[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.26,0:00:02.23,Default,,0000,0000,0000,,- [Voiceover] Pause the\Nvideo and try to add
Dialogue: 0,0:00:02.23,0:00:04.60,Default,,0000,0000,0000,,these two rational expressions.
Dialogue: 0,0:00:04.60,0:00:06.34,Default,,0000,0000,0000,,Okay, I'm assuming you've had a go at it.
Dialogue: 0,0:00:06.34,0:00:09.08,Default,,0000,0000,0000,,Now we can work through this together.
Dialogue: 0,0:00:09.08,0:00:10.78,Default,,0000,0000,0000,,So the first thing that you might have hit
Dialogue: 0,0:00:10.78,0:00:12.10,Default,,0000,0000,0000,,when you tried to do it,
Dialogue: 0,0:00:12.10,0:00:14.49,Default,,0000,0000,0000,,is you realized that they\Nhave different denominators
Dialogue: 0,0:00:14.49,0:00:15.93,Default,,0000,0000,0000,,and it's hard to add fractions
Dialogue: 0,0:00:15.93,0:00:17.77,Default,,0000,0000,0000,,when they have different denominators.
Dialogue: 0,0:00:17.77,0:00:18.70,Default,,0000,0000,0000,,You need to rewrite them
Dialogue: 0,0:00:18.70,0:00:21.02,Default,,0000,0000,0000,,so that you have a common denominator.
Dialogue: 0,0:00:21.02,0:00:23.27,Default,,0000,0000,0000,,And the easiest way to\Nget a common denominator
Dialogue: 0,0:00:23.27,0:00:25.34,Default,,0000,0000,0000,,is you can just multiply\Nthe two denominators,
Dialogue: 0,0:00:25.34,0:00:26.47,Default,,0000,0000,0000,,especially in case like this
Dialogue: 0,0:00:26.47,0:00:28.96,Default,,0000,0000,0000,,where they don't seem\Nto share any factors.
Dialogue: 0,0:00:28.96,0:00:31.26,Default,,0000,0000,0000,,Both of these are about\Nas factor as you can get
Dialogue: 0,0:00:31.26,0:00:33.49,Default,,0000,0000,0000,,and they don't share anything in common.
Dialogue: 0,0:00:33.49,0:00:36.67,Default,,0000,0000,0000,,And so let's set up a common denominator.
Dialogue: 0,0:00:36.67,0:00:39.22,Default,,0000,0000,0000,,So this is going to be equal to
Dialogue: 0,0:00:39.22,0:00:42.68,Default,,0000,0000,0000,,it's going to be equal to something
Dialogue: 0,0:00:42.68,0:00:44.26,Default,,0000,0000,0000,,let's see, it's going\Nto be equal to something
Dialogue: 0,0:00:44.26,0:00:46.12,Default,,0000,0000,0000,,over our common denominator.
Dialogue: 0,0:00:46.12,0:00:47.57,Default,,0000,0000,0000,,Let's make it
Dialogue: 0,0:00:47.57,0:00:49.45,Default,,0000,0000,0000,,let's make it 2x,
Dialogue: 0,0:00:49.45,0:00:50.99,Default,,0000,0000,0000,,I'm going to do this in another color.
Dialogue: 0,0:00:50.99,0:00:54.90,Default,,0000,0000,0000,,So we're going to make it 2x-3
Dialogue: 0,0:00:54.90,0:00:57.67,Default,,0000,0000,0000,,times 3x+1
Dialogue: 0,0:00:57.67,0:01:00.96,Default,,0000,0000,0000,,times 3x+1
Dialogue: 0,0:01:00.96,0:01:03.77,Default,,0000,0000,0000,,and then plus
Dialogue: 0,0:01:03.77,0:01:07.32,Default,,0000,0000,0000,,plus something else
Dialogue: 0,0:01:07.32,0:01:10.24,Default,,0000,0000,0000,,over 2x-3
Dialogue: 0,0:01:10.24,0:01:13.10,Default,,0000,0000,0000,,2x-3
Dialogue: 0,0:01:13.10,0:01:14.87,Default,,0000,0000,0000,,times 3x+1.
Dialogue: 0,0:01:14.87,0:01:19.20,Default,,0000,0000,0000,,Times 3x+1.
Dialogue: 0,0:01:19.20,0:01:21.29,Default,,0000,0000,0000,,And so to go from 2x,
Dialogue: 0,0:01:21.29,0:01:24.33,Default,,0000,0000,0000,,to go from just a 2x-3\Nhere the denominator
Dialogue: 0,0:01:24.33,0:01:27.42,Default,,0000,0000,0000,,to a (2x-3)(3x+1)
Dialogue: 0,0:01:27.42,0:01:29.84,Default,,0000,0000,0000,,we multiply the denominator by 3x+1.
Dialogue: 0,0:01:29.84,0:01:31.08,Default,,0000,0000,0000,,So if we do that to the denominator,
Dialogue: 0,0:01:31.08,0:01:32.46,Default,,0000,0000,0000,,we don't want to change the value
Dialogue: 0,0:01:32.46,0:01:33.59,Default,,0000,0000,0000,,of the rational expression.
Dialogue: 0,0:01:33.59,0:01:36.22,Default,,0000,0000,0000,,We'd also have to do\Nthat to the numerator.
Dialogue: 0,0:01:36.22,0:01:39.85,Default,,0000,0000,0000,,So the original numerator was 5x.
Dialogue: 0,0:01:39.85,0:01:41.68,Default,,0000,0000,0000,,I'll do that in blue color.
Dialogue: 0,0:01:41.68,0:01:45.12,Default,,0000,0000,0000,,So the original numerator was 5x
Dialogue: 0,0:01:45.12,0:01:47.95,Default,,0000,0000,0000,,and now we're going to\Nmultiply it by the 3x+1,
Dialogue: 0,0:01:47.95,0:01:50.83,Default,,0000,0000,0000,,so times 3x+1.
Dialogue: 0,0:01:50.83,0:01:53.52,Default,,0000,0000,0000,,Notice I didn't change the\Nvalue of this expression.
Dialogue: 0,0:01:53.52,0:01:57.51,Default,,0000,0000,0000,,I multiplied it by 3x+1 over 3x+1,
Dialogue: 0,0:01:57.51,0:02:02.23,Default,,0000,0000,0000,,which is 1 as long as\N3x+1 does not equal zero.
Dialogue: 0,0:02:02.23,0:02:04.44,Default,,0000,0000,0000,,So let's do the same thing over here.
Dialogue: 0,0:02:04.44,0:02:08.64,Default,,0000,0000,0000,,Over here I have a denominator of 3x+1,
Dialogue: 0,0:02:08.64,0:02:10.96,Default,,0000,0000,0000,,I multiplied it by 2x-3,
Dialogue: 0,0:02:10.96,0:02:12.29,Default,,0000,0000,0000,,so I would take my numerator,
Dialogue: 0,0:02:12.29,0:02:15.65,Default,,0000,0000,0000,,which is -4x²
Dialogue: 0,0:02:15.65,0:02:19.32,Default,,0000,0000,0000,,and I would also multiply it by 2x-3.
Dialogue: 0,0:02:19.32,0:02:22.27,Default,,0000,0000,0000,,2x-3.
Dialogue: 0,0:02:22.27,0:02:24.08,Default,,0000,0000,0000,,Let me put parentheses around this
Dialogue: 0,0:02:24.08,0:02:27.73,Default,,0000,0000,0000,,so it doesn't look like\NI'm subtracting 4x².
Dialogue: 0,0:02:27.73,0:02:30.09,Default,,0000,0000,0000,,And so then I can rewrite\Nall of this business
Dialogue: 0,0:02:30.09,0:02:33.04,Default,,0000,0000,0000,,as being equal to,
Dialogue: 0,0:02:33.04,0:02:35.12,Default,,0000,0000,0000,,well, in the numerator,
Dialogue: 0,0:02:35.12,0:02:38.65,Default,,0000,0000,0000,,in the numerator I'm\Ngoing to have 5x times 3x
Dialogue: 0,0:02:38.65,0:02:42.27,Default,,0000,0000,0000,,which is 15x²
Dialogue: 0,0:02:42.27,0:02:47.22,Default,,0000,0000,0000,,5x times 1, which is + 5x,
Dialogue: 0,0:02:47.22,0:02:49.41,Default,,0000,0000,0000,,and then over here,
Dialogue: 0,0:02:49.41,0:02:51.11,Default,,0000,0000,0000,,let me do this in green,
Dialogue: 0,0:02:51.11,0:02:55.73,Default,,0000,0000,0000,,let's see, I could do -4x times 2x
Dialogue: 0,0:02:55.73,0:02:59.86,Default,,0000,0000,0000,,which would be -8x²
Dialogue: 0,0:02:59.86,0:03:02.54,Default,,0000,0000,0000,,and then -4x times -3
Dialogue: 0,0:03:02.54,0:03:05.59,Default,,0000,0000,0000,,which is +12x².
Dialogue: 0,0:03:05.59,0:03:06.92,Default,,0000,0000,0000,,Did I do that right?
Dialogue: 0,0:03:06.92,0:03:08.22,Default,,0000,0000,0000,,Negative...
Dialogue: 0,0:03:08.22,0:03:09.55,Default,,0000,0000,0000,,Oh, let me be very careful.
Dialogue: 0,0:03:09.55,0:03:12.98,Default,,0000,0000,0000,,- 4, my spider sense could tell\Nthat I did something shady.
Dialogue: 0,0:03:12.98,0:03:14.93,Default,,0000,0000,0000,,In fact, if you want to pause\Nthe video you could see,
Dialogue: 0,0:03:14.93,0:03:17.40,Default,,0000,0000,0000,,try to figure out what\NI just did that's wrong.
Dialogue: 0,0:03:17.40,0:03:20.79,Default,,0000,0000,0000,,So -4x² times 2x
Dialogue: 0,0:03:20.79,0:03:23.72,Default,,0000,0000,0000,,is -8x to the third power.
Dialogue: 0,0:03:23.72,0:03:28.53,Default,,0000,0000,0000,,- 8x³
Dialogue: 0,0:03:28.53,0:03:32.72,Default,,0000,0000,0000,,and then -4x² times -3 is 12x²
Dialogue: 0,0:03:32.72,0:03:36.06,Default,,0000,0000,0000,,and then our entire denominator
Dialogue: 0,0:03:36.06,0:03:37.20,Default,,0000,0000,0000,,our entire denominator
Dialogue: 0,0:03:37.20,0:03:38.55,Default,,0000,0000,0000,,we have a common denominator now
Dialogue: 0,0:03:38.55,0:03:40.78,Default,,0000,0000,0000,,so we were able to just add everything
Dialogue: 0,0:03:40.78,0:03:43.43,Default,,0000,0000,0000,,is 2x-3
Dialogue: 0,0:03:43.43,0:03:46.06,Default,,0000,0000,0000,,2x-3
Dialogue: 0,0:03:46.06,0:03:49.00,Default,,0000,0000,0000,,times 3x+1
Dialogue: 0,0:03:49.00,0:03:51.87,Default,,0000,0000,0000,,times 3x+1
Dialogue: 0,0:03:51.87,0:03:54.12,Default,,0000,0000,0000,,and let's see, how can we simplify this?
Dialogue: 0,0:03:54.12,0:03:57.35,Default,,0000,0000,0000,,So this is all going to be equal to,
Dialogue: 0,0:03:57.35,0:03:59.02,Default,,0000,0000,0000,,let me draw,
Dialogue: 0,0:03:59.02,0:04:01.76,Default,,0000,0000,0000,,make sure we recognize\Nit's a rational expression,
Dialogue: 0,0:04:01.76,0:04:05.29,Default,,0000,0000,0000,,and so let's see, we can look at, we can,
Dialogue: 0,0:04:05.29,0:04:09.80,Default,,0000,0000,0000,,our highest degree term here is the -8x³
Dialogue: 0,0:04:09.80,0:04:11.89,Default,,0000,0000,0000,,so it's -8
Dialogue: 0,0:04:11.89,0:04:15.40,Default,,0000,0000,0000,,- 8x³
Dialogue: 0,0:04:15.40,0:04:18.69,Default,,0000,0000,0000,,and then we have a 15x²
Dialogue: 0,0:04:18.69,0:04:21.06,Default,,0000,0000,0000,,and we also have a 12x².
Dialogue: 0,0:04:21.06,0:04:23.14,Default,,0000,0000,0000,,We could add those two together
Dialogue: 0,0:04:23.14,0:04:26.85,Default,,0000,0000,0000,,to get a 27x²
Dialogue: 0,0:04:26.85,0:04:30.63,Default,,0000,0000,0000,,so we've already taken care of this,
Dialogue: 0,0:04:30.63,0:04:32.100,Default,,0000,0000,0000,,we've taken care, let me\Ndo it in that green color,
Dialogue: 0,0:04:32.100,0:04:35.90,Default,,0000,0000,0000,,so we've taken of this,
Dialogue: 0,0:04:35.90,0:04:38.85,Default,,0000,0000,0000,,we've taken care of those two\Nand we're just left with a 5x,
Dialogue: 0,0:04:38.85,0:04:42.89,Default,,0000,0000,0000,,so + 5x
Dialogue: 0,0:04:42.89,0:04:45.68,Default,,0000,0000,0000,,and then all of that is over
Dialogue: 0,0:04:45.68,0:04:50.68,Default,,0000,0000,0000,,2x-3 times 3x+1
Dialogue: 0,0:04:51.98,0:04:53.98,Default,,0000,0000,0000,,3x+1
Dialogue: 0,0:04:53.98,0:04:55.63,Default,,0000,0000,0000,,and we are
Dialogue: 0,0:04:55.63,0:04:58.16,Default,,0000,0000,0000,,and we are all done.
Dialogue: 0,0:04:58.16,0:05:00.06,Default,,0000,0000,0000,,It doesn't seem like there's any easy way
Dialogue: 0,0:05:00.06,0:05:01.81,Default,,0000,0000,0000,,to simplify this further.
Dialogue: 0,0:05:01.81,0:05:03.62,Default,,0000,0000,0000,,You could factor out an\Nx out of the numerator
Dialogue: 0,0:05:03.62,0:05:05.06,Default,,0000,0000,0000,,but that's not going to cancel out
Dialogue: 0,0:05:05.06,0:05:06.36,Default,,0000,0000,0000,,with anything in the denominator
Dialogue: 0,0:05:06.36,0:05:09.12,Default,,0000,0000,0000,,and it looks like we are all done.