[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.90,0:00:04.54,Default,,0000,0000,0000,,Find the greatest common factor of these monomials.
Dialogue: 0,0:00:04.54,0:00:07.02,Default,,0000,0000,0000,,Now the greatest common factor of anything
Dialogue: 0,0:00:07.02,0:00:11.76,Default,,0000,0000,0000,,is the largest factor that's divisible into both --
Dialogue: 0,0:00:11.76,0:00:13.54,Default,,0000,0000,0000,,if we're just talking about pure numbers:
Dialogue: 0,0:00:13.54,0:00:14.94,Default,,0000,0000,0000,,into both numbers,
Dialogue: 0,0:00:14.94,0:00:16.64,Default,,0000,0000,0000,,or in this case into both monomials.
Dialogue: 0,0:00:16.64,0:00:18.42,Default,,0000,0000,0000,,Now we have to be a little bit careful
Dialogue: 0,0:00:18.42,0:00:20.21,Default,,0000,0000,0000,,when we talk about 'greatest'
Dialogue: 0,0:00:20.21,0:00:22.94,Default,,0000,0000,0000,,in the context of algebraic expressions like this
Dialogue: 0,0:00:22.94,0:00:25.02,Default,,0000,0000,0000,,because it's 'greatest' from the point of view that
Dialogue: 0,0:00:25.02,0:00:27.20,Default,,0000,0000,0000,,it includes the most factors
Dialogue: 0,0:00:27.20,0:00:29.87,Default,,0000,0000,0000,,for each of these monomials,
Dialogue: 0,0:00:29.87,0:00:32.68,Default,,0000,0000,0000,,it's not necessarily the greatest possible number
Dialogue: 0,0:00:32.68,0:00:35.20,Default,,0000,0000,0000,,because maybe some of these variables
Dialogue: 0,0:00:35.20,0:00:36.76,Default,,0000,0000,0000,,can take on negative values;
Dialogue: 0,0:00:36.76,0:00:38.84,Default,,0000,0000,0000,,maybe they are taking on values less than one
Dialogue: 0,0:00:38.84,0:00:41.27,Default,,0000,0000,0000,,so if squared they actually become a smaller number
Dialogue: 0,0:00:41.27,0:00:42.54,Default,,0000,0000,0000,,but I think,
Dialogue: 0,0:00:42.54,0:00:44.27,Default,,0000,0000,0000,,without getting too much into the weeds there,
Dialogue: 0,0:00:44.27,0:00:46.62,Default,,0000,0000,0000,,I think if we just kind of run through the process of it
Dialogue: 0,0:00:46.62,0:00:48.52,Default,,0000,0000,0000,,you'll understand it a little bit better.
Dialogue: 0,0:00:48.52,0:00:50.42,Default,,0000,0000,0000,,So to find the greatest common factor.
Dialogue: 0,0:00:50.42,0:00:51.84,Default,,0000,0000,0000,,Let's just essentially break down
Dialogue: 0,0:00:51.84,0:00:53.62,Default,,0000,0000,0000,,each of these numbers into
Dialogue: 0,0:00:53.62,0:00:55.88,Default,,0000,0000,0000,,what we could call their prime factorization
Dialogue: 0,0:00:55.88,0:00:57.18,Default,,0000,0000,0000,,but it's kind of a combination
Dialogue: 0,0:00:57.18,0:00:58.36,Default,,0000,0000,0000,,of the prime factorization
Dialogue: 0,0:00:58.36,0:01:00.01,Default,,0000,0000,0000,,of the numeric parts of the number
Dialogue: 0,0:01:00.01,0:01:02.84,Default,,0000,0000,0000,,plus essentially the factorization of the variable part.
Dialogue: 0,0:01:02.84,0:01:04.61,Default,,0000,0000,0000,,So if we wanted to write 10,
Dialogue: 0,0:01:04.61,0:01:08.14,Default,,0000,0000,0000,,or if we wanted to write 10cd^2
Dialogue: 0,0:01:08.14,0:01:09.54,Default,,0000,0000,0000,,we can rewrite that
Dialogue: 0,0:01:09.54,0:01:12.02,Default,,0000,0000,0000,,as the product of the prime factors of 10
Dialogue: 0,0:01:12.02,0:01:14.94,Default,,0000,0000,0000,,- which is just 2 * 5 -
Dialogue: 0,0:01:14.94,0:01:16.67,Default,,0000,0000,0000,,those are both prime numbers
Dialogue: 0,0:01:16.67,0:01:18.47,Default,,0000,0000,0000,,So 10 can be broken down
Dialogue: 0,0:01:18.47,0:01:20.27,Default,,0000,0000,0000,,as 2 times 5.
Dialogue: 0,0:01:20.27,0:01:22.87,Default,,0000,0000,0000,,C can only be broken down by c.
Dialogue: 0,0:01:22.87,0:01:24.08,Default,,0000,0000,0000,,We don't know anything else
Dialogue: 0,0:01:24.08,0:01:26.20,Default,,0000,0000,0000,,that c can be broken into.
Dialogue: 0,0:01:26.20,0:01:28.84,Default,,0000,0000,0000,,So 2 times 5 times c
Dialogue: 0,0:01:28.84,0:01:31.27,Default,,0000,0000,0000,,But then the d^2 can be rewritten
Dialogue: 0,0:01:31.27,0:01:34.45,Default,,0000,0000,0000,,as d times d.
Dialogue: 0,0:01:34.66,0:01:36.21,Default,,0000,0000,0000,,This is what I mean
Dialogue: 0,0:01:36.21,0:01:38.06,Default,,0000,0000,0000,,by writing this monomial
Dialogue: 0,0:01:38.06,0:01:41.13,Default,,0000,0000,0000,,as the product of its constituants.
Dialogue: 0,0:01:41.13,0:01:42.95,Default,,0000,0000,0000,,For the numeric part of it,
Dialogue: 0,0:01:42.95,0:01:45.13,Default,,0000,0000,0000,,it's the constituants of the prime factors
Dialogue: 0,0:01:45.13,0:01:46.66,Default,,0000,0000,0000,,and for the rest of it
Dialogue: 0,0:01:46.66,0:01:48.79,Default,,0000,0000,0000,,we are just expanding out the exponents.
Dialogue: 0,0:01:48.79,0:01:50.05,Default,,0000,0000,0000,,Now, let's do this for
Dialogue: 0,0:01:50.05,0:01:52.65,Default,,0000,0000,0000,,25 c to the third, d squared.
Dialogue: 0,0:01:52.65,0:01:55.28,Default,,0000,0000,0000,,So 25, that is 5 * 5.
Dialogue: 0,0:01:55.28,0:01:58.39,Default,,0000,0000,0000,,So this is equal to 5 * 5.
Dialogue: 0,0:01:58.39,0:02:00.53,Default,,0000,0000,0000,,And then c^3, that is
Dialogue: 0,0:02:00.53,0:02:04.39,Default,,0000,0000,0000,,c times c times c.
Dialogue: 0,0:02:04.39,0:02:06.79,Default,,0000,0000,0000,,And then d-squared,
Dialogue: 0,0:02:06.79,0:02:11.44,Default,,0000,0000,0000,,that is d times d.
Dialogue: 0,0:02:11.44,0:02:13.79,Default,,0000,0000,0000,,So what is their greatest common factor
Dialogue: 0,0:02:13.79,0:02:15.88,Default,,0000,0000,0000,,in this context?
Dialogue: 0,0:02:15.88,0:02:21.12,Default,,0000,0000,0000,,Well, they both have at least one 5,
Dialogue: 0,0:02:21.12,0:02:26.40,Default,,0000,0000,0000,,and they both have at least one c
Dialogue: 0,0:02:26.40,0:02:31.63,Default,,0000,0000,0000,,and then they both have two d's.
Dialogue: 0,0:02:31.63,0:02:34.79,Default,,0000,0000,0000,,So the greatest common factor in this context
Dialogue: 0,0:02:34.79,0:02:36.12,Default,,0000,0000,0000,,the greatest common factor
Dialogue: 0,0:02:36.12,0:02:37.59,Default,,0000,0000,0000,,of these two monomials,
Dialogue: 0,0:02:37.59,0:02:39.79,Default,,0000,0000,0000,,will be the factors that they have in common.
Dialogue: 0,0:02:39.79,0:02:41.28,Default,,0000,0000,0000,,It will be equal to
Dialogue: 0,0:02:41.28,0:02:43.62,Default,,0000,0000,0000,,this five, times
Dialogue: 0,0:02:43.62,0:02:45.48,Default,,0000,0000,0000,,we only have one c in common,
Dialogue: 0,0:02:45.48,0:02:48.37,Default,,0000,0000,0000,,times - we have two d's in common.
Dialogue: 0,0:02:48.37,0:02:50.12,Default,,0000,0000,0000,,So this is equal to
Dialogue: 0,0:02:50.12,0:02:53.88,Default,,0000,0000,0000,,5 times c times d-squared.
Dialogue: 0,0:02:53.88,0:02:55.94,Default,,0000,0000,0000,,So 5 c d-squared
Dialogue: 0,0:02:55.94,0:02:57.39,Default,,0000,0000,0000,,we can view as the greatest --
Dialogue: 0,0:02:57.39,0:02:58.92,Default,,0000,0000,0000,,I'll put that in quotes,
Dialogue: 0,0:02:58.92,0:03:00.28,Default,,0000,0000,0000,,you know, depending on whether c
Dialogue: 0,0:03:00.28,0:03:01.40,Default,,0000,0000,0000,,is negative or positive -
Dialogue: 0,0:03:01.40,0:03:02.70,Default,,0000,0000,0000,,and d is greater than
Dialogue: 0,0:03:02.70,0:03:03.100,Default,,0000,0000,0000,,or less than zero.
Dialogue: 0,0:03:03.100,0:03:05.62,Default,,0000,0000,0000,,But this is the "greatest" common factor
Dialogue: 0,0:03:05.62,0:03:07.22,Default,,0000,0000,0000,,of these two monomials.
Dialogue: 0,0:03:07.22,0:03:09.06,Default,,0000,0000,0000,,It's devisable into both of them
Dialogue: 0,0:03:09.06,0:03:10.06,Default,,0000,0000,0000,,and it uses
Dialogue: 0,0:03:10.06,0:03:11.58,Default,,0000,0000,0000,,the most factors possible.