0:00:00.245,0:00:01.059 - [Voiceover] Pause this video 0:00:01.059,0:00:03.101 and see if you can subtract this magenta 0:00:03.101,0:00:06.502 rational expression from this yellow one. 0:00:06.502,0:00:08.406 Alright, now let's do this together. 0:00:08.406,0:00:09.928 And the first thing that jumps out at you 0:00:09.928,0:00:11.101 is that you realize these don't 0:00:11.101,0:00:12.563 have the same denominator and you would 0:00:12.563,0:00:14.118 like them to have the same denominator. 0:00:14.118,0:00:16.148 And so you might say,[br]well, let me rewrite them 0:00:16.148,0:00:18.584 so that they have a common denominator. 0:00:18.584,0:00:20.710 And a common denominator that will work 0:00:20.710,0:00:25.710 will be one that is divisible[br]by each of these denominators. 0:00:26.142,0:00:28.163 So it has all the factors of each 0:00:28.163,0:00:29.799 of these denominators and lucky for us, 0:00:29.799,0:00:32.260 each of these denominators[br]are already factored. 0:00:32.260,0:00:34.245 So let me just write[br]the common denominator, 0:00:34.245,0:00:37.462 I'll start rewriting[br]the yellow expression. 0:00:37.462,0:00:39.237 So, you have the yellow expression, 0:00:39.237,0:00:40.200 actually, let me just make it clear, 0:00:40.200,0:00:42.186 I'm going to write both, the yellow one, 0:00:42.186,0:00:45.600 and then you're going to[br]subtract the magenta one. 0:00:45.600,0:00:49.743 Whoops. I'm saying yellow[br]but drawing in magenta. 0:00:49.743,0:00:52.901 So you have the yellow expression 0:00:52.901,0:00:54.202 which I'm about to rewrite, actually, 0:00:54.202,0:00:55.263 I'm going to make a longer line, 0:00:55.263,0:01:00.263 so the yellow expression[br]minus the magenta one, 0:01:02.391,0:01:06.013 minus the magenta one, right over there. 0:01:06.013,0:01:07.673 Now, as I mentioned, we[br]want to have a denominator 0:01:07.673,0:01:09.716 that has all, the common[br]denominator has to have, 0:01:09.716,0:01:12.537 be divisible by both,[br]this yellow denominator 0:01:12.537,0:01:14.313 and this magenta one. 0:01:14.313,0:01:19.270 So it's got to have[br]the Z plus eight in it. 0:01:19.270,0:01:23.635 It's got to have the 9z minus five in it. 0:01:23.635,0:01:25.912 And it's also got to have both of these. 0:01:25.912,0:01:28.813 Well, I already, we already[br]accounted for the 9z minus five. 0:01:28.813,0:01:33.813 So it has to have, be divisible[br]by Z plus six. Z plus six. 0:01:35.234,0:01:36.928 Notice just by multiplying the denominator 0:01:36.928,0:01:40.387 by Z plus six, we're not[br]divisible by both of these factors 0:01:40.387,0:01:43.929 AND both of these factors[br]because 9z minus five 0:01:43.929,0:01:46.831 was the factor common to both of them. 0:01:46.831,0:01:47.887 And if you were just dealing with numbers 0:01:47.887,0:01:49.628 when you were just adding[br]or subtracting fractions, 0:01:49.628,0:01:52.530 it works the exact same way. 0:01:52.530,0:01:54.667 Alright, so what will[br]the numerator become? 0:01:54.667,0:01:58.312 Well, we multiply the denominator times 0:01:58.312,0:01:59.729 Z plus six, so we have to do the 0:01:59.729,0:02:01.470 same thing to the numerator. 0:02:01.470,0:02:05.672 It's going to be negative Z[br]to the third times Z plus six. 0:02:05.672,0:02:07.436 Now let's focus over here. 0:02:07.436,0:02:09.932 We had, well, we want[br]the same denominator, 0:02:09.932,0:02:13.392 so we can write this as Z plus eight 0:02:13.392,0:02:18.392 Z plus eight times Z plus six, 0:02:19.499,0:02:24.499 times Z plus six times 9z minus five. 0:02:29.935,0:02:31.514 And these are equivalent. 0:02:31.514,0:02:33.371 I've just changed the order[br]that we multiply in it, 0:02:33.371,0:02:35.077 that doesn't change their value. 0:02:35.077,0:02:39.639 And if we multiplied the, so[br]we had a three on top before 0:02:39.639,0:02:42.135 and if we multiply the[br]denominator times Z plus eight, 0:02:42.135,0:02:47.135 we also have to multiply the[br]numerator times Z plus eight. 0:02:48.181,0:02:49.364 So there you go. 0:02:49.364,0:02:52.047 And so, this is going to be equal to, 0:02:52.047,0:02:54.287 this is going to be equal[br]to, actually, I'll just make 0:02:54.287,0:02:58.188 a big line right over here. 0:02:58.188,0:03:00.545 This is all going to be equal to. 0:03:00.545,0:03:02.947 We have our, probably[br]don't need that much space, 0:03:02.947,0:03:06.046 let me see, maybe that,[br]maybe about that much. 0:03:06.046,0:03:08.102 So I'm going to have the same denominator 0:03:08.102,0:03:10.343 and I'll just write it[br]in a neutral color now. 0:03:10.343,0:03:15.343 Z plus eight times 9z minus[br]five times Z plus six. 0:03:20.035,0:03:23.496 So over here, just in this blue color, 0:03:23.496,0:03:26.339 we want to distribute this[br]negative Z to the third. 0:03:26.339,0:03:30.924 Negative Z to the third times[br]Z is negative Z to the fourth. 0:03:30.924,0:03:35.924 Negative Z to the third times[br]six is minus 6z to the third. 0:03:37.238,0:03:39.652 And now this negative[br]sign, right over here, 0:03:39.652,0:03:42.938 actually, instead of saying negative Z, 0:03:42.938,0:03:44.413 negative of this entire thing, 0:03:44.413,0:03:48.313 we could just say plus[br]the negative of this. 0:03:48.313,0:03:49.649 Or another of thinking about it, 0:03:49.649,0:03:54.129 you could view this as negative[br]three times Z plus eight. 0:03:54.129,0:03:55.859 So we could just distribute that. 0:03:55.859,0:03:57.298 So let's do that. 0:03:57.298,0:04:02.298 So negative three times Z is negative 3z 0:04:02.975,0:04:07.975 and negative three times[br]eight is negative 24. 0:04:08.037,0:04:09.116 And there you go. 0:04:09.116,0:04:11.752 We are, we are done. 0:04:11.752,0:04:13.295 We found a common denominator. 0:04:13.295,0:04:14.353 And once you have a common denominator, 0:04:14.353,0:04:17.056 you could just subtract[br]or add the numerators, 0:04:17.056,0:04:20.388 and instead of doing this[br]as minus this entire thing, 0:04:20.388,0:04:23.187 I viewed it as adding and then having 0:04:23.187,0:04:24.886 a negative three in the numerator, 0:04:24.886,0:04:26.721 distributing that and then these, 0:04:26.721,0:04:28.021 I can't simplify it any further. 0:04:28.021,0:04:29.227 Sometimes you'll do one of these types 0:04:29.227,0:04:30.469 of exercises and you might have 0:04:30.469,0:04:33.929 two second-degree terms[br]or two first-degree terms 0:04:33.929,0:04:35.601 or two constants or something like that 0:04:35.601,0:04:37.121 and then you might want[br]to add or subtract them 0:04:37.121,0:04:40.128 to simplify it but here, these[br]all have different degrees 0:04:40.128,0:04:45.128 so I can't simplify it any[br]further and so we are all done.