[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:11.80,0:00:14.41,Default,,0000,0000,0000,,College Algebra students, \NDillon here.
Dialogue: 0,0:00:14.41,0:00:17.33,Default,,0000,0000,0000,,We are continuing on\Nwith Section 3.2.
Dialogue: 0,0:00:17.33,0:00:21.03,Default,,0000,0000,0000,,Yes, it's a long one.\NLet's get started.
Dialogue: 0,0:00:27.48,0:00:35.96,Default,,0000,0000,0000,,We've learned that there's a connection \Nbetween the x-intercepts, also called zeros,
Dialogue: 0,0:00:35.96,0:00:42.82,Default,,0000,0000,0000,,and the factors that make up \Na particular polynomial.
Dialogue: 0,0:00:42.82,0:00:46.100,Default,,0000,0000,0000,,We've learned that if the
Dialogue: 0,0:00:48.25,0:00:51.61,Default,,0000,0000,0000,,polynomial has the factor, \Nthere's a corresponding zero.
Dialogue: 0,0:00:51.61,0:00:53.88,Default,,0000,0000,0000,,Here's the last example we did.
Dialogue: 0,0:00:53.88,0:00:59.68,Default,,0000,0000,0000,,We started with this \Nfourth-degree polynomial.
Dialogue: 0,0:01:00.98,0:01:05.18,Default,,0000,0000,0000,,We factored it; we factored out \Na negative x squared.
Dialogue: 0,0:01:05.18,0:01:08.48,Default,,0000,0000,0000,,We're left with a quadratic, \Nwhich we then factored.
Dialogue: 0,0:01:08.48,0:01:10.33,Default,,0000,0000,0000,,We end up with four factors.
Dialogue: 0,0:01:10.33,0:01:15.54,Default,,0000,0000,0000,,Two factors of x: \N2x plus 3 and x minus 1.
Dialogue: 0,0:01:15.54,0:01:20.63,Default,,0000,0000,0000,,Each one of these contributed \Na zero (an x-intercept),
Dialogue: 0,0:01:20.63,0:01:24.30,Default,,0000,0000,0000,,and we learned\Nthat the zero—
Dialogue: 0,0:01:26.02,0:01:29.44,Default,,0000,0000,0000,,that the factor to the second degree\Nhas some interesting behavior.
Dialogue: 0,0:01:29.44,0:01:31.36,Default,,0000,0000,0000,,That's the last thing \Nwe finish up here.
Dialogue: 0,0:01:31.36,0:01:37.71,Default,,0000,0000,0000,,We're going to study how these \Npolynomials behave around their zeros.
Dialogue: 0,0:01:38.67,0:01:41.02,Default,,0000,0000,0000,,There are two types of behavior:
Dialogue: 0,0:01:41.02,0:01:45.09,Default,,0000,0000,0000,,one where it actually goes through\Nthe zero, one where it's tangent.
Dialogue: 0,0:01:45.09,0:01:51.74,Default,,0000,0000,0000,,I think it'll be instructive to\Njust look at this in Desmos.
Dialogue: 0,0:01:53.72,0:01:56.77,Default,,0000,0000,0000,,Not that my graph is not—
Dialogue: 0,0:01:57.01,0:01:59.96,Default,,0000,0000,0000,,it shows all the important details, \Nbut it looks a little nicer in Desmos.
Dialogue: 0,0:01:59.96,0:02:04.67,Default,,0000,0000,0000,,Here we go: y equals negative\N2x to the fourth power.
Dialogue: 0,0:02:06.11,0:02:09.88,Default,,0000,0000,0000,,That's our leading term, our \Nleading coefficient is negative 2.
Dialogue: 0,0:02:09.88,0:02:14.72,Default,,0000,0000,0000,,And remember, these monomial terms, \Nthese monomial leading terms,
Dialogue: 0,0:02:14.72,0:02:19.90,Default,,0000,0000,0000,,they determine the \Nend behavior of the polynomial.
Dialogue: 0,0:02:21.42,0:02:24.12,Default,,0000,0000,0000,,They dominate the end behavior.
Dialogue: 0,0:02:27.93,0:02:32.68,Default,,0000,0000,0000,,Okay, there it is, a slightly more\Nattractive version of the graph.
Dialogue: 0,0:02:32.68,0:02:36.66,Default,,0000,0000,0000,,We can see that the \Ntwo types of behavior here
Dialogue: 0,0:02:36.66,0:02:39.62,Default,,0000,0000,0000,,that we're going to study a bit\Nmore to finish up this section.
Dialogue: 0,0:02:39.62,0:02:42.93,Default,,0000,0000,0000,,Where it goes— the graph—
Dialogue: 0,0:02:42.93,0:02:49.16,Default,,0000,0000,0000,,Excuse me, the polynomial goes \Nthrough the x-intercept (the zero),
Dialogue: 0,0:02:49.16,0:02:52.75,Default,,0000,0000,0000,,if the degree of that factor is odd.
Dialogue: 0,0:02:52.75,0:02:58.13,Default,,0000,0000,0000,,So, in this case the \Ndegree of 2x plus 3 is 1.
Dialogue: 0,0:02:58.13,0:02:59.62,Default,,0000,0000,0000,,That's an odd number.
Dialogue: 0,0:02:59.62,0:03:03.100,Default,,0000,0000,0000,,The degree of x minus 1 \Nis to the first power.
Dialogue: 0,0:03:03.100,0:03:11.48,Default,,0000,0000,0000,,So, notice for both of those zeros, \Nthe graph goes through the points.
Dialogue: 0,0:03:11.72,0:03:16.11,Default,,0000,0000,0000,,The zero that had an\Neven degree (second degree),
Dialogue: 0,0:03:16.11,0:03:19.02,Default,,0000,0000,0000,,we call this multiplicity two.
Dialogue: 0,0:03:19.02,0:03:22.18,Default,,0000,0000,0000,,The other two factors \Nhave multiplicity one.
Dialogue: 0,0:03:22.18,0:03:25.64,Default,,0000,0000,0000,,This factor x has \Nmultiplicity two.
Dialogue: 0,0:03:25.64,0:03:28.91,Default,,0000,0000,0000,,And you'll notice that the \Ngraph is tangent at that point,
Dialogue: 0,0:03:28.91,0:03:35.78,Default,,0000,0000,0000,,so the degree of the factor, \Nalso called the multiplicity,
Dialogue: 0,0:03:35.78,0:03:39.13,Default,,0000,0000,0000,,is going to determine \Nhow it behaves.
Dialogue: 0,0:03:39.13,0:03:44.61,Default,,0000,0000,0000,,Okay, so just to practice this \Nsketching the graph one more time,
Dialogue: 0,0:03:44.61,0:03:46.11,Default,,0000,0000,0000,,let's do another one.
Dialogue: 0,0:03:46.11,0:03:49.29,Default,,0000,0000,0000,,And I like this one because the \Nfactoring is a little different.
Dialogue: 0,0:03:49.29,0:03:52.51,Default,,0000,0000,0000,,Something we haven't\Nseen yet in this section.
Dialogue: 0,0:03:53.99,0:03:57.22,Default,,0000,0000,0000,,Okay, right off the bat,
Dialogue: 0,0:03:57.76,0:04:03.11,Default,,0000,0000,0000,,you notice there's no greatest \Ncommon factor to take out here.
Dialogue: 0,0:04:03.11,0:04:09.81,Default,,0000,0000,0000,,And you have to go back to the\Npolynomial factoring methods
Dialogue: 0,0:04:09.81,0:04:12.71,Default,,0000,0000,0000,,that we learned probably \Nin the previous class
Dialogue: 0,0:04:12.71,0:04:16.04,Default,,0000,0000,0000,,and realize that you\Nhave four terms here.
Dialogue: 0,0:04:16.04,0:04:19.09,Default,,0000,0000,0000,,You'll notice that there's \Na common term in each.
Dialogue: 0,0:04:19.09,0:04:22.04,Default,,0000,0000,0000,,If we group these\Ntwo together,
Dialogue: 0,0:04:22.04,0:04:23.81,Default,,0000,0000,0000,,and group these two together,
Dialogue: 0,0:04:23.81,0:04:25.86,Default,,0000,0000,0000,,there's a common term\Nyou can take out.
Dialogue: 0,0:04:25.86,0:04:27.17,Default,,0000,0000,0000,,Let's take that out.
Dialogue: 0,0:04:27.17,0:04:29.64,Default,,0000,0000,0000,,I'm going to take an x squared \Nout of the first two terms.
Dialogue: 0,0:04:29.64,0:04:32.14,Default,,0000,0000,0000,,This is called factoring \Nby grouping, by the way.
Dialogue: 0,0:04:32.14,0:04:36.19,Default,,0000,0000,0000,,Then what's left in just\Nthe first two terms?
Dialogue: 0,0:04:36.19,0:04:38.03,Default,,0000,0000,0000,,I'm ignoring these last two.
Dialogue: 0,0:04:38.03,0:04:40.58,Default,,0000,0000,0000,,I pull that out and \NI get x minus 2.
Dialogue: 0,0:04:40.58,0:04:43.69,Default,,0000,0000,0000,,Notice that there's a common \Nfactor we can pull out here.
Dialogue: 0,0:04:43.69,0:04:46.76,Default,,0000,0000,0000,,When the leading coefficient is negative, \Nyou should always pull out a negative.
Dialogue: 0,0:04:46.76,0:04:49.50,Default,,0000,0000,0000,,So we're going to\Npull out a negative four.
Dialogue: 0,0:04:49.50,0:04:52.41,Default,,0000,0000,0000,,What's left?\NX minus 2.
Dialogue: 0,0:04:52.91,0:04:54.71,Default,,0000,0000,0000,,And that's what \Nwe're looking for.
Dialogue: 0,0:04:54.71,0:04:58.52,Default,,0000,0000,0000,,We needed a common factor, \Na common binomial factor
Dialogue: 0,0:04:58.52,0:05:02.62,Default,,0000,0000,0000,,that we can now take out of both \Nof those, so take out the x minus 2
Dialogue: 0,0:05:02.62,0:05:05.63,Default,,0000,0000,0000,,and we're left with\Nx squared minus 4.
Dialogue: 0,0:05:05.63,0:05:08.42,Default,,0000,0000,0000,,I hope you recognize \Nthat the second term here
Dialogue: 0,0:05:08.42,0:05:10.85,Default,,0000,0000,0000,,is actually the difference\Nof two squares.
Dialogue: 0,0:05:10.85,0:05:13.18,Default,,0000,0000,0000,,We're going to\Nhave this x minus 2
Dialogue: 0,0:05:13.18,0:05:20.00,Default,,0000,0000,0000,,and then the difference of two squares \Nfactors to x plus 2 and x minus 2.
Dialogue: 0,0:05:20.83,0:05:22.73,Default,,0000,0000,0000,,We're going to \Nrewrite it like this
Dialogue: 0,0:05:22.73,0:05:26.57,Default,,0000,0000,0000,,because we're going to talk more \Nabout this idea of multiplicity.
Dialogue: 0,0:05:26.57,0:05:29.77,Default,,0000,0000,0000,,The x minus 2 is actually\Nto the second power.
Dialogue: 0,0:05:29.77,0:05:32.09,Default,,0000,0000,0000,,This—
Dialogue: 0,0:05:33.100,0:05:36.40,Default,,0000,0000,0000,,The x-intercept \Nhas multiplicity two.
Dialogue: 0,0:05:36.40,0:05:39.26,Default,,0000,0000,0000,,We just learned how\Nthat's going to behave.
Dialogue: 0,0:05:39.26,0:05:41.93,Default,,0000,0000,0000,,It's going to be tangent \Nat the point x equals 2.
Dialogue: 0,0:05:41.93,0:05:45.36,Default,,0000,0000,0000,,And then we have the \Nsecond zero here: x plus 2.
Dialogue: 0,0:05:46.13,0:05:49.87,Default,,0000,0000,0000,,Okay, Let's draw a graph,\Nsketch a graph,
Dialogue: 0,0:05:51.04,0:05:52.86,Default,,0000,0000,0000,,somewhat accurate.
Dialogue: 0,0:05:58.58,0:06:00.66,Default,,0000,0000,0000,,Then we'll do it\Nin Desmos, also.
Dialogue: 0,0:06:00.66,0:06:07.78,Default,,0000,0000,0000,,Okay, so the interesting points are— \Nfrom this factor, we get x equals 2.
Dialogue: 0,0:06:07.78,0:06:11.48,Default,,0000,0000,0000,,We know it's going to be tangent there; \Nwe'll figure that out in a second.
Dialogue: 0,0:06:11.48,0:06:14.26,Default,,0000,0000,0000,,The second factor\Ngives us x minus 2.
Dialogue: 0,0:06:14.26,0:06:16.63,Default,,0000,0000,0000,,And then let's look at \Nthis leading term.
Dialogue: 0,0:06:16.63,0:06:19.12,Default,,0000,0000,0000,,It’s an odd degree.
Dialogue: 0,0:06:19.12,0:06:21.26,Default,,0000,0000,0000,,The leading coefficient\Nis positive.
Dialogue: 0,0:06:21.26,0:06:26.79,Default,,0000,0000,0000,,So, you know, the end behavior\Nis like this, in this direction,
Dialogue: 0,0:06:26.79,0:06:29.74,Default,,0000,0000,0000,,and like this \Nin this direction.
Dialogue: 0,0:06:29.74,0:06:34.02,Default,,0000,0000,0000,,As x goes to positive infinity, \Nwe know y goes to positive infinity.
Dialogue: 0,0:06:34.02,0:06:37.84,Default,,0000,0000,0000,,As x goes to negative infinity, \Nbecause it's odd degree,
Dialogue: 0,0:06:37.84,0:06:40.56,Default,,0000,0000,0000,,the y value is going to \Ngo to negative infinity.
Dialogue: 0,0:06:40.56,0:06:42.53,Default,,0000,0000,0000,,That's the end behavior.
Dialogue: 0,0:06:42.53,0:06:44.41,Default,,0000,0000,0000,,Then what happens in \Nbetween these two?
Dialogue: 0,0:06:44.41,0:06:46.79,Default,,0000,0000,0000,,Now remember, these are \Nthe only two x-intercepts.
Dialogue: 0,0:06:46.79,0:06:47.85,Default,,0000,0000,0000,,That's important.
Dialogue: 0,0:06:47.85,0:06:50.51,Default,,0000,0000,0000,,We know the graph must\Ncome up and go through this one
Dialogue: 0,0:06:50.51,0:06:55.32,Default,,0000,0000,0000,,(this has odd multiplicity: \Nx plus 2 to the first power),
Dialogue: 0,0:06:55.32,0:06:57.40,Default,,0000,0000,0000,,and then it's going to come up;
Dialogue: 0,0:06:57.40,0:07:00.28,Default,,0000,0000,0000,,we know it has to go through this point, \Nbut it doesn't actually go through.
Dialogue: 0,0:07:00.28,0:07:02.96,Default,,0000,0000,0000,,It's going to come down \Nhere and it's tangent,
Dialogue: 0,0:07:02.96,0:07:05.08,Default,,0000,0000,0000,,and then it takes off\Nin that direction.
Dialogue: 0,0:07:05.08,0:07:08.24,Default,,0000,0000,0000,,It's going to look \Nsomething like that.
Dialogue: 0,0:07:08.49,0:07:09.98,Default,,0000,0000,0000,,Let's just graph it in a Desmos,
Dialogue: 0,0:07:09.98,0:07:13.36,Default,,0000,0000,0000,,so we have a slightly \Nmore attractive version of it.
Dialogue: 0,0:07:17.97,0:07:21.73,Default,,0000,0000,0000,,Let's see… we're \Ngoing to get rid of this.
Dialogue: 0,0:07:22.80,0:07:24.62,Default,,0000,0000,0000,,Change that to \Nthe third power,
Dialogue: 0,0:07:24.62,0:07:34.52,Default,,0000,0000,0000,,so minus 2x to the \Nsecond power minus 4x plus 8.
Dialogue: 0,0:07:40.05,0:07:42.94,Default,,0000,0000,0000,,Our scale is a \Nlittle bit off here.
Dialogue: 0,0:07:42.94,0:07:44.67,Default,,0000,0000,0000,,Let's just fix this.
Dialogue: 0,0:07:44.67,0:07:49.98,Default,,0000,0000,0000,,We’re going to fix this maximum \Ny value to something a little bigger.
Dialogue: 0,0:07:49.98,0:07:51.43,Default,,0000,0000,0000,,I don't really know where it is.
Dialogue: 0,0:07:51.43,0:07:53.41,Default,,0000,0000,0000,,I'm going to try 50 and see.\NThat's too big.
Dialogue: 0,0:07:53.41,0:07:55.80,Default,,0000,0000,0000,,How about 20?\NThere we go.
Dialogue: 0,0:07:57.60,0:08:01.48,Default,,0000,0000,0000,,Okay, so that's similar to \Nwhat we sketched here.
Dialogue: 0,0:08:03.16,0:08:07.33,Default,,0000,0000,0000,,All right, a little\Nprettier in Desmos.
Dialogue: 0,0:08:07.33,0:08:14.16,Default,,0000,0000,0000,,And we can see that the zero \Nthat has multiplicity two—
Dialogue: 0,0:08:14.16,0:08:15.65,Default,,0000,0000,0000,,that's our vocabulary:
Dialogue: 0,0:08:15.65,0:08:19.21,Default,,0000,0000,0000,,"multiplicity two" because that factor\Nis raised to the second power—
Dialogue: 0,0:08:19.21,0:08:22.23,Default,,0000,0000,0000,,the polynomial is\Ntangent at that point.
Dialogue: 0,0:08:22.23,0:08:25.97,Default,,0000,0000,0000,,Whereas the zero with \Nmultiplicity one (an odd number),
Dialogue: 0,0:08:25.97,0:08:28.70,Default,,0000,0000,0000,,it actually goes through\Nthe x-intercept.
Dialogue: 0,0:08:29.07,0:08:31.92,Default,,0000,0000,0000,,Okay, we're going to do a\Nfew more examples of that
Dialogue: 0,0:08:31.92,0:08:33.47,Default,,0000,0000,0000,,to finish up the section here.
Dialogue: 0,0:08:33.47,0:08:34.90,Default,,0000,0000,0000,,Here we go.
Dialogue: 0,0:08:39.44,0:08:41.65,Default,,0000,0000,0000,,Everything I just said is \Nsummarized in this box.
Dialogue: 0,0:08:41.65,0:08:44.13,Default,,0000,0000,0000,,You might want to pause the \Nvideo and take a closer look.
Dialogue: 0,0:08:44.13,0:08:46.24,Default,,0000,0000,0000,,If the multiplicity—
Dialogue: 0,0:08:46.24,0:08:50.27,Default,,0000,0000,0000,,and the multiplicity is determined\Nby the power of the factor.
Dialogue: 0,0:08:50.27,0:08:53.03,Default,,0000,0000,0000,,Each factor has a \Npower one or greater.
Dialogue: 0,0:08:53.03,0:08:59.83,Default,,0000,0000,0000,,If it's an odd multiplicity, \Nthen the x-intercept is
Dialogue: 0,0:08:59.83,0:09:02.57,Default,,0000,0000,0000,,going to behave like\Nthis for the polynomial.
Dialogue: 0,0:09:02.57,0:09:04.76,Default,,0000,0000,0000,,If it's even multiplicity \Nit behaves like this:
Dialogue: 0,0:09:04.76,0:09:08.10,Default,,0000,0000,0000,,either tangent from the top \Nor tangent from the bottom.
Dialogue: 0,0:09:09.98,0:09:15.42,Default,,0000,0000,0000,,Then you might ask, “Well how do\Nyou decide?” Well, look up here.
Dialogue: 0,0:09:15.42,0:09:18.95,Default,,0000,0000,0000,,We knew what the end behavior\Nof the polynomial was, okay?
Dialogue: 0,0:09:18.95,0:09:20.53,Default,,0000,0000,0000,,We knew it was\Ncoming up here.
Dialogue: 0,0:09:20.53,0:09:27.06,Default,,0000,0000,0000,,It has to go through this intercept, \Nand then we're above the x-axis here.
Dialogue: 0,0:09:27.06,0:09:29.01,Default,,0000,0000,0000,,And we know it's got to\Nbe tangent at that point.
Dialogue: 0,0:09:29.01,0:09:32.66,Default,,0000,0000,0000,,This is the only option, we know it \Ndoesn't have any other x-intercepts,
Dialogue: 0,0:09:32.66,0:09:35.15,Default,,0000,0000,0000,,so it can't go \Nthrough the x-axis.
Dialogue: 0,0:09:35.15,0:09:37.75,Default,,0000,0000,0000,,You just use a \Nlittle bit of logic.
Dialogue: 0,0:09:38.62,0:09:40.40,Default,,0000,0000,0000,,Okay, let's graph this one.
Dialogue: 0,0:09:40.40,0:09:42.30,Default,,0000,0000,0000,,I think these are kinda fun.
Dialogue: 0,0:09:42.30,0:09:47.22,Default,,0000,0000,0000,,The first thing we usually do is— \Nso notice this is factored for us.
Dialogue: 0,0:09:47.22,0:09:48.86,Default,,0000,0000,0000,,That's nice, right?
Dialogue: 0,0:09:48.86,0:09:58.76,Default,,0000,0000,0000,,We're going to list the zero and \Nthe multiplicity, then we'll graph it.
Dialogue: 0,0:09:59.12,0:10:05.27,Default,,0000,0000,0000,,The first zero is x equals zero, \Nmultiplicity four.
Dialogue: 0,0:10:05.27,0:10:07.54,Default,,0000,0000,0000,,And notice that that's \Nan even number.
Dialogue: 0,0:10:07.54,0:10:12.91,Default,,0000,0000,0000,,The second zero from this factor, \Nwe get x equals two, multiplicity three.
Dialogue: 0,0:10:12.91,0:10:16.37,Default,,0000,0000,0000,,That's odd, so it's going\Nto go through that point.
Dialogue: 0,0:10:16.37,0:10:19.57,Default,,0000,0000,0000,,Then x equals negative 1.
Dialogue: 0,0:10:20.66,0:10:25.01,Default,,0000,0000,0000,,Again, even multiplicity, so we know\Nit's going to be tangent at that point.
Dialogue: 0,0:10:25.01,0:10:26.93,Default,,0000,0000,0000,,This is a pretty funky graph.
Dialogue: 0,0:10:26.93,0:10:32.74,Default,,0000,0000,0000,,Now, what is the leading term\Ngoing to look like here?
Dialogue: 0,0:10:32.74,0:10:34.69,Default,,0000,0000,0000,,Well, if you multiply this out—
Dialogue: 0,0:10:34.69,0:10:37.81,Default,,0000,0000,0000,,this is kind of a crazy polynomial, \Nbut we can do it.
Dialogue: 0,0:10:37.81,0:10:42.06,Default,,0000,0000,0000,,If you multiply this out, it's going\Nto be a third-degree polynomial
Dialogue: 0,0:10:42.06,0:10:46.22,Default,,0000,0000,0000,,times the fourth-degree term\Nis going to be seventh degree.
Dialogue: 0,0:10:47.40,0:10:51.70,Default,,0000,0000,0000,,Multiply this out, it's \Ngoing to be a second degree,
Dialogue: 0,0:10:51.70,0:10:53.95,Default,,0000,0000,0000,,so seventh degree \Ntimes second degree.
Dialogue: 0,0:10:53.95,0:10:56.64,Default,,0000,0000,0000,,We're going to\Nhave x to the ninth.
Dialogue: 0,0:10:56.64,0:10:59.47,Default,,0000,0000,0000,,Yikes, that's crazy, right?
Dialogue: 0,0:10:59.92,0:11:04.79,Default,,0000,0000,0000,,This is a third-degree polynomial \Ntimes the fourth-degree term.
Dialogue: 0,0:11:04.79,0:11:06.98,Default,,0000,0000,0000,,That gives us\Nx to the seventh.
Dialogue: 0,0:11:06.98,0:11:09.62,Default,,0000,0000,0000,,Then if this is multiplied out, \Nit's going to be squared.
Dialogue: 0,0:11:09.62,0:11:12.02,Default,,0000,0000,0000,,So we have x to the ninth.
Dialogue: 0,0:11:12.02,0:11:14.34,Default,,0000,0000,0000,,Notice the leading coefficient\Nis going to be 1,
Dialogue: 0,0:11:14.34,0:11:17.64,Default,,0000,0000,0000,,because all of the leading \Ncoefficients here are 1.
Dialogue: 0,0:11:17.64,0:11:23.54,Default,,0000,0000,0000,,And we know if this is odd degree,\Nit's going to behave like this,
Dialogue: 0,0:11:23.54,0:11:25.78,Default,,0000,0000,0000,,and like this with\Nstuff in the middle.
Dialogue: 0,0:11:25.78,0:11:31.86,Default,,0000,0000,0000,,Odd degree, just like x to the third, \Nx to the ninth behaves like x to the third.
Dialogue: 0,0:11:31.86,0:11:33.49,Default,,0000,0000,0000,,Okay, here we go.
Dialogue: 0,0:11:33.49,0:11:38.58,Default,,0000,0000,0000,,This is a pretty \Nfancy polynomial.
Dialogue: 0,0:11:38.58,0:11:41.26,Default,,0000,0000,0000,,I'm not sure why I'm using\Nblue here all of a sudden.
Dialogue: 0,0:11:41.26,0:11:44.03,Default,,0000,0000,0000,,Should’ve changed color.
Dialogue: 0,0:11:45.53,0:11:46.66,Default,,0000,0000,0000,,Okay.
Dialogue: 0,0:11:46.66,0:11:55.24,Default,,0000,0000,0000,,So here are intercepts: zero, \Npositive 2 and negative 1.
Dialogue: 0,0:11:56.35,0:11:58.78,Default,,0000,0000,0000,,Okay, and then we know\Nthe end behavior
Dialogue: 0,0:11:58.78,0:12:02.77,Default,,0000,0000,0000,,because we know the leading\Nterm and the leading coefficient.
Dialogue: 0,0:12:02.77,0:12:05.65,Default,,0000,0000,0000,,End behavior has to be like this
Dialogue: 0,0:12:07.34,0:12:09.32,Default,,0000,0000,0000,,and like this.
Dialogue: 0,0:12:10.44,0:12:14.88,Default,,0000,0000,0000,,Okay, so what happens \Nat this leftmost zero?
Dialogue: 0,0:12:14.88,0:12:18.25,Default,,0000,0000,0000,,Well, it's got to be tangent there\Nbecause it's even multiplicity.
Dialogue: 0,0:12:18.25,0:12:22.49,Default,,0000,0000,0000,,So it has to come up and touch\Nhere and then come back down.
Dialogue: 0,0:12:22.49,0:12:24.29,Default,,0000,0000,0000,,What happens at zero?
Dialogue: 0,0:12:24.29,0:12:28.48,Default,,0000,0000,0000,,Well, it's also even multiplicity.
Dialogue: 0,0:12:28.48,0:12:32.22,Default,,0000,0000,0000,,It's going to come \Ndown and be like this.
Dialogue: 0,0:12:32.22,0:12:35.25,Default,,0000,0000,0000,,It's got to be tangent here, \Nand we're below the x-axis.
Dialogue: 0,0:12:35.25,0:12:36.86,Default,,0000,0000,0000,,And we know it's tangent there.
Dialogue: 0,0:12:36.86,0:12:39.09,Default,,0000,0000,0000,,So it can't go through and \Ncome back down and touch,
Dialogue: 0,0:12:39.09,0:12:40.84,Default,,0000,0000,0000,,it’s gotta be something like that.
Dialogue: 0,0:12:40.84,0:12:44.64,Default,,0000,0000,0000,,And then the zero of\Nmultiplicity three is at 2.
Dialogue: 0,0:12:44.64,0:12:48.66,Default,,0000,0000,0000,,Then we know this has to come\Ndown and go back up and out.
Dialogue: 0,0:12:48.66,0:12:50.84,Default,,0000,0000,0000,,It's going to look \Nsomething like that.
Dialogue: 0,0:12:50.84,0:12:52.53,Default,,0000,0000,0000,,We don't know how\Nbig this loop is here,
Dialogue: 0,0:12:52.53,0:12:54.86,Default,,0000,0000,0000,,but let's put it in Desmos.
Dialogue: 0,0:12:58.79,0:13:01.83,Default,,0000,0000,0000,,Should be quite interesting, \NI would think.
Dialogue: 0,0:13:03.39,0:13:06.89,Default,,0000,0000,0000,,If you're looking for an\Nidea for a tattoo, here it is.
Dialogue: 0,0:13:07.31,0:13:09.89,Default,,0000,0000,0000,,This is a pretty\Ncool polynomial.
Dialogue: 0,0:13:11.38,0:13:14.41,Default,,0000,0000,0000,,Just trying to get \Nboth on the screen.
Dialogue: 0,0:13:22.91,0:13:27.94,Default,,0000,0000,0000,,Okay, so x minus 2\Nraised to the third power.
Dialogue: 0,0:13:29.39,0:13:31.98,Default,,0000,0000,0000,,It's a pretty fancy polynomial.
Dialogue: 0,0:13:31.98,0:13:36.76,Default,,0000,0000,0000,,X plus 1 is to the second power.
Dialogue: 0,0:13:37.02,0:13:39.39,Default,,0000,0000,0000,,Ooh, all right, so…
Dialogue: 0,0:13:41.04,0:13:44.06,Default,,0000,0000,0000,,Our graph, it's close.
Dialogue: 0,0:13:44.24,0:13:45.71,Default,,0000,0000,0000,,Okay, let's see.
Dialogue: 0,0:13:45.71,0:13:49.43,Default,,0000,0000,0000,,We didn't know how— we knew— \NLet's see if we can—
Dialogue: 0,0:13:50.11,0:13:53.08,Default,,0000,0000,0000,,I'm going to change the scale a little bit\Nso we can see this detail a little better.
Dialogue: 0,0:13:53.08,0:13:54.86,Default,,0000,0000,0000,,Negative 2 to 3.
Dialogue: 0,0:13:54.86,0:13:57.36,Default,,0000,0000,0000,,That's not going to \Nchange it much.
Dialogue: 0,0:14:01.36,0:14:03.28,Default,,0000,0000,0000,,Then we're going to\Nchange the y scale.
Dialogue: 0,0:14:03.28,0:14:04.89,Default,,0000,0000,0000,,Let's make this…
Dialogue: 0,0:14:06.25,0:14:08.53,Default,,0000,0000,0000,,Let's see, on the negative side,
Dialogue: 0,0:14:08.53,0:14:15.19,Default,,0000,0000,0000,,we could go to negative 6 \Nmaybe, and let's go with 2.
Dialogue: 0,0:14:15.38,0:14:16.61,Default,,0000,0000,0000,,Yeah, there we go.
Dialogue: 0,0:14:16.61,0:14:17.61,Default,,0000,0000,0000,,Okay.
Dialogue: 0,0:14:17.61,0:14:23.43,Default,,0000,0000,0000,,It's different but it \Nhas the same features.
Dialogue: 0,0:14:23.43,0:14:25.91,Default,,0000,0000,0000,,It’s doing the same \Nthings we expected.
Dialogue: 0,0:14:25.91,0:14:28.76,Default,,0000,0000,0000,,We come up here,\Nwe’re tangent at negative 1;
Dialogue: 0,0:14:28.76,0:14:31.46,Default,,0000,0000,0000,,we come back down, \Nwe’re tangent at zero.
Dialogue: 0,0:14:31.46,0:14:32.56,Default,,0000,0000,0000,,If we click on the graph,
Dialogue: 0,0:14:32.56,0:14:35.18,Default,,0000,0000,0000,,we can prove that what \Nwe have is accurate.
Dialogue: 0,0:14:35.18,0:14:37.77,Default,,0000,0000,0000,,Okay, tangent at negative 1, \Ntangent at zero.
Dialogue: 0,0:14:37.77,0:14:39.13,Default,,0000,0000,0000,,We have little dip here,
Dialogue: 0,0:14:39.13,0:14:42.03,Default,,0000,0000,0000,,we didn't know how far that went, \Nbut that's what it looks like.
Dialogue: 0,0:14:42.03,0:14:43.42,Default,,0000,0000,0000,,And then we have another dip,
Dialogue: 0,0:14:43.42,0:14:48.98,Default,,0000,0000,0000,,and then it goes up\Nthrough 2 (x equals 2).
Dialogue: 0,0:14:49.82,0:14:53.85,Default,,0000,0000,0000,,So, our sketch was pretty accurate, \Nbut not as pretty, of course.
Dialogue: 0,0:14:59.58,0:15:03.53,Default,,0000,0000,0000,,Okay, I would like you to \Npause the recording please,
Dialogue: 0,0:15:03.53,0:15:06.58,Default,,0000,0000,0000,,and try number 29.
Dialogue: 0,0:15:07.84,0:15:10.42,Default,,0000,0000,0000,,And then we'll do it together.
Dialogue: 0,0:15:16.62,0:15:17.33,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:15:17.33,0:15:19.27,Default,,0000,0000,0000,,I hope you tried it.
Dialogue: 0,0:15:19.27,0:15:22.45,Default,,0000,0000,0000,,First of all, what's the\Nleading term here?
Dialogue: 0,0:15:22.67,0:15:24.53,Default,,0000,0000,0000,,Let's figure that out.
Dialogue: 0,0:15:24.92,0:15:26.48,Default,,0000,0000,0000,,The leading term, let's see.
Dialogue: 0,0:15:26.48,0:15:30.23,Default,,0000,0000,0000,,This is going to be a second-degree\Npolynomial times another first-degree
Dialogue: 0,0:15:30.23,0:15:34.02,Default,,0000,0000,0000,,(so that’s going to be third), \Nmultiplied by third, so this gonna be—
Dialogue: 0,0:15:34.02,0:15:39.37,Default,,0000,0000,0000,,The leading coefficient is 1; \Nthe leading term is x to the sixth.
Dialogue: 0,0:15:39.67,0:15:41.90,Default,,0000,0000,0000,,Let me just repeat that, \Nyou multiply this out,
Dialogue: 0,0:15:41.90,0:15:43.62,Default,,0000,0000,0000,,that's going to be\Nsecond-degree,
Dialogue: 0,0:15:43.62,0:15:46.55,Default,,0000,0000,0000,,times another x is \Ngoing to be third-degree,
Dialogue: 0,0:15:46.55,0:15:49.49,Default,,0000,0000,0000,,times x to the third is \Ngoing to be x to the sixth.
Dialogue: 0,0:15:49.49,0:15:55.35,Default,,0000,0000,0000,,So, we know the end behavior\Nof this stuff in the middle is
Dialogue: 0,0:15:55.35,0:15:58.90,Default,,0000,0000,0000,,going to look like that.\NIt's going to go up:
Dialogue: 0,0:15:58.90,0:16:02.50,Default,,0000,0000,0000,,as x goes to positive infinity,\Ny is going to go to positive infinity
Dialogue: 0,0:16:02.50,0:16:07.17,Default,,0000,0000,0000,,[and] as x goes to negative infinity, \Ny is going to go to positive infinity,
Dialogue: 0,0:16:07.17,0:16:11.48,Default,,0000,0000,0000,,because this is a six-degree term and\Nthe leading coefficient is positive 1.
Dialogue: 0,0:16:13.45,0:16:15.75,Default,,0000,0000,0000,,Okay, what about the zeros?
Dialogue: 0,0:16:15.75,0:16:18.93,Default,,0000,0000,0000,,Just write those out, \Nand the multiplicity.
Dialogue: 0,0:16:22.60,0:16:25.19,Default,,0000,0000,0000,,So, the first zero is zero.
Dialogue: 0,0:16:27.22,0:16:29.63,Default,,0000,0000,0000,,A multiplicity three, that's odd.
Dialogue: 0,0:16:29.63,0:16:35.83,Default,,0000,0000,0000,,The next one is x equals negative 2, \Nmultiplicity there is one, it's odd,
Dialogue: 0,0:16:35.83,0:16:39.02,Default,,0000,0000,0000,,so it's going to go through \Nboth of those points.
Dialogue: 0,0:16:39.02,0:16:43.36,Default,,0000,0000,0000,,And then x equals 3, \Nand that's even.
Dialogue: 0,0:16:43.36,0:16:46.79,Default,,0000,0000,0000,,So, it's going to be \Ntangent of x equals 3.
Dialogue: 0,0:16:48.46,0:16:50.68,Default,,0000,0000,0000,,Let's sketch a graph.
Dialogue: 0,0:16:54.87,0:16:56.91,Default,,0000,0000,0000,,All right, so let's see…
Dialogue: 0,0:16:56.91,0:17:01.11,Default,,0000,0000,0000,,how about at x equals zero,\Nthere's our point.
Dialogue: 0,0:17:01.47,0:17:05.81,Default,,0000,0000,0000,,Negative 2, there's our point, \Nand positive 3.
Dialogue: 0,0:17:06.58,0:17:10.17,Default,,0000,0000,0000,,Then our end behavior, \Nwe know it's got to be
Dialogue: 0,0:17:10.17,0:17:14.24,Default,,0000,0000,0000,,something like this and\Nsomething like this.
Dialogue: 0,0:17:15.79,0:17:20.19,Default,,0000,0000,0000,,Then what happens here at negative—\Nlet's take care of the leftmost one first—
Dialogue: 0,0:17:20.19,0:17:22.94,Default,,0000,0000,0000,,negative 2 is odd multiplicity, right?
Dialogue: 0,0:17:22.94,0:17:27.66,Default,,0000,0000,0000,,So we know it must come down\Nand go through this point.
Dialogue: 0,0:17:29.07,0:17:30.69,Default,,0000,0000,0000,,What happens at zero?
Dialogue: 0,0:17:30.69,0:17:35.91,Default,,0000,0000,0000,,It's odd multiplicity, so it's going to \Ncome down and then go back up.
Dialogue: 0,0:17:36.82,0:17:40.49,Default,,0000,0000,0000,,I don't know what's going on\Nwith my Wacom board here.
Dialogue: 0,0:17:40.49,0:17:42.72,Default,,0000,0000,0000,,We might have to reboot.
Dialogue: 0,0:17:43.17,0:17:46.04,Default,,0000,0000,0000,,Okay, it’s misbehaving.
Dialogue: 0,0:17:47.37,0:17:49.40,Default,,0000,0000,0000,,Okay, so it's down and \Nit's going to go up.
Dialogue: 0,0:17:49.40,0:17:51.62,Default,,0000,0000,0000,,Why? Because this\Nis odd multiplicity.
Dialogue: 0,0:17:51.62,0:17:53.58,Default,,0000,0000,0000,,Then what about this zero?
Dialogue: 0,0:17:53.58,0:17:56.07,Default,,0000,0000,0000,,Well, there are no \Nother zeros, okay?
Dialogue: 0,0:17:56.07,0:17:59.53,Default,,0000,0000,0000,,And at 3, it has \Nmultiplicity two.
Dialogue: 0,0:17:59.53,0:18:01.68,Default,,0000,0000,0000,,So, you know it \Nmust come down—
Dialogue: 0,0:18:01.68,0:18:06.72,Default,,0000,0000,0000,,excuse me, it must come down to this, \Ntouch, and then zoom back up.
Dialogue: 0,0:18:07.13,0:18:09.67,Default,,0000,0000,0000,,It's going to look \Nsomething like that.
Dialogue: 0,0:18:11.22,0:18:13.12,Default,,0000,0000,0000,,Aren't these fun?
Dialogue: 0,0:18:13.35,0:18:19.17,Default,,0000,0000,0000,,Okay, let's try that in Desmos\Nto see if we're close.
Dialogue: 0,0:18:19.72,0:18:30.70,Default,,0000,0000,0000,,So, x to the third, x plus 2—\Ngotta get rid of this exponent here—
Dialogue: 0,0:18:32.77,0:18:36.23,Default,,0000,0000,0000,,and then x— gotta get rid of that\Nexponent— I don't know, yeah,
Dialogue: 0,0:18:36.23,0:18:45.20,Default,,0000,0000,0000,,you shouldn't have that in there— \Nx minus 3, that's to the second power.
Dialogue: 0,0:18:46.98,0:18:50.24,Default,,0000,0000,0000,,Okay, let's see \Nwhat we have here.
Dialogue: 0,0:18:50.65,0:18:52.13,Default,,0000,0000,0000,,Do I have that typed in right?
Dialogue: 0,0:18:52.13,0:18:56.92,Default,,0000,0000,0000,,X to the third times x plus \N2 times x minus 3 squared.
Dialogue: 0,0:18:56.92,0:19:00.68,Default,,0000,0000,0000,,Okay, let's see if \Nwe're in the ball park.
Dialogue: 0,0:19:02.34,0:19:05.57,Default,,0000,0000,0000,,Okay so, there appears to\Nbe something wrong.
Dialogue: 0,0:19:05.57,0:19:07.52,Default,,0000,0000,0000,,Oh, I just don’t— \Nyou can't see enough of it.
Dialogue: 0,0:19:07.52,0:19:10.51,Default,,0000,0000,0000,,We have to change the scale so \Nwe can see more of the graph,
Dialogue: 0,0:19:10.51,0:19:11.99,Default,,0000,0000,0000,,so let's fix this.
Dialogue: 0,0:19:11.99,0:19:13.58,Default,,0000,0000,0000,,The problem is we’re \Nzoomed in too close.
Dialogue: 0,0:19:13.58,0:19:15.14,Default,,0000,0000,0000,,There we go.
Dialogue: 0,0:19:15.14,0:19:19.37,Default,,0000,0000,0000,,Okay, look, everything interesting \Nhappens between negative 3 and 4,
Dialogue: 0,0:19:19.37,0:19:22.08,Default,,0000,0000,0000,,so let's change that.
Dialogue: 0,0:19:22.40,0:19:26.43,Default,,0000,0000,0000,,Just trying to get this so\Nwe get a better picture.
Dialogue: 0,0:19:27.96,0:19:31.50,Default,,0000,0000,0000,,Then we need this\Ny and x to be bigger.
Dialogue: 0,0:19:31.50,0:19:33.93,Default,,0000,0000,0000,,So, how about—\Nagain, I don't know how big—
Dialogue: 0,0:19:33.93,0:19:35.46,Default,,0000,0000,0000,,So, let’s go to negative—
Dialogue: 0,0:19:35.46,0:19:37.90,Default,,0000,0000,0000,,Nope, gotta go bigger\Nthan that, negative 40.
Dialogue: 0,0:19:37.90,0:19:39.30,Default,,0000,0000,0000,,There we go.
Dialogue: 0,0:19:39.30,0:19:42.91,Default,,0000,0000,0000,,See how you just gotta play around with it, \Nso you can see what you need to see.
Dialogue: 0,0:19:42.91,0:19:45.72,Default,,0000,0000,0000,,Let's go 50 here.\NOh, there we go.
Dialogue: 0,0:19:45.72,0:19:47.80,Default,,0000,0000,0000,,Look at that.\NBeautiful.
Dialogue: 0,0:19:48.85,0:19:49.48,Default,,0000,0000,0000,,Okay.
Dialogue: 0,0:19:49.48,0:19:52.02,Default,,0000,0000,0000,,Does that look like \Nwhat we drew?
Dialogue: 0,0:19:52.02,0:19:53.61,Default,,0000,0000,0000,,Sure does.
Dialogue: 0,0:19:53.61,0:19:54.80,Default,,0000,0000,0000,,Pretty close.
Dialogue: 0,0:19:54.80,0:19:58.78,Default,,0000,0000,0000,,It comes down, \Nodd multiplicity,
Dialogue: 0,0:19:58.78,0:20:00.43,Default,,0000,0000,0000,,odd multiplicity,
Dialogue: 0,0:20:00.43,0:20:04.00,Default,,0000,0000,0000,,and then even multiplicity, \Nso it's tangent here.
Dialogue: 0,0:20:04.12,0:20:06.22,Default,,0000,0000,0000,,Isn’t it beautiful?
Dialogue: 0,0:20:06.70,0:20:10.90,Default,,0000,0000,0000,,Polynomials are beautiful because \Nthey're smooth and continuous.
Dialogue: 0,0:20:15.40,0:20:21.91,Default,,0000,0000,0000,,Okay, so believe it or not,\Nthere's 3.2 part F.
Dialogue: 0,0:20:21.91,0:20:23.62,Default,,0000,0000,0000,,There's another part to this.
Dialogue: 0,0:20:23.62,0:20:27.28,Default,,0000,0000,0000,,We're going to stop right there\Nand we'll have one last part.
Dialogue: 0,0:20:27.28,0:20:29.60,Default,,0000,0000,0000,,Okay, bye, see you shortly.