1 00:00:00,900 --> 00:00:04,540 Find the greatest common factor of these monomials. 2 00:00:04,540 --> 00:00:07,015 Now the greatest common factor of anything 3 00:00:07,015 --> 00:00:11,757 is the largest factor that's divisible into both -- 4 00:00:11,757 --> 00:00:13,540 if we're just talking about pure numbers: 5 00:00:13,540 --> 00:00:14,938 into both numbers, 6 00:00:14,938 --> 00:00:16,637 or in this case into both monomials. 7 00:00:16,637 --> 00:00:18,420 Now we have to be a little bit careful 8 00:00:18,420 --> 00:00:20,206 when we talk about 'greatest' 9 00:00:20,206 --> 00:00:22,937 in the context of algebraic expressions like this 10 00:00:22,937 --> 00:00:25,022 because it's 'greatest' from the point of view that 11 00:00:25,022 --> 00:00:27,203 it includes the most factors 12 00:00:27,203 --> 00:00:29,869 for each of these monomials, 13 00:00:29,869 --> 00:00:32,676 it's not necessarily the greatest possible number 14 00:00:32,676 --> 00:00:35,205 because maybe some of these variables 15 00:00:35,205 --> 00:00:36,756 can take on negative values; 16 00:00:36,756 --> 00:00:38,840 maybe they are taking on values less than one 17 00:00:38,840 --> 00:00:41,267 so if squared they actually become a smaller number 18 00:00:41,267 --> 00:00:42,537 but I think, 19 00:00:42,537 --> 00:00:44,272 without getting too much into the weeds there, 20 00:00:44,272 --> 00:00:46,616 I think if we just kind of run through the process of it 21 00:00:46,616 --> 00:00:48,520 you'll understand it a little bit better. 22 00:00:48,520 --> 00:00:50,420 So to find the greatest common factor. 23 00:00:50,420 --> 00:00:51,842 Let's just essentially break down 24 00:00:51,842 --> 00:00:53,619 each of these numbers into 25 00:00:53,619 --> 00:00:55,876 what we could call their prime factorization 26 00:00:55,876 --> 00:00:57,175 but it's kind of a combination 27 00:00:57,175 --> 00:00:58,365 of the prime factorization 28 00:00:58,365 --> 00:01:00,014 of the numeric parts of the number 29 00:01:00,014 --> 00:01:02,837 plus essentially the factorization of the variable part. 30 00:01:02,837 --> 00:01:04,607 So if we wanted to write 10, 31 00:01:04,607 --> 00:01:08,143 or if we wanted to write 10cd^2 32 00:01:08,143 --> 00:01:09,536 we can rewrite that 33 00:01:09,536 --> 00:01:12,020 as the product of the prime factors of 10 34 00:01:12,020 --> 00:01:14,937 - which is just 2 * 5 - 35 00:01:14,937 --> 00:01:16,666 those are both prime numbers 36 00:01:16,666 --> 00:01:18,466 So 10 can be broken down 37 00:01:18,466 --> 00:01:20,270 as 2 times 5. 38 00:01:20,270 --> 00:01:22,870 C can only be broken down by c. 39 00:01:22,870 --> 00:01:24,085 We don't know anything else 40 00:01:24,085 --> 00:01:26,200 that c can be broken into. 41 00:01:26,200 --> 00:01:28,842 So 2 times 5 times c 42 00:01:28,842 --> 00:01:31,271 But then the d^2 can be rewritten 43 00:01:31,271 --> 00:01:34,453 as d times d. 44 00:01:34,661 --> 00:01:36,213 This is what I mean 45 00:01:36,213 --> 00:01:38,055 by writing this monomial 46 00:01:38,055 --> 00:01:41,126 as the product of its constituants. 47 00:01:41,126 --> 00:01:42,949 For the numeric part of it, 48 00:01:42,949 --> 00:01:45,129 it's the constituants of the prime factors 49 00:01:45,129 --> 00:01:46,656 and for the rest of it 50 00:01:46,656 --> 00:01:48,790 we are just expanding out the exponents. 51 00:01:48,790 --> 00:01:50,054 Now, let's do this for 52 00:01:50,054 --> 00:01:52,653 25 c to the third, d squared. 53 00:01:52,653 --> 00:01:55,279 So 25, that is 5 * 5. 54 00:01:55,279 --> 00:01:58,390 So this is equal to 5 * 5. 55 00:01:58,390 --> 00:02:00,534 And then c^3, that is 56 00:02:00,534 --> 00:02:04,390 c times c times c. 57 00:02:04,390 --> 00:02:06,790 And then d-squared, 58 00:02:06,790 --> 00:02:11,444 that is d times d. 59 00:02:11,444 --> 00:02:13,793 So what is their greatest common factor 60 00:02:13,793 --> 00:02:15,876 in this context? 61 00:02:15,876 --> 00:02:21,123 Well, they both have at least one 5, 62 00:02:21,123 --> 00:02:26,395 and they both have at least one c 63 00:02:26,395 --> 00:02:31,629 and then they both have two d's. 64 00:02:31,629 --> 00:02:34,789 So the greatest common factor in this context 65 00:02:34,789 --> 00:02:36,123 the greatest common factor 66 00:02:36,123 --> 00:02:37,591 of these two monomials, 67 00:02:37,591 --> 00:02:39,789 will be the factors that they have in common. 68 00:02:39,789 --> 00:02:41,285 It will be equal to 69 00:02:41,285 --> 00:02:43,624 this five, times 70 00:02:43,624 --> 00:02:45,480 we only have one c in common, 71 00:02:45,480 --> 00:02:48,371 times - we have two d's in common. 72 00:02:48,371 --> 00:02:50,125 So this is equal to 73 00:02:50,125 --> 00:02:53,876 5 times c times d-squared. 74 00:02:53,876 --> 00:02:55,944 So 5 c d-squared 75 00:02:55,944 --> 00:02:57,394 we can view as the greatest -- 76 00:02:57,394 --> 00:02:58,925 I'll put that in quotes, 77 00:02:58,925 --> 00:03:00,279 you know, depending on whether c 78 00:03:00,279 --> 00:03:01,395 is negative or positive - 79 00:03:01,395 --> 00:03:02,701 and d is greater than 80 00:03:02,701 --> 00:03:03,998 or less than zero. 81 00:03:03,998 --> 00:03:05,623 But this is the "greatest" common factor 82 00:03:05,623 --> 00:03:07,225 of these two monomials. 83 00:03:07,225 --> 00:03:09,062 It's devisable into both of them 84 00:03:09,062 --> 00:03:10,060 and it uses 85 00:03:10,060 --> 00:03:11,583 the most factors possible.