[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.29,0:00:01.32,Default,,0000,0000,0000,,- [Voiceover] Let's see\Nif we can figure out
Dialogue: 0,0:00:01.32,0:00:05.02,Default,,0000,0000,0000,,the product of x minus\Nfour and x plus seven.
Dialogue: 0,0:00:05.02,0:00:08.42,Default,,0000,0000,0000,,And we want to write that product\Nin standard quadratic form
Dialogue: 0,0:00:08.42,0:00:10.55,Default,,0000,0000,0000,,which is just a fancy way of saying a form
Dialogue: 0,0:00:10.55,0:00:13.95,Default,,0000,0000,0000,,where you have some coefficient\Non the second degree term,
Dialogue: 0,0:00:13.95,0:00:16.61,Default,,0000,0000,0000,,a x squared plus some coefficient b
Dialogue: 0,0:00:16.61,0:00:19.52,Default,,0000,0000,0000,,on the first degree term\Nplus the constant term.
Dialogue: 0,0:00:19.52,0:00:22.42,Default,,0000,0000,0000,,So this right over here would\Nbe standard quadratic form.
Dialogue: 0,0:00:22.42,0:00:24.95,Default,,0000,0000,0000,,So that's the form that we\Nwant to express this product in
Dialogue: 0,0:00:24.95,0:00:26.15,Default,,0000,0000,0000,,and encourage you to pause the video
Dialogue: 0,0:00:26.15,0:00:28.26,Default,,0000,0000,0000,,and try to work through it on your own.
Dialogue: 0,0:00:28.26,0:00:30.29,Default,,0000,0000,0000,,Alright, now let's work through this.
Dialogue: 0,0:00:30.29,0:00:32.89,Default,,0000,0000,0000,,And the key when we're multiplying\Ntwo binomials like this,
Dialogue: 0,0:00:32.89,0:00:34.82,Default,,0000,0000,0000,,or actually when you're\Nmultiplying any polynomials,
Dialogue: 0,0:00:34.82,0:00:37.42,Default,,0000,0000,0000,,is just to remember the\Ndistributive property
Dialogue: 0,0:00:37.42,0:00:39.69,Default,,0000,0000,0000,,that we all by this point know quite well.
Dialogue: 0,0:00:39.69,0:00:41.62,Default,,0000,0000,0000,,So what we could view this is as
Dialogue: 0,0:00:41.62,0:00:43.92,Default,,0000,0000,0000,,is we could distribute this x minus four,
Dialogue: 0,0:00:43.92,0:00:46.72,Default,,0000,0000,0000,,this entire expression\Nover the x and the seven.
Dialogue: 0,0:00:46.72,0:00:48.72,Default,,0000,0000,0000,,So we could say that\Nthis is the same thing
Dialogue: 0,0:00:48.72,0:00:51.61,Default,,0000,0000,0000,,as x minus four times x
Dialogue: 0,0:00:51.61,0:00:55.22,Default,,0000,0000,0000,,plus x minus four times seven.
Dialogue: 0,0:00:55.22,0:00:56.59,Default,,0000,0000,0000,,So let's write that.
Dialogue: 0,0:00:56.59,0:00:58.82,Default,,0000,0000,0000,,So x minus four times x,\Nor we could write this
Dialogue: 0,0:00:58.82,0:01:03.82,Default,,0000,0000,0000,,as x times x minus four.
Dialogue: 0,0:01:04.52,0:01:07.65,Default,,0000,0000,0000,,That's distributing, or multiplying\Nthe x minus four times x
Dialogue: 0,0:01:07.65,0:01:08.85,Default,,0000,0000,0000,,that's right there.
Dialogue: 0,0:01:08.85,0:01:13.59,Default,,0000,0000,0000,,Plus seven times x minus four.
Dialogue: 0,0:01:13.59,0:01:15.89,Default,,0000,0000,0000,,Times x minus four.
Dialogue: 0,0:01:15.89,0:01:19.05,Default,,0000,0000,0000,,Notice all we did is\Ndistribute the x minus four.
Dialogue: 0,0:01:19.05,0:01:20.99,Default,,0000,0000,0000,,We took this whole thing\Nand we multiplied it
Dialogue: 0,0:01:20.99,0:01:22.82,Default,,0000,0000,0000,,by each term over here.
Dialogue: 0,0:01:22.82,0:01:24.89,Default,,0000,0000,0000,,We multiplied x by x minus four
Dialogue: 0,0:01:24.89,0:01:27.45,Default,,0000,0000,0000,,and we multiplied seven by x minus four.
Dialogue: 0,0:01:27.45,0:01:29.95,Default,,0000,0000,0000,,Now, we see that we have these,
Dialogue: 0,0:01:29.95,0:01:31.88,Default,,0000,0000,0000,,I guess you can call\Nthem two seperate terms.
Dialogue: 0,0:01:31.88,0:01:34.61,Default,,0000,0000,0000,,And to simplify each of them,\Nor to multiply them out,
Dialogue: 0,0:01:34.61,0:01:35.49,Default,,0000,0000,0000,,we just have to distribute.
Dialogue: 0,0:01:35.49,0:01:37.79,Default,,0000,0000,0000,,In this first we're going to\Nhave to distribute this blue x.
Dialogue: 0,0:01:37.79,0:01:40.05,Default,,0000,0000,0000,,And over here we have to\Ndistribute this blue seven.
Dialogue: 0,0:01:40.05,0:01:41.62,Default,,0000,0000,0000,,So let's do that.
Dialogue: 0,0:01:41.62,0:01:46.62,Default,,0000,0000,0000,,So here we can say x times\Nx is going to be x squared.
Dialogue: 0,0:01:46.72,0:01:49.72,Default,,0000,0000,0000,,X times, we have a negative here,
Dialogue: 0,0:01:49.72,0:01:52.25,Default,,0000,0000,0000,,so we can say negative four is\Ngoing to be negative four x.
Dialogue: 0,0:01:52.25,0:01:55.35,Default,,0000,0000,0000,,And just like, that we get\Nx squared minus four x.
Dialogue: 0,0:01:55.35,0:02:00.15,Default,,0000,0000,0000,,And then over here we have seven times x
Dialogue: 0,0:02:00.15,0:02:03.29,Default,,0000,0000,0000,,so that's going to be plus seven x.
Dialogue: 0,0:02:03.29,0:02:06.82,Default,,0000,0000,0000,,And then we have seven\Ntimes the negative four
Dialogue: 0,0:02:06.82,0:02:09.72,Default,,0000,0000,0000,,which is negative 28.
Dialogue: 0,0:02:09.72,0:02:11.75,Default,,0000,0000,0000,,And we are almost done.
Dialogue: 0,0:02:11.75,0:02:13.22,Default,,0000,0000,0000,,We can simplify it a little bit more.
Dialogue: 0,0:02:13.22,0:02:15.02,Default,,0000,0000,0000,,We have two first degree terms here.
Dialogue: 0,0:02:15.02,0:02:18.99,Default,,0000,0000,0000,,If I have negative four xs\Nand to that I add seven xs,
Dialogue: 0,0:02:18.99,0:02:20.39,Default,,0000,0000,0000,,what is that going to be?
Dialogue: 0,0:02:20.39,0:02:21.72,Default,,0000,0000,0000,,Well those two terms together,
Dialogue: 0,0:02:21.72,0:02:24.59,Default,,0000,0000,0000,,these two terms together are going to be
Dialogue: 0,0:02:24.59,0:02:27.92,Default,,0000,0000,0000,,negative four plus seven xs.
Dialogue: 0,0:02:27.92,0:02:32.92,Default,,0000,0000,0000,,Negative four plus, plus seven.
Dialogue: 0,0:02:33.15,0:02:37.12,Default,,0000,0000,0000,,Negative four plus seven xs.
Dialogue: 0,0:02:37.12,0:02:38.51,Default,,0000,0000,0000,,So all I'm doing here,\NI'm making it very clear
Dialogue: 0,0:02:38.51,0:02:40.22,Default,,0000,0000,0000,,that I'm adding these two coefficients,
Dialogue: 0,0:02:40.22,0:02:42.05,Default,,0000,0000,0000,,and then we have all of the other terms.
Dialogue: 0,0:02:42.05,0:02:43.22,Default,,0000,0000,0000,,We have the x squared.
Dialogue: 0,0:02:43.22,0:02:46.19,Default,,0000,0000,0000,,X squared plus this and then we have,
Dialogue: 0,0:02:46.19,0:02:48.42,Default,,0000,0000,0000,,and then we have the minus,
Dialogue: 0,0:02:48.42,0:02:50.48,Default,,0000,0000,0000,,and then we have the minus 28.
Dialogue: 0,0:02:50.48,0:02:51.99,Default,,0000,0000,0000,,And we're at the home stretch!
Dialogue: 0,0:02:51.99,0:02:55.02,Default,,0000,0000,0000,,This would simply to x squared.
Dialogue: 0,0:02:55.02,0:02:57.65,Default,,0000,0000,0000,,Now negative four plus seven is three,
Dialogue: 0,0:02:57.65,0:03:00.79,Default,,0000,0000,0000,,so this is going to be plus three x.
Dialogue: 0,0:03:00.79,0:03:04.59,Default,,0000,0000,0000,,That's what these two middle\Nterms simplify to, to three x.
Dialogue: 0,0:03:04.59,0:03:06.62,Default,,0000,0000,0000,,And then we have minus 28.
Dialogue: 0,0:03:06.62,0:03:09.05,Default,,0000,0000,0000,,Minus 28.
Dialogue: 0,0:03:09.05,0:03:11.92,Default,,0000,0000,0000,,And just like that, we are done!
Dialogue: 0,0:03:11.92,0:03:15.25,Default,,0000,0000,0000,,And a fun thing to think about,\Nsince it's in the same form.
Dialogue: 0,0:03:15.25,0:03:18.40,Default,,0000,0000,0000,,If we were to compare a is one,
Dialogue: 0,0:03:18.40,0:03:21.65,Default,,0000,0000,0000,,b is three, and c is -28,
Dialogue: 0,0:03:21.65,0:03:23.95,Default,,0000,0000,0000,,but it's interesting here\Nto look at the pattern
Dialogue: 0,0:03:23.95,0:03:26.59,Default,,0000,0000,0000,,when we multiplied these two binomials.
Dialogue: 0,0:03:26.59,0:03:28.79,Default,,0000,0000,0000,,Especially these two binomials\Nwhere the coefficient
Dialogue: 0,0:03:28.79,0:03:31.65,Default,,0000,0000,0000,,on the x term was a one.
Dialogue: 0,0:03:31.65,0:03:34.02,Default,,0000,0000,0000,,Notice we have x times x,
Dialogue: 0,0:03:34.02,0:03:36.71,Default,,0000,0000,0000,,that what actually forms the\Nx squared term over here.
Dialogue: 0,0:03:36.71,0:03:39.65,Default,,0000,0000,0000,,We have negative four, let\Nme do this in a new color.
Dialogue: 0,0:03:39.65,0:03:43.35,Default,,0000,0000,0000,,We have negative four times,\Nthat's not a new color.
Dialogue: 0,0:03:43.35,0:03:45.74,Default,,0000,0000,0000,,We have,
Dialogue: 0,0:03:45.74,0:03:50.42,Default,,0000,0000,0000,,we have negative four times seven,
Dialogue: 0,0:03:50.42,0:03:53.59,Default,,0000,0000,0000,,which is going to be negative 28.
Dialogue: 0,0:03:53.59,0:03:55.48,Default,,0000,0000,0000,,And then how did we get this middle term?
Dialogue: 0,0:03:55.48,0:03:57.02,Default,,0000,0000,0000,,How did we get this three x?
Dialogue: 0,0:03:57.02,0:04:01.22,Default,,0000,0000,0000,,Well, you had the negative\Nfour x plus the seven x.
Dialogue: 0,0:04:01.22,0:04:04.38,Default,,0000,0000,0000,,Or the negative four\Nplus the seven times x.
Dialogue: 0,0:04:04.38,0:04:07.61,Default,,0000,0000,0000,,You had the negative four plus the seven,
Dialogue: 0,0:04:07.61,0:04:10.69,Default,,0000,0000,0000,,plus the seven times x.
Dialogue: 0,0:04:10.69,0:04:12.39,Default,,0000,0000,0000,,So I hope you see a little\Nbit of a pattern here.
Dialogue: 0,0:04:12.39,0:04:13.82,Default,,0000,0000,0000,,If you're multiplying two binomials
Dialogue: 0,0:04:13.82,0:04:16.25,Default,,0000,0000,0000,,where the coefficients on\Nthe x term are both one.
Dialogue: 0,0:04:16.25,0:04:17.95,Default,,0000,0000,0000,,It's going to be x squared.
Dialogue: 0,0:04:17.95,0:04:20.19,Default,,0000,0000,0000,,And then the last term, the constant term,
Dialogue: 0,0:04:20.19,0:04:21.92,Default,,0000,0000,0000,,is going to be the product\Nof these two constants.
Dialogue: 0,0:04:21.92,0:04:23.72,Default,,0000,0000,0000,,Negative four and seven.
Dialogue: 0,0:04:23.72,0:04:26.52,Default,,0000,0000,0000,,And then the first degree\Nterm right over here,
Dialogue: 0,0:04:26.52,0:04:28.52,Default,,0000,0000,0000,,it's coefficient is going to be the sum
Dialogue: 0,0:04:28.52,0:04:31.79,Default,,0000,0000,0000,,of these two constants,\Nnegative four and seven.
Dialogue: 0,0:04:31.79,0:04:33.29,Default,,0000,0000,0000,,Now this might, you could do
Dialogue: 0,0:04:33.29,0:04:34.95,Default,,0000,0000,0000,,this pattern if you practice it.
Dialogue: 0,0:04:34.95,0:04:36.25,Default,,0000,0000,0000,,It's just something that will help you
Dialogue: 0,0:04:36.25,0:04:37.98,Default,,0000,0000,0000,,multiply binomials a little bit faster.
Dialogue: 0,0:04:37.98,0:04:39.25,Default,,0000,0000,0000,,But it's super important that
Dialogue: 0,0:04:39.25,0:04:40.79,Default,,0000,0000,0000,,you realize where this came from.
Dialogue: 0,0:04:40.79,0:04:42.28,Default,,0000,0000,0000,,This came from nothing more than
Dialogue: 0,0:04:42.28,0:04:44.87,Default,,0000,0000,0000,,applying the distributive property twice.