WEBVTT 00:00:00.900 --> 00:00:04.540 Find the greatest common factor of these monomials. 00:00:04.540 --> 00:00:07.015 Now the greatest common factor of anything 00:00:07.015 --> 00:00:11.757 is the largest factor that's divisible into both -- 00:00:11.757 --> 00:00:13.540 if we're just talking about pure numbers: 00:00:13.540 --> 00:00:14.938 into both numbers, 00:00:14.938 --> 00:00:16.637 or in this case into both monomials. 00:00:16.637 --> 00:00:18.420 Now we have to be a little bit careful 00:00:18.420 --> 00:00:20.206 when we talk about 'greatest' 00:00:20.206 --> 00:00:22.937 in the context of algebraic expressions like this 00:00:22.937 --> 00:00:25.022 because it's 'greatest' from the point of view that 00:00:25.022 --> 00:00:27.203 it includes the most factors 00:00:27.203 --> 00:00:29.869 for each of these monomials, 00:00:29.869 --> 00:00:32.676 it's not necessarily the greatest possible number 00:00:32.676 --> 00:00:35.205 because maybe some of these variables 00:00:35.205 --> 00:00:36.756 can take on negative values; 00:00:36.756 --> 00:00:38.840 maybe they are taking on values less than one 00:00:38.840 --> 00:00:41.267 so if squared they actually become a smaller number 00:00:41.267 --> 00:00:42.537 but I think, 00:00:42.537 --> 00:00:44.272 without getting too much into the weeds there, 00:00:44.272 --> 00:00:46.616 I think if we just kind of run through the process of it 00:00:46.616 --> 00:00:48.520 you'll understand it a little bit better. 00:00:48.520 --> 00:00:50.420 So to find the greatest common factor. 00:00:50.420 --> 00:00:51.842 Let's just essentially break down 00:00:51.842 --> 00:00:53.619 each of these numbers into 00:00:53.619 --> 00:00:55.876 what we could call their prime factorization 00:00:55.876 --> 00:00:57.175 but it's kind of a combination 00:00:57.175 --> 00:00:58.365 of the prime factorization 00:00:58.365 --> 00:01:00.014 of the numeric parts of the number 00:01:00.014 --> 00:01:02.837 plus essentially the factorization of the variable part. 00:01:02.837 --> 00:01:04.607 So if we wanted to write 10, 00:01:04.607 --> 00:01:08.143 or if we wanted to write 10cd^2 00:01:08.143 --> 00:01:09.536 we can rewrite that 00:01:09.536 --> 00:01:12.020 as the product of the prime factors of 10 00:01:12.020 --> 00:01:14.937 - which is just 2 * 5 - 00:01:14.937 --> 00:01:16.666 those are both prime numbers 00:01:16.666 --> 00:01:18.466 So 10 can be broken down 00:01:18.466 --> 00:01:20.270 as 2 times 5. 00:01:20.270 --> 00:01:22.870 C can only be broken down by c. 00:01:22.870 --> 00:01:24.085 We don't know anything else 00:01:24.085 --> 00:01:26.200 that c can be broken into. 00:01:26.200 --> 00:01:28.842 So 2 times 5 times c 00:01:28.842 --> 00:01:31.271 But then the d^2 can be rewritten 00:01:31.271 --> 00:01:34.453 as d times d. 00:01:34.661 --> 00:01:36.213 This is what I mean 00:01:36.213 --> 00:01:38.055 by writing this monomial 00:01:38.055 --> 00:01:41.126 as the product of its constituants. 00:01:41.126 --> 00:01:42.949 For the numeric part of it, 00:01:42.949 --> 00:01:45.129 it's the constituants of the prime factors 00:01:45.129 --> 00:01:46.656 and for the rest of it 00:01:46.656 --> 00:01:48.790 we are just expanding out the exponents. 00:01:48.790 --> 00:01:50.054 Now, let's do this for 00:01:50.054 --> 00:01:52.653 25 c to the third, d squared. 00:01:52.653 --> 00:01:55.279 So 25, that is 5 * 5. 00:01:55.279 --> 00:01:58.390 So this is equal to 5 * 5. 00:01:58.390 --> 00:02:00.534 And then c^3, that is 00:02:00.534 --> 00:02:04.390 c times c times c. 00:02:04.390 --> 00:02:06.790 And then d-squared, 00:02:06.790 --> 00:02:11.444 that is d times d. 00:02:11.444 --> 00:02:13.793 So what is their greatest common factor 00:02:13.793 --> 00:02:15.876 in this context? 00:02:15.876 --> 00:02:21.123 Well, they both have at least one 5, 00:02:21.123 --> 00:02:26.395 and they both have at least one c 00:02:26.395 --> 00:02:31.629 and then they both have two d's. 00:02:31.629 --> 00:02:34.789 So the greatest common factor in this context 00:02:34.789 --> 00:02:36.123 the greatest common factor 00:02:36.123 --> 00:02:37.591 of these two monomials, 00:02:37.591 --> 00:02:39.789 will be the factors that they have in common. 00:02:39.789 --> 00:02:41.285 It will be equal to 00:02:41.285 --> 00:02:43.624 this five, times 00:02:43.624 --> 00:02:45.480 we only have one c in common, 00:02:45.480 --> 00:02:48.371 times - we have two d's in common. 00:02:48.371 --> 00:02:50.125 So this is equal to 00:02:50.125 --> 00:02:53.876 5 times c times d-squared. 00:02:53.876 --> 00:02:55.944 So 5 c d-squared 00:02:55.944 --> 00:02:57.394 we can view as the greatest -- 00:02:57.394 --> 00:02:58.925 I'll put that in quotes, 00:02:58.925 --> 00:03:00.279 you know, depending on whether c 00:03:00.279 --> 00:03:01.395 is negative or positive - 00:03:01.395 --> 00:03:02.701 and d is greater than 00:03:02.701 --> 00:03:03.998 or less than zero. 00:03:03.998 --> 00:03:05.623 But this is the "greatest" common factor 00:03:05.623 --> 00:03:07.225 of these two monomials. 00:03:07.225 --> 00:03:09.062 It's devisable into both of them 00:03:09.062 --> 00:03:10.060 and it uses 00:03:10.060 --> 00:03:11.583 the most factors possible.