[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.24,0:00:01.06,Default,,0000,0000,0000,,- [Voiceover] Pause this video
Dialogue: 0,0:00:01.06,0:00:03.10,Default,,0000,0000,0000,,and see if you can subtract this magenta
Dialogue: 0,0:00:03.10,0:00:06.50,Default,,0000,0000,0000,,rational expression from this yellow one.
Dialogue: 0,0:00:06.50,0:00:08.41,Default,,0000,0000,0000,,Alright, now let's do this together.
Dialogue: 0,0:00:08.41,0:00:09.93,Default,,0000,0000,0000,,And the first thing that jumps out at you
Dialogue: 0,0:00:09.93,0:00:11.10,Default,,0000,0000,0000,,is that you realize these don't
Dialogue: 0,0:00:11.10,0:00:12.56,Default,,0000,0000,0000,,have the same denominator and you would
Dialogue: 0,0:00:12.56,0:00:14.12,Default,,0000,0000,0000,,like them to have the same denominator.
Dialogue: 0,0:00:14.12,0:00:16.15,Default,,0000,0000,0000,,And so you might say,\Nwell, let me rewrite them
Dialogue: 0,0:00:16.15,0:00:18.58,Default,,0000,0000,0000,,so that they have a common denominator.
Dialogue: 0,0:00:18.58,0:00:20.71,Default,,0000,0000,0000,,And a common denominator that will work
Dialogue: 0,0:00:20.71,0:00:25.71,Default,,0000,0000,0000,,will be one that is divisible\Nby each of these denominators.
Dialogue: 0,0:00:26.14,0:00:28.16,Default,,0000,0000,0000,,So it has all the factors of each
Dialogue: 0,0:00:28.16,0:00:29.80,Default,,0000,0000,0000,,of these denominators and lucky for us,
Dialogue: 0,0:00:29.80,0:00:32.26,Default,,0000,0000,0000,,each of these denominators\Nare already factored.
Dialogue: 0,0:00:32.26,0:00:34.24,Default,,0000,0000,0000,,So let me just write\Nthe common denominator,
Dialogue: 0,0:00:34.24,0:00:37.46,Default,,0000,0000,0000,,I'll start rewriting\Nthe yellow expression.
Dialogue: 0,0:00:37.46,0:00:39.24,Default,,0000,0000,0000,,So, you have the yellow expression,
Dialogue: 0,0:00:39.24,0:00:40.20,Default,,0000,0000,0000,,actually, let me just make it clear,
Dialogue: 0,0:00:40.20,0:00:42.19,Default,,0000,0000,0000,,I'm going to write both, the yellow one,
Dialogue: 0,0:00:42.19,0:00:45.60,Default,,0000,0000,0000,,and then you're going to\Nsubtract the magenta one.
Dialogue: 0,0:00:45.60,0:00:49.74,Default,,0000,0000,0000,,Whoops. I'm saying yellow\Nbut drawing in magenta.
Dialogue: 0,0:00:49.74,0:00:52.90,Default,,0000,0000,0000,,So you have the yellow expression
Dialogue: 0,0:00:52.90,0:00:54.20,Default,,0000,0000,0000,,which I'm about to rewrite, actually,
Dialogue: 0,0:00:54.20,0:00:55.26,Default,,0000,0000,0000,,I'm going to make a longer line,
Dialogue: 0,0:00:55.26,0:01:00.26,Default,,0000,0000,0000,,so the yellow expression\Nminus the magenta one,
Dialogue: 0,0:01:02.39,0:01:06.01,Default,,0000,0000,0000,,minus the magenta one, right over there.
Dialogue: 0,0:01:06.01,0:01:07.67,Default,,0000,0000,0000,,Now, as I mentioned, we\Nwant to have a denominator
Dialogue: 0,0:01:07.67,0:01:09.72,Default,,0000,0000,0000,,that has all, the common\Ndenominator has to have,
Dialogue: 0,0:01:09.72,0:01:12.54,Default,,0000,0000,0000,,be divisible by both,\Nthis yellow denominator
Dialogue: 0,0:01:12.54,0:01:14.31,Default,,0000,0000,0000,,and this magenta one.
Dialogue: 0,0:01:14.31,0:01:19.27,Default,,0000,0000,0000,,So it's got to have\Nthe Z plus eight in it.
Dialogue: 0,0:01:19.27,0:01:23.64,Default,,0000,0000,0000,,It's got to have the 9z minus five in it.
Dialogue: 0,0:01:23.64,0:01:25.91,Default,,0000,0000,0000,,And it's also got to have both of these.
Dialogue: 0,0:01:25.91,0:01:28.81,Default,,0000,0000,0000,,Well, I already, we already\Naccounted for the 9z minus five.
Dialogue: 0,0:01:28.81,0:01:33.81,Default,,0000,0000,0000,,So it has to have, be divisible\Nby Z plus six. Z plus six.
Dialogue: 0,0:01:35.23,0:01:36.93,Default,,0000,0000,0000,,Notice just by multiplying the denominator
Dialogue: 0,0:01:36.93,0:01:40.39,Default,,0000,0000,0000,,by Z plus six, we're not\Ndivisible by both of these factors
Dialogue: 0,0:01:40.39,0:01:43.93,Default,,0000,0000,0000,,AND both of these factors\Nbecause 9z minus five
Dialogue: 0,0:01:43.93,0:01:46.83,Default,,0000,0000,0000,,was the factor common to both of them.
Dialogue: 0,0:01:46.83,0:01:47.89,Default,,0000,0000,0000,,And if you were just dealing with numbers
Dialogue: 0,0:01:47.89,0:01:49.63,Default,,0000,0000,0000,,when you were just adding\Nor subtracting fractions,
Dialogue: 0,0:01:49.63,0:01:52.53,Default,,0000,0000,0000,,it works the exact same way.
Dialogue: 0,0:01:52.53,0:01:54.67,Default,,0000,0000,0000,,Alright, so what will\Nthe numerator become?
Dialogue: 0,0:01:54.67,0:01:58.31,Default,,0000,0000,0000,,Well, we multiply the denominator times
Dialogue: 0,0:01:58.31,0:01:59.73,Default,,0000,0000,0000,,Z plus six, so we have to do the
Dialogue: 0,0:01:59.73,0:02:01.47,Default,,0000,0000,0000,,same thing to the numerator.
Dialogue: 0,0:02:01.47,0:02:05.67,Default,,0000,0000,0000,,It's going to be negative Z\Nto the third times Z plus six.
Dialogue: 0,0:02:05.67,0:02:07.44,Default,,0000,0000,0000,,Now let's focus over here.
Dialogue: 0,0:02:07.44,0:02:09.93,Default,,0000,0000,0000,,We had, well, we want\Nthe same denominator,
Dialogue: 0,0:02:09.93,0:02:13.39,Default,,0000,0000,0000,,so we can write this as Z plus eight
Dialogue: 0,0:02:13.39,0:02:18.39,Default,,0000,0000,0000,,Z plus eight times Z plus six,
Dialogue: 0,0:02:19.50,0:02:24.50,Default,,0000,0000,0000,,times Z plus six times 9z minus five.
Dialogue: 0,0:02:29.94,0:02:31.51,Default,,0000,0000,0000,,And these are equivalent.
Dialogue: 0,0:02:31.51,0:02:33.37,Default,,0000,0000,0000,,I've just changed the order\Nthat we multiply in it,
Dialogue: 0,0:02:33.37,0:02:35.08,Default,,0000,0000,0000,,that doesn't change their value.
Dialogue: 0,0:02:35.08,0:02:39.64,Default,,0000,0000,0000,,And if we multiplied the, so\Nwe had a three on top before
Dialogue: 0,0:02:39.64,0:02:42.14,Default,,0000,0000,0000,,and if we multiply the\Ndenominator times Z plus eight,
Dialogue: 0,0:02:42.14,0:02:47.14,Default,,0000,0000,0000,,we also have to multiply the\Nnumerator times Z plus eight.
Dialogue: 0,0:02:48.18,0:02:49.36,Default,,0000,0000,0000,,So there you go.
Dialogue: 0,0:02:49.36,0:02:52.05,Default,,0000,0000,0000,,And so, this is going to be equal to,
Dialogue: 0,0:02:52.05,0:02:54.29,Default,,0000,0000,0000,,this is going to be equal\Nto, actually, I'll just make
Dialogue: 0,0:02:54.29,0:02:58.19,Default,,0000,0000,0000,,a big line right over here.
Dialogue: 0,0:02:58.19,0:03:00.54,Default,,0000,0000,0000,,This is all going to be equal to.
Dialogue: 0,0:03:00.54,0:03:02.95,Default,,0000,0000,0000,,We have our, probably\Ndon't need that much space,
Dialogue: 0,0:03:02.95,0:03:06.05,Default,,0000,0000,0000,,let me see, maybe that,\Nmaybe about that much.
Dialogue: 0,0:03:06.05,0:03:08.10,Default,,0000,0000,0000,,So I'm going to have the same denominator
Dialogue: 0,0:03:08.10,0:03:10.34,Default,,0000,0000,0000,,and I'll just write it\Nin a neutral color now.
Dialogue: 0,0:03:10.34,0:03:15.34,Default,,0000,0000,0000,,Z plus eight times 9z minus\Nfive times Z plus six.
Dialogue: 0,0:03:20.04,0:03:23.50,Default,,0000,0000,0000,,So over here, just in this blue color,
Dialogue: 0,0:03:23.50,0:03:26.34,Default,,0000,0000,0000,,we want to distribute this\Nnegative Z to the third.
Dialogue: 0,0:03:26.34,0:03:30.92,Default,,0000,0000,0000,,Negative Z to the third times\NZ is negative Z to the fourth.
Dialogue: 0,0:03:30.92,0:03:35.92,Default,,0000,0000,0000,,Negative Z to the third times\Nsix is minus 6z to the third.
Dialogue: 0,0:03:37.24,0:03:39.65,Default,,0000,0000,0000,,And now this negative\Nsign, right over here,
Dialogue: 0,0:03:39.65,0:03:42.94,Default,,0000,0000,0000,,actually, instead of saying negative Z,
Dialogue: 0,0:03:42.94,0:03:44.41,Default,,0000,0000,0000,,negative of this entire thing,
Dialogue: 0,0:03:44.41,0:03:48.31,Default,,0000,0000,0000,,we could just say plus\Nthe negative of this.
Dialogue: 0,0:03:48.31,0:03:49.65,Default,,0000,0000,0000,,Or another of thinking about it,
Dialogue: 0,0:03:49.65,0:03:54.13,Default,,0000,0000,0000,,you could view this as negative\Nthree times Z plus eight.
Dialogue: 0,0:03:54.13,0:03:55.86,Default,,0000,0000,0000,,So we could just distribute that.
Dialogue: 0,0:03:55.86,0:03:57.30,Default,,0000,0000,0000,,So let's do that.
Dialogue: 0,0:03:57.30,0:04:02.30,Default,,0000,0000,0000,,So negative three times Z is negative 3z
Dialogue: 0,0:04:02.98,0:04:07.98,Default,,0000,0000,0000,,and negative three times\Neight is negative 24.
Dialogue: 0,0:04:08.04,0:04:09.12,Default,,0000,0000,0000,,And there you go.
Dialogue: 0,0:04:09.12,0:04:11.75,Default,,0000,0000,0000,,We are, we are done.
Dialogue: 0,0:04:11.75,0:04:13.30,Default,,0000,0000,0000,,We found a common denominator.
Dialogue: 0,0:04:13.30,0:04:14.35,Default,,0000,0000,0000,,And once you have a common denominator,
Dialogue: 0,0:04:14.35,0:04:17.06,Default,,0000,0000,0000,,you could just subtract\Nor add the numerators,
Dialogue: 0,0:04:17.06,0:04:20.39,Default,,0000,0000,0000,,and instead of doing this\Nas minus this entire thing,
Dialogue: 0,0:04:20.39,0:04:23.19,Default,,0000,0000,0000,,I viewed it as adding and then having
Dialogue: 0,0:04:23.19,0:04:24.89,Default,,0000,0000,0000,,a negative three in the numerator,
Dialogue: 0,0:04:24.89,0:04:26.72,Default,,0000,0000,0000,,distributing that and then these,
Dialogue: 0,0:04:26.72,0:04:28.02,Default,,0000,0000,0000,,I can't simplify it any further.
Dialogue: 0,0:04:28.02,0:04:29.23,Default,,0000,0000,0000,,Sometimes you'll do one of these types
Dialogue: 0,0:04:29.23,0:04:30.47,Default,,0000,0000,0000,,of exercises and you might have
Dialogue: 0,0:04:30.47,0:04:33.93,Default,,0000,0000,0000,,two second-degree terms\Nor two first-degree terms
Dialogue: 0,0:04:33.93,0:04:35.60,Default,,0000,0000,0000,,or two constants or something like that
Dialogue: 0,0:04:35.60,0:04:37.12,Default,,0000,0000,0000,,and then you might want\Nto add or subtract them
Dialogue: 0,0:04:37.12,0:04:40.13,Default,,0000,0000,0000,,to simplify it but here, these\Nall have different degrees
Dialogue: 0,0:04:40.13,0:04:45.13,Default,,0000,0000,0000,,so I can't simplify it any\Nfurther and so we are all done.