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