial? 00:31:50.760 --> 00:31:56.420 Tenth degree. Okay, and there's only two terms we call it a tenth degree 00:31:59.700 --> 00:32:00.600 Binomial 00:32:07.420 --> 00:32:08.320 Okay 00:32:08.740 --> 00:32:11.380 Again, the degree is the most important thing 00:32:11.380 --> 00:32:13.240 however many terms 00:32:13.940 --> 00:32:19.647 Is less important and if you use a monomial binomial trying to use to replace the 00:32:19.647 --> 00:32:20.397 word polynomial 00:32:21.820 --> 00:32:25.280 Let's take we're gonna pause we still got a lot more to learn 00:32:26.340 --> 00:32:28.200 But we're gonna practice these 00:32:29.520 --> 00:32:30.920 Let me pause 00:32:35.260 --> 00:32:39.532 Now we're talking about adding and subtracting polynomials, but you've probably done 00:32:39.532 --> 00:32:40.600 in the past 00:32:41.220 --> 00:32:42.580 But let's just review 00:33:13.240 --> 00:33:21.680 So we're adding two quadratic polynomials to quadratic trinomials to be precise. 00:33:24.520 --> 00:33:26.080 Do we need these parentheses here? 00:33:27.320 --> 00:33:31.040 No, right, because this addition is the same as this addition, same as this. 00:33:42.040 --> 00:33:44.480 So, and how are we going to add these? 00:33:44.940 --> 00:33:46.060 We've done this in the past I believe. 00:33:47.040 --> 00:33:48.220 We're going to add like terms. 00:33:48.680 --> 00:33:49.980 We could rearrange it. 00:33:51.080 --> 00:33:52.420 You don't need to do this. 00:33:52.420 --> 00:33:54.740 we usually do this part in our head. 00:34:02.840 --> 00:34:05.340 Okay and what are like terms? 00:34:09.920 --> 00:34:18.202 Terms with the same degree, okay, are like terms. So they have to have the same 00:34:18.202 --> 00:34:18.720 variables 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 00:34:24.076 --> 00:34:24.826 like terms? 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 00:34:40.248 --> 00:34:41.120 plus 4 00:34:41.120 --> 00:34:45.245 it's really because of the distributive property but again we do this part in our 00:34:45.245 --> 00:34:45.630 head. 00:34:45.630 --> 00:34:48.910 But same thing here, these are like terms. 00:34:55.930 --> 00:34:58.710 And we get, what is that, negative 5x. 00:34:59.070 --> 00:35:01.090 And the negative 5 to 6 is 1. 00:35:02.950 --> 00:35:05.130 And we usually don't write plus and negative, 00:35:05.250 --> 00:35:09.670 so we just rewrite as 5x squared minus 5x plus 1. 00:35:11.710 --> 00:35:11.810 OK. 00:35:11.810 --> 00:35:16.750 And typically we write it in, it's called descending order. 00:35:19.890 --> 00:35:22.570 Descending from the term with the largest degree 00:35:22.570 --> 00:35:24.310 down to the term with the smallest degree. 00:35:24.450 --> 00:35:27.990 So we've got a quadratic, a linear, a constant monomial. 00:35:28.850 --> 00:35:30.530 So we write it in descending order. 00:35:54.330 --> 00:35:55.170 I 00:35:55.170 --> 00:35:59.070 Will subtracting polynomials 00:36:01.430 --> 00:36:06.750 First up do I need these parentheses here, or can I just remove them? 00:36:08.390 --> 00:36:09.510 Definitely not. 00:36:10.490 --> 00:36:13.190 Because we're subtracting this entire thing. 00:36:13.690 --> 00:36:14.590 Not subtract. 00:36:14.670 --> 00:36:18.670 Without this, it would just be a minus 5x squared, which is not the case. 00:36:19.770 --> 00:36:21.570 So this is very important. 00:36:21.870 --> 00:36:28.970 We need to subtract, we add the opposite, and the opposite is we just distribute the 00:36:28.970 --> 00:36:32.930 negative is what I call it, even though technically it's not that. 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 00:36:50.508 --> 00:36:50.850 after 00:36:50.850 --> 00:36:57.356 this fraction or you just think of it as distributing the negative or the negative 00:36:57.356 --> 00:36:58.106 one. 00:36:59.310 --> 00:37:01.150 Does X cubed have any like terms? 00:37:01.150 --> 00:37:11.110 nope 4x negative 4x squared plus negative 5x squared is negative 9x 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 00:37:27.730 --> 00:37:32.930 distribute that negative or we add the opposite of a polynomial which is all 00:37:32.930 --> 00:37:42.010 the flip signs not too complicated right okay you've done that in the past okay 00:37:42.010 --> 00:37:45.750 so that should be a review now let's talk about multiplying polynomials 00:37:57.730 --> 00:37:59.570 I think I know where it is. 00:38:07.510 --> 00:38:11.730 So here we're multiplying a monomial times a trinomial. 00:38:13.050 --> 00:38:15.290 We're multiplying one quadratic times another quadratic. 00:38:17.150 --> 00:38:19.050 And what property do we have to use? 00:38:20.110 --> 00:38:23.330 The distributive property of multiplication over a distribution. 00:38:23.350 --> 00:38:27.070 Very good. The distributive property of multiplication, this multiplication. 00:38:27.730 --> 00:38:39.810 over addition we simply distribute just like we did with radicals the other day 00:38:39.810 --> 00:38:41.330 Two, let's draw. 00:38:47.110 --> 00:38:48.010 Three. 00:38:48.690 --> 00:38:49.590 Four. 00:38:50.590 --> 00:38:51.190 Six. 00:38:51.190 --> 00:38:51.990 Seven. 00:38:51.990 --> 00:38:52.710 Eight. 00:38:52.710 --> 00:38:53.610 Ten. 00:38:53.870 --> 00:38:54.710 Ten. 00:38:54.710 --> 00:38:57.850 And then we just use our rules of x on it that we've done. 00:38:58.930 --> 00:39:00.830 Three x squared times x squared. 00:39:01.910 --> 00:39:03.470 X to the two plus two. 00:39:04.030 --> 00:39:05.930 X to the fourth. 00:39:09.790 --> 00:39:11.390 minus 12. 00:39:13.510 --> 00:39:22.650 x squared times x is x to the 2 plus 1, which is x cubed. 00:39:25.270 --> 00:39:26.270 15x squared. 00:39:30.530 --> 00:39:32.630 It's just the rules of exponents. 00:39:34.170 --> 00:39:35.810 And we end up with this. 00:39:38.830 --> 00:39:40.270 What is the degree here? 00:39:41.570 --> 00:39:42.010 Four. 00:39:42.010 --> 00:39:42.910 Four. 00:39:43.510 --> 00:39:44.710 How many terms? 00:39:45.390 --> 00:39:46.010 Three. 00:39:46.010 --> 00:39:46.910 Do we call this a? 00:39:48.090 --> 00:39:49.510 Four degrees. 00:39:49.510 --> 00:39:51.270 He's trying to rattle me. 00:40:11.750 --> 00:40:13.570 Go ahead and distribute. 00:40:18.990 --> 00:40:20.390 you. 00:41:14.430 --> 00:41:19.250 42x to the fourth. 00:41:19.470 --> 00:41:23.390 What about that? 00:41:25.490 --> 00:41:27.510 Did we all get that? Yes, sir. Yeah. 00:41:28.110 --> 00:41:31.150 Multiplying is all about just using the distributive property. 00:41:32.390 --> 00:41:34.670 Because polynomials have multiple terms. 00:41:36.890 --> 00:41:40.030 Now, so that's multiplying monomials 00:41:40.030 --> 00:41:46.510 times any polynomial you just distribute the monomial to every term and then 00:41:46.510 --> 00:42:05.610 simplify now let's talk about a binomial times a binomial is a linear binomial 00:42:05.850 --> 00:42:10.050 The linear binomial what property we have to use 00:42:14.130 --> 00:42:17.850 The distributive property of multiplication over 00:42:18.370 --> 00:42:19.270 addition 00:42:19.450 --> 00:42:23.650 Just like we use radicals. We have to distribute the whole 00:42:24.930 --> 00:42:29.210 Binomial the whole polynomial to every term in the other 00:42:32.290 --> 00:42:34.750 We get 2x plus 3 00:42:34.750 --> 00:42:41.750 times x plus 2x plus 3 times negative 5. 00:42:50.410 --> 00:42:52.450 And now what do we have to do? 00:42:53.590 --> 00:42:56.350 Use the distributive property two more times. 00:42:59.670 --> 00:43:01.430 Just like we did with radicals. 00:43:28.070 --> 00:43:32.130 And then the last step. 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 00:43:40.628 --> 00:43:41.690 squared 3 00:43:41.690 --> 00:44:02.870 minus 10 minus 7x. Okay is there a shortcut where we could just bypass this 00:44:02.870 --> 00:44:07.870 whole step. Why don't you talk to your neighbor, what's a shortcut? 00:44:07.870 --> 00:44:09.690 it. 00:44:25.550 --> 00:44:31.510 Uh, anybody got a shortcut? 00:44:37.490 --> 00:44:38.390 No shortcuts? 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 00:44:44.962 --> 00:44:45.712 another one. 00:45:01.510 --> 00:45:02.410 I don't know. 00:45:02.450 --> 00:45:03.350 Can I spell it? 00:45:03.790 --> 00:45:04.690 Go. 00:45:05.490 --> 00:45:06.390 They can't. 00:45:06.430 --> 00:45:07.310 They can't. 00:45:07.310 --> 00:45:08.210 They can't. 00:45:08.270 --> 00:45:08.950 They can't. 00:45:08.950 --> 00:45:11.030 They can't. 00:45:11.030 --> 00:45:12.290 So what do we do? 00:45:15.430 --> 00:45:16.330 We distribute. 00:45:18.470 --> 00:45:19.370 OK. 00:45:19.970 --> 00:45:22.430 Does it matter if I distribute this here 00:45:22.430 --> 00:45:26.350 or if I distribute this here? 00:45:26.950 --> 00:45:27.850 It doesn't matter. 00:45:27.890 --> 00:45:28.950 It works out at the end the same. 00:45:32.010 --> 00:45:35.230 Let's just do that. I always go left to right. 00:45:57.870 --> 00:46:02.070 So we distribute the 4x cubed minus 3x squared. 00:46:03.330 --> 00:46:04.650 and now what do we gotta do? 00:46:06.350 --> 00:46:07.270 distribute again 00:46:10.370 --> 00:46:12.220 I wish there was an easier way 00:46:12.220 --> 00:46:14.230 In what order do we write our final answering? 00:46:14.850 --> 00:46:19.719 What's it called descending descending the term with the largest degree down to the 00:46:19.719 --> 00:46:21.110 term of a degree? 00:46:22.370 --> 00:46:23.990 So let's go back 00:46:25.950 --> 00:46:32.350 Is there a shortcut where we can skip this entire step, yes, what's the shortcut? 00:46:32.350 --> 00:46:34.250 aluminum foil method 00:46:34.250 --> 00:46:35.150 and 00:46:35.350 --> 00:46:36.310 well, five 00:46:37.210 --> 00:46:40.170 why are you guys talking about aluminum foil? 00:46:41.450 --> 00:46:43.310 I think like you multiply the front 00:46:43.310 --> 00:46:44.930 two terms together 00:46:44.930 --> 00:46:47.430 then you multiply the outside terms 00:46:47.430 --> 00:46:50.310 and then you multiply the inside terms 00:46:50.310 --> 00:46:52.150 and the last terms 00:46:52.150 --> 00:46:55.250 okay, I like that 00:46:55.250 --> 00:46:58.690 except what happens when we have 00:47:06.630 --> 00:47:09.130 Two terms times three terms. 00:47:19.010 --> 00:47:24.610 Okay so let's do a shortcut without any weird words like foil. 00:47:31.070 --> 00:47:34.170 Let's look at this. Where did these two terms come from? 00:47:39.610 --> 00:47:49.330 Was this times this. Okay. Which let me erase this. 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 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 00:48:18.410 --> 00:48:28.610 squared and the 5x there okay so we can skip this entire step by just taking 00:48:28.610 --> 00:48:38.370 each term and distribute it to the other terms in the other polynomial and then 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 00:48:43.070 --> 00:48:49.210 it's just a bunch of distributing we have a binomial times a binomial some 00:48:49.210 --> 00:48:57.610 people like the acronym foil okay first outside inside last but I don't mess with 00:48:57.610 --> 00:49:03.710 that because once you get two times three terms a binomial times a trinomial 00:49:04.750 --> 00:49:13.050 you just take the first term just distribute it to every term and then 00:49:13.050 --> 00:49:15.710 And you go to the next term, distribute to the next term. 00:49:21.370 --> 00:49:33.930 We got 2x times x squared plus 2x times 4x plus 2x times negative 5. 00:49:36.650 --> 00:49:37.770 Those should all be in purple. 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 00:50:01.476 --> 00:50:02.530 of 00:50:04.640 --> 00:50:13.420 3 times x squared plus 3 times 4x plus 3 times maybe 5. 00:50:14.060 --> 00:50:17.760 Eventually we even do this in our head and we start writing all 6 terms. 00:50:17.980 --> 00:50:22.400 So 2 terms times 3 terms, 2 times 3, we always get that number of terms. 00:50:22.400 --> 00:50:35.760 2x cubed plus 8x squared minus 10x 3x squared plus 12x minus 15. 00:50:37.080 --> 00:50:41.560 And once we do that, we just combine like terms. 00:50:43.860 --> 00:50:51.500 This guy likes this term 11x squared. 00:50:53.120 --> 00:50:54.680 That's two. 00:50:57.620 --> 00:50:58.520 Minus 15. 00:51:02.250 --> 00:51:04.000 I think it's a cubic polynomial. 00:51:07.660 --> 00:51:10.100 Cubic is correct, the degree is three, 00:51:11.100 --> 00:51:13.540 but the number of terms is four. 00:51:13.780 --> 00:51:17.460 So we spell it cubic polynomial with four terms. 00:51:19.380 --> 00:51:20.280 Okay. 00:51:22.040 --> 00:51:23.840 Try another one. 00:51:40.060 --> 00:51:42.200 Just take each term in any one of them. 00:51:42.520 --> 00:51:43.460 I always go left right. 00:51:44.820 --> 00:51:48.500 We distribute to every term in the other polynomial. 00:51:55.700 --> 00:52:00.880 Start distributing and then simplifying and then combining like turns. 00:52:43.560 --> 00:52:47.375 A trinomial times a trinomial will have to redistribute how many terms you're going 00:52:47.375 --> 00:52:48.120 to have. 00:52:48.120 --> 00:52:49.440 All I have is nine. 00:52:50.200 --> 00:52:50.580 Nine. 00:52:50.580 --> 00:52:51.940 Three times three is nine. 00:52:52.820 --> 00:52:54.620 So write out all nine terms. 00:52:55.860 --> 00:52:57.060 And then start simplifying. 00:53:23.400 --> 00:53:26.180 Once we get good at this, you can just multiply in your head. 00:53:27.720 --> 00:53:30.340 Expert times expert. 00:54:24.560 --> 00:54:25.360 And then... 00:54:25.360 --> 00:54:26.800 I'm always like... 00:54:26.800 --> 00:54:28.920 Still being wrong? 00:54:31.780 --> 00:54:32.680 Minus 7x... 00:54:33.020 --> 00:54:34.540 Plus 20, there we go. 00:54:34.940 --> 00:54:35.840 Oh my goodness. 00:54:36.700 --> 00:54:37.600 Wait, you didn't like it. 00:54:37.800 --> 00:54:39.160 You told me it was wrong. 00:54:41.180 --> 00:54:42.220 I said his was wrong. 00:54:42.860 --> 00:54:44.200 Why are you looking at the statement from the guy? 00:54:44.200 --> 00:54:45.500 Are you telling me to do it, bro? 00:54:46.880 --> 00:54:47.780 Wait a minute. 00:54:49.700 --> 00:54:51.960 Okay, what questions do we have on this? 00:54:53.140 --> 00:54:55.780 It's just a lot of distributing 00:54:56.500 --> 00:54:58.840 using the distributive property of addition, 00:55:00.100 --> 00:55:01.200 of multiplication over addition. 00:55:01.800 --> 00:55:02.840 Okay, we're gonna stop here. 00:55:03.400 --> 00:55:04.250 Yes.