WEBVTT

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College Algebra students, 
Dillon here.

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We are continuing on
with Section 3.2.

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Yes, it's a long one.
Let's get started.

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We've learned that there's a connection 
between the x-intercepts, also called zeros,

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and the factors that make up 
a particular polynomial.

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We've learned that if the

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polynomial has the factor, 
there's a corresponding zero.

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Here's the last example we did.

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We started with this 
fourth-degree polynomial.

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We factored it; we factored out 
a negative x squared.

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We're left with a quadratic, 
which we then factored.

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We end up with four factors.

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Two factors of x: 
2x plus 3 and x minus 1.

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Each one of these contributed 
a zero (an x-intercept),

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and we learned
that the zero—

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that the factor to the second degree
has some interesting behavior.

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That's the last thing 
we finish up here.

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We're going to study how these 
polynomials behave around their zeros.

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There are two types of behavior:

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one where it actually goes through
the zero, one where it's tangent.

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I think it'll be instructive to
just look at this in Desmos.

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Not that my graph is not—

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it shows all the important details, 
but it looks a little nicer in Desmos.

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Here we go: y equals negative
2x to the fourth power.

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That's our leading term, our 
leading coefficient is negative 2.

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And remember, these monomial terms, 
these monomial leading terms,

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they determine the 
end behavior of the polynomial.

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They dominate the end behavior.

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Okay, there it is, a slightly more
attractive version of the graph.

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We can see that the 
two types of behavior here

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that we're going to study a bit
more to finish up this section.

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Where it goes— the graph—

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Excuse me, the polynomial goes 
through the x-intercept (the zero),

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if the degree of that factor is odd.

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So, in this case the 
degree of 2x plus 3 is 1.

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That's an odd number.

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The degree of x minus 1 
is to the first power.

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So, notice for both of those zeros, 
the graph goes through the points.

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The zero that had an
even degree (second degree),

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we call this multiplicity two.

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The other two factors 
have multiplicity one.

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This factor x has 
multiplicity two.

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And you'll notice that the 
graph is tangent at that point,

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so the degree of the factor, 
also called the multiplicity,

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is going to determine 
how it behaves.

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Okay, so just to practice this 
sketching the graph one more time,

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let's do another one.

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And I like this one because the 
factoring is a little different.

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Something we haven't
seen yet in this section.

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Okay, right off the bat,

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you notice there's no greatest 
common factor to take out here.

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And you have to go back to the
polynomial factoring methods

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that we learned probably 
in the previous class

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and realize that you
have four terms here.

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You'll notice that there's 
a common term in each.

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If we group these
two together,

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and group these two together,

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there's a common term
you can take out.

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Let's take that out.

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I'm going to take an x squared 
out of the first two terms.

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This is called factoring 
by grouping, by the way.

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Then what's left in just
the first two terms?

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I'm ignoring these last two.

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I pull that out and 
I get x minus 2.

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Notice that there's a common 
factor we can pull out here.

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When the leading coefficient is negative, 
you should always pull out a negative.

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So we're going to
pull out a negative four.

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What's left?
X minus 2.

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And that's what 
we're looking for.

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We needed a common factor, 
a common binomial factor

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that we can now take out of both 
of those, so take out the x minus 2

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and we're left with
x squared minus 4.

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I hope you recognize 
that the second term here

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is actually the difference
of two squares.

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We're going to
have this x minus 2

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and then the difference of two squares 
factors to x plus 2 and x minus 2.

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We're going to 
rewrite it like this

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because we're going to talk more 
about this idea of multiplicity.

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The x minus 2 is actually
to the second power.

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This—

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The x-intercept 
has multiplicity two.

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We just learned how
that's going to behave.

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It's going to be tangent 
at the point x equals 2.

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And then we have the 
second zero here: x plus 2.

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Okay, Let's draw a graph,
sketch a graph,

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somewhat accurate.

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Then we'll do it
in Desmos, also.

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Okay, so the interesting points are— 
from this factor, we get x equals 2.

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We know it's going to be tangent there; 
we'll figure that out in a second.

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The second factor
gives us x minus 2.

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And then let's look at 
this leading term.

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It’s an odd degree.

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The leading coefficient
is positive.

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So, you know, the end behavior
is like this, in this direction,

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and like this 
in this direction.

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As x goes to positive infinity, 
we know y goes to positive infinity.

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As x goes to negative infinity, 
because it's odd degree,

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the y value is going to 
go to negative infinity.

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That's the end behavior.

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Then what happens in 
between these two?

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Now remember, these are 
the only two x-intercepts.

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That's important.

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We know the graph must
come up and go through this one

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(this has odd multiplicity: 
x plus 2 to the first power),

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and then it's going to come up;

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we know it has to go through this point, 
but it doesn't actually go through.

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It's going to come down 
here and it's tangent,

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and then it takes off
in that direction.

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It's going to look 
something like that.

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Let's just graph it in a Desmos,

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so we have a slightly 
more attractive version of it.

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Let's see… we're 
going to get rid of this.

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Change that to 
the third power,

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so minus 2x to the 
second power minus 4x plus 8.

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Our scale is a 
little bit off here.

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Let's just fix this.

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We’re going to fix this maximum 
y value to something a little bigger.

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I don't really know where it is.

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I'm going to try 50 and see.
That's too big.

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How about 20?
There we go.

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Okay, so that's similar to 
what we sketched here.

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All right, a little
prettier in Desmos.

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And we can see that the zero 
that has multiplicity two—

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that's our vocabulary:

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"multiplicity two" because that factor
is raised to the second power—

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the polynomial is
tangent at that point.

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Whereas the zero with 
multiplicity one (an odd number),

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it actually goes through
the x-intercept.

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Okay, we're going to do a
few more examples of that

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to finish up the section here.

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Here we go.

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Everything I just said is 
summarized in this box.

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You might want to pause the 
video and take a closer look.

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If the multiplicity—

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and the multiplicity is determined
by the power of the factor.

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Each factor has a 
power one or greater.

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If it's an odd multiplicity, 
then the x-intercept is

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going to behave like
this for the polynomial.

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If it's even multiplicity 
it behaves like this:

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either tangent from the top 
or tangent from the bottom.

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Then you might ask, “Well how do
you decide?” Well, look up here.

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We knew what the end behavior
of the polynomial was, okay?

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We knew it was
coming up here.

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It has to go through this intercept, 
and then we're above the x-axis here.

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And we know it's got to
be tangent at that point.

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This is the only option, we know it 
doesn't have any other x-intercepts,

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so it can't go 
through the x-axis.

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You just use a 
little bit of logic.

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Okay, let's graph this one.

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I think these are kinda fun.

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The first thing we usually do is— 
so notice this is factored for us.

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That's nice, right?

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We're going to list the zero and 
the multiplicity, then we'll graph it.

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The first zero is x equals zero, 
multiplicity four.

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And notice that that's 
an even number.

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The second zero from this factor, 
we get x equals two, multiplicity three.

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That's odd, so it's going
to go through that point.

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Then x equals negative 1.

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Again, even multiplicity, so we know
it's going to be tangent at that point.

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This is a pretty funky graph.

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Now, what is the leading term
going to look like here?

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Well, if you multiply this out—

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this is kind of a crazy polynomial, 
but we can do it.

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If you multiply this out, it's going
to be a third-degree polynomial

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times the fourth-degree term
is going to be seventh degree.

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Multiply this out, it's 
going to be a second degree,

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so seventh degree 
times second degree.

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We're going to
have x to the ninth.

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Yikes, that's crazy, right?

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This is a third-degree polynomial 
times the fourth-degree term.

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That gives us
x to the seventh.

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Then if this is multiplied out, 
it's going to be squared.

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So we have x to the ninth.

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Notice the leading coefficient
is going to be 1,

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because all of the leading 
coefficients here are 1.

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And we know if this is odd degree,
it's going to behave like this,

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and like this with
stuff in the middle.

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Odd degree, just like x to the third, 
x to the ninth behaves like x to the third.

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Okay, here we go.

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This is a pretty 
fancy polynomial.

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I'm not sure why I'm using
blue here all of a sudden.

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Should’ve changed color.

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Okay.

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So here are intercepts: zero, 
positive 2 and negative 1.

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Okay, and then we know
the end behavior

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because we know the leading
term and the leading coefficient.

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End behavior has to be like this

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and like this.

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Okay, so what happens 
at this leftmost zero?

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Well, it's got to be tangent there
because it's even multiplicity.

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So it has to come up and touch
here and then come back down.

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What happens at zero?

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Well, it's also even multiplicity.

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It's going to come 
down and be like this.

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It's got to be tangent here, 
and we're below the x-axis.

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And we know it's tangent there.

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So it can't go through and 
come back down and touch,

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it’s gotta be something like that.

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And then the zero of
multiplicity three is at 2.

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Then we know this has to come
down and go back up and out.

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It's going to look 
something like that.

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We don't know how
big this loop is here,

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but let's put it in Desmos.

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Should be quite interesting, 
I would think.

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If you're looking for an
idea for a tattoo, here it is.

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This is a pretty
cool polynomial.

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Just trying to get 
both on the screen.

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Okay, so x minus 2
raised to the third power.

00:13:29.393 --> 00:13:31.976
It's a pretty fancy polynomial.

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X plus 1 is to the second power.

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Ooh, all right, so…

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Our graph, it's close.

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Okay, let's see.

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We didn't know how— we knew— 
Let's see if we can—

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I'm going to change the scale a little bit
so we can see this detail a little better.

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Negative 2 to 3.

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That's not going to 
change it much.

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Then we're going to
change the y scale.

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Let's make this…

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Let's see, on the negative side,

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we could go to negative 6 
maybe, and let's go with 2.

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Yeah, there we go.

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Okay.

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It's different but it 
has the same features.

00:14:23.429 --> 00:14:25.913
It’s doing the same 
things we expected.

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We come up here,
we’re tangent at negative 1;

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we come back down, 
we’re tangent at zero.

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If we click on the graph,

00:14:32.560 --> 00:14:35.180
we can prove that what 
we have is accurate.

00:14:35.180 --> 00:14:37.767
Okay, tangent at negative 1, 
tangent at zero.

00:14:37.767 --> 00:14:39.130
We have little dip here,

00:14:39.130 --> 00:14:42.033
we didn't know how far that went, 
but that's what it looks like.

00:14:42.033 --> 00:14:43.415
And then we have another dip,

00:14:43.415 --> 00:14:48.981
and then it goes up
through 2 (x equals 2).

00:14:49.816 --> 00:14:53.847
So, our sketch was pretty accurate, 
but not as pretty, of course.

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Okay, I would like you to 
pause the recording please,

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and try number 29.

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And then we'll do it together.

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All right.

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I hope you tried it.

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First of all, what's the
leading term here?

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Let's figure that out.

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The leading term, let's see.

00:15:26.482 --> 00:15:30.233
This is going to be a second-degree
polynomial times another first-degree

00:15:30.233 --> 00:15:34.016
(so that’s going to be third), 
multiplied by third, so this gonna be—

00:15:34.016 --> 00:15:39.368
The leading coefficient is 1; 
the leading term is x to the sixth.

00:15:39.668 --> 00:15:41.905
Let me just repeat that, 
you multiply this out,

00:15:41.905 --> 00:15:43.621
that's going to be
second-degree,

00:15:43.621 --> 00:15:46.553
times another x is 
going to be third-degree,

00:15:46.553 --> 00:15:49.488
times x to the third is 
going to be x to the sixth.

00:15:49.488 --> 00:15:55.352
So, we know the end behavior
of this stuff in the middle is

00:15:55.352 --> 00:15:58.903
going to look like that.
It's going to go up:

00:15:58.903 --> 00:16:02.502
as x goes to positive infinity,
y is going to go to positive infinity

00:16:02.502 --> 00:16:07.169
[and] as x goes to negative infinity, 
y is going to go to positive infinity,

00:16:07.169 --> 00:16:11.484
because this is a six-degree term and
the leading coefficient is positive 1.

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Okay, what about the zeros?

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Just write those out, 
and the multiplicity.

00:16:22.600 --> 00:16:25.186
So, the first zero is zero.

00:16:27.217 --> 00:16:29.633
A multiplicity three, that's odd.

00:16:29.633 --> 00:16:35.832
The next one is x equals negative 2, 
multiplicity there is one, it's odd,

00:16:35.832 --> 00:16:39.017
so it's going to go through 
both of those points.

00:16:39.017 --> 00:16:43.357
And then x equals 3, 
and that's even.

00:16:43.357 --> 00:16:46.790
So, it's going to be 
tangent of x equals 3.

00:16:48.457 --> 00:16:50.682
Let's sketch a graph.

00:16:54.874 --> 00:16:56.906
All right, so let's see…

00:16:56.906 --> 00:17:01.106
how about at x equals zero,
there's our point.

00:17:01.473 --> 00:17:05.809
Negative 2, there's our point, 
and positive 3.

00:17:06.575 --> 00:17:10.173
Then our end behavior, 
we know it's got to be

00:17:10.173 --> 00:17:14.239
something like this and
something like this.

00:17:15.789 --> 00:17:20.190
Then what happens here at negative—
let's take care of the leftmost one first—

00:17:20.190 --> 00:17:22.941
negative 2 is odd multiplicity, right?

00:17:22.941 --> 00:17:27.658
So we know it must come down
and go through this point.

00:17:29.074 --> 00:17:30.691
What happens at zero?

00:17:30.691 --> 00:17:35.908
It's odd multiplicity, so it's going to 
come down and then go back up.

00:17:36.824 --> 00:17:40.487
I don't know what's going on
with my Wacom board here.

00:17:40.487 --> 00:17:42.724
We might have to reboot.

00:17:43.173 --> 00:17:46.043
Okay, it’s misbehaving.

00:17:47.371 --> 00:17:49.405
Okay, so it's down and 
it's going to go up.

00:17:49.405 --> 00:17:51.625
Why? Because this
is odd multiplicity.

00:17:51.625 --> 00:17:53.580
Then what about this zero?

00:17:53.580 --> 00:17:56.069
Well, there are no 
other zeros, okay?

00:17:56.069 --> 00:17:59.534
And at 3, it has 
multiplicity two.

00:17:59.534 --> 00:18:01.685
So, you know it 
must come down—

00:18:01.685 --> 00:18:06.718
excuse me, it must come down to this, 
touch, and then zoom back up.

00:18:07.133 --> 00:18:09.667
It's going to look 
something like that.

00:18:11.217 --> 00:18:13.118
Aren't these fun?

00:18:13.354 --> 00:18:19.168
Okay, let's try that in Desmos
to see if we're close.

00:18:19.718 --> 00:18:30.700
So, x to the third, x plus 2—
gotta get rid of this exponent here—

00:18:32.767 --> 00:18:36.228
and then x— gotta get rid of that
exponent— I don't know, yeah,

00:18:36.228 --> 00:18:45.198
you shouldn't have that in there— 
x minus 3, that's to the second power.

00:18:46.980 --> 00:18:50.245
Okay, let's see 
what we have here.

00:18:50.652 --> 00:18:52.128
Do I have that typed in right?

00:18:52.128 --> 00:18:56.915
X to the third times x plus 
2 times x minus 3 squared.

00:18:56.915 --> 00:19:00.681
Okay, let's see if 
we're in the ball park.

00:19:02.336 --> 00:19:05.568
Okay so, there appears to
be something wrong.

00:19:05.568 --> 00:19:07.520
Oh, I just don’t— 
you can't see enough of it.

00:19:07.520 --> 00:19:10.510
We have to change the scale so 
we can see more of the graph,

00:19:10.510 --> 00:19:11.993
so let's fix this.

00:19:11.993 --> 00:19:13.584
The problem is we’re 
zoomed in too close.

00:19:13.584 --> 00:19:15.141
There we go.

00:19:15.141 --> 00:19:19.369
Okay, look, everything interesting 
happens between negative 3 and 4,

00:19:19.369 --> 00:19:22.078
so let's change that.

00:19:22.400 --> 00:19:26.431
Just trying to get this so
we get a better picture.

00:19:27.965 --> 00:19:31.498
Then we need this
y and x to be bigger.

00:19:31.498 --> 00:19:33.932
So, how about—
again, I don't know how big—

00:19:33.932 --> 00:19:35.465
So, let’s go to negative—

00:19:35.465 --> 00:19:37.898
Nope, gotta go bigger
than that, negative 40.

00:19:37.898 --> 00:19:39.300
There we go.

00:19:39.300 --> 00:19:42.907
See how you just gotta play around with it, 
so you can see what you need to see.

00:19:42.907 --> 00:19:45.716
Let's go 50 here.
Oh, there we go.

00:19:45.716 --> 00:19:47.800
Look at that.
Beautiful.

00:19:48.849 --> 00:19:49.483
Okay.

00:19:49.483 --> 00:19:52.015
Does that look like 
what we drew?

00:19:52.015 --> 00:19:53.614
Sure does.

00:19:53.614 --> 00:19:54.798
Pretty close.

00:19:54.798 --> 00:19:58.775
It comes down, 
odd multiplicity,

00:19:58.775 --> 00:20:00.430
odd multiplicity,

00:20:00.430 --> 00:20:04.003
and then even multiplicity, 
so it's tangent here.

00:20:04.118 --> 00:20:06.218
Isn’t it beautiful?

00:20:06.702 --> 00:20:10.899
Polynomials are beautiful because 
they're smooth and continuous.

00:20:15.398 --> 00:20:21.914
Okay, so believe it or not,
there's 3.2 part F.

00:20:21.914 --> 00:20:23.615
There's another part to this.

00:20:23.615 --> 00:20:27.281
We're going to stop right there
and we'll have one last part.

00:20:27.281 --> 00:20:29.598
Okay, bye, see you shortly.