Flux and magnetic flux

WEBVTT
Kind: captions
Language: en

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- [Voiceover] What I
wanna do in this video

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is give ourselves an introduction

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or an intuition for the
term Flux in general.

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And then think about how it applies

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to the idea of magnetic

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Magnetic Flux.

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So, first of all, when people
are just talking about flux,

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and this is the easiest way that I know

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how to conceptualize it.

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They're talking about
how much of something

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is flowing through a surface
in a given amount of time.

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So if you imagine that this is,

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this is, I'm just defining a
volume of air right over here.

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And let's say the air is
denser near the bottom

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of this volume of air,

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so there's more air down here

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than there is up here,

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which is generally true.

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Air density goes down as
you increase altitude.

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So there's very low density up here.

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This is in between.

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I don't have to draw
all the air particles,

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but you get the sense.

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Lot of air, lot of air down here.

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And the air is moving.

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And so let's say that the air,

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let's say the air is...

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Let me do the velocity
vectors in a different color.

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So, these are some of the
velocity vectors of the air.

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Let's say the air on this side is moving

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is moving at let's say, a medium velocity.

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So right over there.

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But as we move more in that direction,

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the air is moving faster.

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So they have larger
velocity vectors like that.

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We see that in general,
all of the air is moving

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in that general direction,
that's the way I'm drawing it.

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And I can draw the
velocity vectors over here.

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The air is less dense.

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But the trend in the velocity vectors,

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when I go from the left to the right is

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roughly the same.

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So that's the flow.

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Now I'm just sampling some of
the velocity vectors there.

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So let's talk about flux.

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And the thing about flux,

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really of any form,

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you have to think about a surface.

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So let's imagine you were
to put some type of a net.

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Let's say you were to put a net right over

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right over here, right over here.

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And if we think about the flux,

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we would say, well how much air

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is traveling through that net
in a certain amount of time?

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We could say how many molecules

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are traveling in say, each second.

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And so this would have some
flux associated with it.

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Relative to the air.

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But what if we were to take that same net.

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And we're assuming it's
some type of theoretical net

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that actually does not
impede the air flow.

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But it helps us visualize a surface.

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What if we were to move that
net a little bit to the right

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where the density is the same,

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but the particles are just moving faster.

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Well now, our flux would increase.

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So this is larger flux.

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Larger, larger flux.

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Why is our flux increase?

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Because, well, the density is the same,

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but in any given amount of time

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I'm gonna have more things
going through that surface.

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Now what if I were to put that same net

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and move it up to this high altitude,

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right over here?

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That high altitude.

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Well, the velocity of the
molecules are the same,

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and they're going in the same direction.

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But there's just fewer of them.

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So you're going to have
less fewer molecules

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traveling through the surface
in a given amount of time.

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So this is going to have smaller flux.

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And this is all relative to my first one.

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Smaller, smaller flux.

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Now, what if you were to take the same net

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and instead of the direction of the air

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being perpendicular to the surface,

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be normal to the surface,

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what if you were to take the net

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and reorientate it so
that the air is going

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in the same direction of the surface.

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So what if you were to take the same net

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and you were to make it like this.

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So that it's the same net.

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I know it doesn't look exactly the same.

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But it's the same net,

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and you're to make it like this.

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Well now how much air is
traveling through that net

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in a given amount of time?

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Well now very little to zero air

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is gonna be traveling through that net

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in any given amount of time.

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The air is going along the surface,

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not through the surface.

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So this one, let's just say that

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all the air is going exactly

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in the same direction as the surface,

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so nothing is going through the surface.

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We would say that this one has zero,

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zero flux.

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Now let's say that this theoretical net

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that actually does not
impede the air flow,

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let's say we can stretch
it or contract it.

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So if we stretch that net,

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in the same, let's say we were to do it,

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let's say we were to
stretch it up like this.

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So it becomes a bigger net.

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So it becomes a bigger net like this.

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Well now this thing's
gonna have larger flux

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because there's just more area.

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There's more to flow through.

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So now, if you said for this surface,

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you're gonna have a larger flux

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because there's just gonna be more air

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is going to go through that

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in any given unit of time.

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So as you can see,

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when we think about how much of something

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just flux in the traditional sense.

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How much of something
goes through a surface

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in a given unit of time.

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It depends in this case, the
density of the substance.

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It depends on

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it's velocity.

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Both the magnitude of the velocity

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and the direction of the velocity.

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We see if we orient the surface

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or if we oriented the velocity.

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So it's not going normal to the surface,

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perpendicular to the surface.

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Well then, we can have our flux go down.

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And you can have things in between.

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You can have a net.

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Let's say we took that same original net.

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But the direction of the
air is neither normal

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nor exactly in the same
direction of the surface.

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Well this flux is going to be in between,

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is going to be in between
that original flux.

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And this one right over here.

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There's gonna be air flowing through.

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There is going to be air

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flowing through

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the actual surface.

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But it's not going exactly
normal to the surface.

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And as we will explore later

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when we get a little
bit more mathy into it.

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We actually care about the component

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of the vector of the air
that is exactly normal

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when we eventually calculate flux.

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But this flux is gonna
be some place in between

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this one and the zero flux

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because the air isn't going
exactly perpendicular.

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Well that's just the more basic,

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or for me the more
intuitive notion of flux.

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But what do we mean by magnetic,

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by magnetic flux?

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Well, like regular flux,

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we're still dealing with
how things are kind of,

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you could say going through a surface.

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But instead of thinking
about air particles or

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water molecules or things like that,

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we're gonna be thinking
about a magnetic field.

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So let me draw a surface.

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So I have, I have a little bar magnet.

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This is the north side,
this is the south side.

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We see our field lines.

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And then I've drawn a
couple of the magnetic

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field vectors in white there.

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And let's say that we have a surface.

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We have a surface like this.

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And so, for this surface

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when we think about the flux

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we wanna care about...

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things are actually moving when
we think about magnetic flux

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we aren't actually thinking about actual

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physical things moving
through the surface.

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The way we did when we thought about,

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I guess you could say,
traditional or more,

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or flow-based flux.

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But it's a similar idea.

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We care about the component

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of the magnetic field.

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And the density of the magnetic field

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that is normal to this surface.

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So let's say that this has a given flux.

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So let's call that, and
the notation is phy.

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And, that's the flux
of the magnetic field.

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And once again, it's based on,

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it's based on the strength
of the magnetic field

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and specially the component
of the magnetic field vectors

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that are perpendicular,

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that are perpendicular
to this actual surface.

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So this would have one magnetic flux.

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But if I were to take the same surface

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and make it parallel,

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make it parallel to the
magnetic field vecotrs

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instead of being normal or...

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Now the magnetic field
vectors are parallel to it

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instead of being normal.

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So now our flux is zero.

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So this would have zero,
or pretty close to zero.

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So approximately zero

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magnetic flux.

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Magnetic flux.

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Flux.

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Now if I were to take this surface

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and instead of orienting this way

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I just moved it further away.

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So instead of putting in here

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I would take it all the way out here

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where the magnetic field is weaker.

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Where the magnetic field is weaker.

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So we have a weaker
magnetic field out here.

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The flux would also decrease.

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So it has a lot of the same properties

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as this more physical flux

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that we were talking about before.

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But instead of talking about, say,

00:08:37.084 --> 00:08:39.058
the velocity of the air molecules,

00:08:39.058 --> 00:08:42.054
no, the velocity vectors
of the air molecules

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and how those relate to the surface,

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here we're talking about
the magnetic field vectors.

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And how those relate to the surface.

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But the metaphor, the
analogy is still the same.

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If they are perpendicular,
you have larger flux.

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If they are going in the same direction,

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you might have zero or very little.

00:09:01.721 --> 00:09:03.590
You might have very little flux.

00:09:03.590 --> 00:09:07.154
If you have a weak
magnetic field out here,

00:09:07.154 --> 00:09:08.292
the flux is going to be lower

00:09:08.292 --> 00:09:11.694
than if you have a strong magnetic field.

00:09:11.694 --> 00:09:12.855
That's analogous to,

00:09:12.855 --> 00:09:15.386
when you had high density
and high velocity,

00:09:15.386 --> 00:09:16.883
that was a lot of flux.

00:09:16.883 --> 00:09:18.683
Versus low density or low velocity

00:09:18.683 --> 00:09:20.657
that is lower flux.

00:09:20.657 --> 00:09:23.884
And also, we can increase
the total surface.

00:09:23.884 --> 00:09:25.394
So if we stretch that surface

00:09:25.394 --> 00:09:27.356
or we had a larger
surface right over here,

00:09:27.356 --> 00:09:29.956
the flux through this
surface is gonna be larger

00:09:29.956 --> 00:09:31.814
than the flux through this surface.

00:09:31.814 --> 00:09:33.521
Let's say this surface is analogous

00:09:33.521 --> 00:09:36.121
to a surface like this right over here.

00:09:36.121 --> 00:09:38.792
Let's say that the magnetic
field is symmetric on both sides

00:09:38.792 --> 00:09:40.742
so the flux through this surface

00:09:40.742 --> 00:09:42.588
and this surface would be the same.

00:09:42.588 --> 00:09:44.260
So if you were to stretch it out

00:09:44.260 --> 00:09:46.222
to be a larger surface,

00:09:46.222 --> 00:09:49.821
well now you just have
more of the magnetic field

00:09:49.821 --> 00:09:52.991
that is now normal to the surface.

00:09:52.991 --> 00:09:55.254
So hopefully this gives you
at least a beginning ideas

00:09:55.254 --> 00:09:57.751
of the notion of magnetic flux.

00:09:57.751 --> 00:09:59.086
And how it relates to,

00:09:59.086 --> 00:10:01.240
I guess you could think
of more physical flux.


Plans for biology

WEBVTT
Kind: captions
Language: en

00:00:00.260 --> 00:00:01.700
- [Voiceover] So this is Sal Khan here

00:00:01.700 --> 00:00:06.204
with Emily Abrash, our
Biology Content Fellow.

00:00:06.204 --> 00:00:07.505
And you have a PhD in Biology,

00:00:07.505 --> 00:00:09.409
so you know what you're talking about.

00:00:09.409 --> 00:00:10.608
- [Voiceover] You'd like to
think that, wouldn't you.

00:00:10.608 --> 00:00:12.251
(laughter)

00:00:12.558 --> 00:00:14.718
- [Voiceover] And so, Khan Academy has had

00:00:14.718 --> 00:00:16.947
biology content for a long time now.

00:00:16.947 --> 00:00:19.385
You know, back in the
day, I think in 2007, 2008

00:00:19.385 --> 00:00:21.660
I had made a handful of biology videos.

00:00:21.660 --> 00:00:24.609
But what are we hoping to do now?

00:00:24.609 --> 00:00:26.095
- [Voiceover] I mean, I
think here what we're really

00:00:26.095 --> 00:00:29.206
going for is having really
comprehensive coverage.

00:00:29.206 --> 00:00:31.598
Really making sure that
we look at all of the

00:00:31.598 --> 00:00:33.409
different branches of biology

00:00:33.409 --> 00:00:36.033
and have great material
that's going to be a resource

00:00:36.033 --> 00:00:39.005
for someone who's looking
at any one of those.

00:00:39.005 --> 00:00:41.257
You know, not just certain
things that happen to be cool,

00:00:41.257 --> 00:00:44.043
but really digging into a
lot of different branches

00:00:44.043 --> 00:00:45.668
of this subject.

00:00:45.668 --> 00:00:47.219
- [Voiceover] And so how
are you thinking about it.

00:00:47.219 --> 00:00:49.964
I mean, you're out there,
trying to figure out

00:00:49.964 --> 00:00:51.891
what we should cover and
what we shouldn't cover.

00:00:51.891 --> 00:00:55.281
What is your hope for the next year?

00:00:55.281 --> 00:00:58.091
- [Voiceover] My hope is
really that we're going to have

00:00:58.091 --> 00:01:00.877
material that will serve
pretty much anybody

00:01:00.877 --> 00:01:03.454
who's in, like, an
advanced high school level

00:01:03.454 --> 00:01:06.124
or an intro college level class.

00:01:06.124 --> 00:01:08.400
And I mean, there's a lot of
diversity in those classes,

00:01:08.400 --> 00:01:10.768
so it's not going to match every class.

00:01:10.768 --> 00:01:13.508
But the hope would be that
if you encounter a topic

00:01:13.508 --> 00:01:15.807
that you're confused
about or interested in

00:01:15.807 --> 00:01:17.084
in one of those classes,

00:01:17.084 --> 00:01:18.918
that you'd be able to come to us

00:01:18.918 --> 00:01:22.026
and really learn more about that.

00:01:22.026 --> 00:01:23.558
- [Voiceover] And the way
we've been working together

00:01:23.558 --> 00:01:25.857
is fun, it's fun to have
people to work with now.

00:01:25.857 --> 00:01:27.134
(laughter)

00:01:27.134 --> 00:01:29.642
They never fit into the walk-in
closet, but now we have...

00:01:29.642 --> 00:01:32.544
And you know, you've been
doing a pretty incredible job.

00:01:32.544 --> 00:01:35.168
You've been obviously
taking a high level view,

00:01:35.168 --> 00:01:37.188
thinking about the different
gaps, what we need to fill in,

00:01:37.188 --> 00:01:38.883
to really have exhaustive coverage for

00:01:38.883 --> 00:01:42.876
intro college biology class,
or an AP biology class.

00:01:42.876 --> 00:01:47.335
You've been writing articles
like this one right over here,

00:01:47.335 --> 00:01:50.841
where you really get a hopefully
intuitive but thoughtful

00:01:50.841 --> 00:01:54.184
explanation and kind of deep explanation

00:01:54.184 --> 00:01:56.878
of what's going on.

00:01:56.878 --> 00:01:59.733
We're also working together
to produce more videos

00:01:59.733 --> 00:02:01.428
so that, once again, we don't just cover

00:02:01.428 --> 00:02:03.704
a lot of the main ideas in biology,

00:02:03.704 --> 00:02:06.165
we're covering all of the
things that you would see

00:02:06.165 --> 00:02:11.165
in an AP biology class or a college class.

00:02:11.529 --> 00:02:14.803
So, what's next? What are
you working on right now?

00:02:14.803 --> 00:02:17.264
- [Voiceover] Right now I am just sort of

00:02:17.264 --> 00:02:19.307
in the middle of genetics.

00:02:19.307 --> 00:02:22.326
You've already got a lot of
great videos on genetics,

00:02:22.326 --> 00:02:24.787
so I'm kind of looking
at what we already have,

00:02:24.787 --> 00:02:27.411
as well as sort of some
extensions of that.

00:02:27.411 --> 00:02:30.011
Like the sort of chromosomal
basis of genetics

00:02:30.011 --> 00:02:31.892
and how that was discovered.

00:02:31.892 --> 00:02:34.748
So that's kind of what's
on my plate right now,

00:02:34.748 --> 00:02:37.952
but after that, it will
kind of be full steam ahead

00:02:37.952 --> 00:02:41.365
into molecular biology, DNA replication,

00:02:41.365 --> 00:02:43.478
transcription and translation.

00:02:43.478 --> 00:02:45.815
Which again, we've already
got some great content on,

00:02:45.815 --> 00:02:47.789
but there are places
where I think we can even

00:02:47.789 --> 00:02:49.669
build it out some more.

00:02:49.669 --> 00:02:51.202
- [Voiceover] I'm excited,
I know I have a backlog

00:02:51.202 --> 00:02:53.385
of four or five videos
that you've been asking for

00:02:53.385 --> 00:02:55.288
for a few days, so they'll be coming soon.

00:02:55.288 --> 00:02:56.357
- [Voiceover] I'm excited for them!

00:02:56.357 --> 00:02:57.518
(laughter)

00:02:57.518 --> 00:02:59.509
- [Voiceover] Thank you.


Factoring perfect squares: missing values

WEBVTT
Kind: captions
Language: en

00:00:00.413 --> 00:00:02.247
- [Voiceover] The quadratic expression

00:00:02.247 --> 00:00:04.371
x-squared plus five x plus c

00:00:04.371 --> 00:00:06.218
is a perfect square.

00:00:06.218 --> 00:00:10.779
It can be factored as x plus d-squared.

00:00:10.779 --> 00:00:15.779
Both c and d are positive
rational numbers.

00:00:16.109 --> 00:00:17.644
What I wanna figure out in this video

00:00:17.644 --> 00:00:20.379
is what is c, given the information

00:00:20.379 --> 00:00:21.848
that we have right over here?

00:00:21.848 --> 00:00:24.111
What is c going to be equal to?

00:00:24.111 --> 00:00:26.619
And what is d going to be equal to?

00:00:26.619 --> 00:00:27.884
Like always, pause the video

00:00:27.884 --> 00:00:30.112
and see if you can figure it out.

00:00:30.182 --> 00:00:31.769
Let's work through this together.

00:00:31.769 --> 00:00:34.497
We're saying that x-squared
plus five x plus c

00:00:34.497 --> 00:00:38.223
can be rewritten as x plus d-squared.

00:00:38.223 --> 00:00:39.744
Let me write that down.

00:00:39.744 --> 00:00:42.414
So this part, this part,

00:00:42.414 --> 00:00:47.360
x-squared plus five x plus c,

00:00:47.360 --> 00:00:48.671
we're saying that,

00:00:48.671 --> 00:00:51.353
that could be written as x plus d-squared.

00:00:51.353 --> 00:00:55.799
This is equal to x plus d-squared.

00:00:55.799 --> 00:00:57.262
Now we can rewrite,

00:00:57.262 --> 00:00:59.863
x plus d-squared is going to be equal to

00:00:59.863 --> 00:01:01.982
x-squared plus

00:01:01.982 --> 00:01:06.982
two dx

00:01:08.032 --> 00:01:09.954
plus d-squared.

00:01:09.954 --> 00:01:12.544
If this step, right over
here, you find strange,

00:01:12.544 --> 00:01:14.689
I encourage you to watch the videos

00:01:14.689 --> 00:01:19.689
on squaring binomials or on
perfect square polynomials,

00:01:19.739 --> 00:01:21.701
either one, so you can see the pattern

00:01:21.701 --> 00:01:23.186
that this is going to be.

00:01:23.186 --> 00:01:26.913
X squared plus two times the product

00:01:26.913 --> 00:01:30.372
of both of these terms plus d-squared.

00:01:30.372 --> 00:01:31.876
When you look at it like this,

00:01:31.876 --> 00:01:35.301
you can start to pattern
match a little bit.

00:01:35.301 --> 00:01:40.270
You can say, alright,
five x, right over here,

00:01:40.270 --> 00:01:44.379
that is going to have
to be equal to two d,

00:01:44.379 --> 00:01:46.550
and then, you can also say,

00:01:46.550 --> 00:01:51.511
that c is going to have
to be equal to d-squared.

00:01:51.981 --> 00:01:56.249
Once again, you can say
two d is equal to five,

00:01:56.289 --> 00:01:58.229
two d is equal to five,

00:01:58.289 --> 00:02:01.927
or that d is equal to five halves.

00:02:01.927 --> 00:02:04.551
We've figured out what d is equal to.

00:02:04.551 --> 00:02:06.431
Now we can figure out what c is,

00:02:06.431 --> 00:02:10.518
because we know that c needs
to be equal to d-squared,

00:02:10.518 --> 00:02:14.630
gimme that orange color, actually,

00:02:14.750 --> 00:02:18.018
so we know that c is equal to d-squared,

00:02:18.018 --> 00:02:20.975
which is the same thing
as five halves, squared.

00:02:20.975 --> 00:02:22.708
We just figured out what d is equal to.

00:02:22.708 --> 00:02:24.577
Gonna be five halves, squared,

00:02:24.577 --> 00:02:27.885
which is going to be 25 over four.

00:02:27.885 --> 00:02:32.210
C is equal to

00:02:32.210 --> 00:02:33.830
25 over four,

00:02:33.830 --> 00:02:35.861
d is equal to five halves.

00:02:35.861 --> 00:02:37.184
We're done.


Factoring perfect squares: shared factors

WEBVTT
Kind: captions
Language: en

00:00:00.455 --> 00:00:02.335
- [Voiceover] The quadratic
expressions 4x squared,

00:00:02.335 --> 00:00:06.637
plus 12x, plus 9, and 4x squared minus 9,

00:00:06.637 --> 00:00:09.601
share a common binomial factor.

00:00:09.601 --> 00:00:11.967
What binomial factor do they share?

00:00:11.967 --> 00:00:13.067
And I encourage you to pause the video,

00:00:13.067 --> 00:00:15.400
see if you can figure it out.

00:00:15.400 --> 00:00:17.645
So, let's do this by taking
each of these expressions,

00:00:17.645 --> 00:00:19.457
and trying to factor them into binomials,

00:00:19.457 --> 00:00:21.878
and then see if they share
a common binomial factor.

00:00:21.878 --> 00:00:23.567
I guess they do share
one, it's to figure out

00:00:23.567 --> 00:00:26.101
which one they actually share.

00:00:26.101 --> 00:00:31.101
So, let's focus first on this
4x squared, plus 12x, plus 9.

00:00:32.951 --> 00:00:36.934
So, the first thing that
might jump out at you is,

00:00:36.934 --> 00:00:39.034
well let's see, I have a
4 here, this coefficient

00:00:39.034 --> 00:00:41.434
on the x squared term,
that's a perfect square.

00:00:41.434 --> 00:00:44.235
I could write the entire 4x squared term,

00:00:44.235 --> 00:00:49.235
I could rewrite that
as 2x to the 2nd power,

00:00:49.901 --> 00:00:54.735
then out here, I have
a constant term, the 9,

00:00:54.735 --> 00:00:56.454
that also is a perfect square.

00:00:56.454 --> 00:01:00.401
I could rewrite that as 3 squared.

00:01:00.401 --> 00:01:03.334
And you could say, "Well,
gee, does this fit the pattern

00:01:03.334 --> 00:01:05.701
of a perfect square polynomial?"

00:01:05.701 --> 00:01:07.034
In order for it to fit the pattern

00:01:07.034 --> 00:01:08.195
of a perfect square polynomial,

00:01:08.195 --> 00:01:10.801
the coefficient, here on the x term,

00:01:10.801 --> 00:01:15.801
would need to be 2 times the
product of this 2 and this 3,

00:01:18.491 --> 00:01:22.034
and it is, indeed, 2 times
the product of 2 and 3.

00:01:22.034 --> 00:01:27.034
It is 2 times 6, so we
could write this part,

00:01:27.168 --> 00:01:28.034
right over here,

00:01:28.034 --> 00:01:33.034
as 2 times 2, times that 2,
times 3, times that 3x, x.

00:01:41.077 --> 00:01:44.301
Then of course, we have to add
these three things together,

00:01:44.301 --> 00:01:45.801
so plus, plus.

00:01:45.801 --> 00:01:47.335
And so just like that,

00:01:47.335 --> 00:01:51.067
we can recognize, hey, this is
a perfect square polynomial,

00:01:51.067 --> 00:01:52.201
right over here.

00:01:52.201 --> 00:01:54.148
And if what I'm saying, right now,

00:01:54.148 --> 00:01:55.435
sounds like a little bit of voodoo,

00:01:55.435 --> 00:01:57.401
I encourage you to
watch some of the videos

00:01:57.401 --> 00:02:00.177
on perfect square trinomials,
perfect square polynomials,

00:02:00.177 --> 00:02:03.539
some of the last two
videos in this progression.

00:02:03.539 --> 00:02:06.735
So this this thing can be rewritten

00:02:06.735 --> 00:02:11.735
as the same thing as 2x
plus 3, 2x plus 3, squared,

00:02:16.924 --> 00:02:18.954
2x plus 3, squared.

00:02:18.954 --> 00:02:21.401
Once again, because it's of the form,

00:02:21.401 --> 00:02:23.911
you have the entire 2x squared here,

00:02:23.911 --> 00:02:26.467
you have the 3 squared here,

00:02:26.467 --> 00:02:29.268
and then this middle term
is 2 times the product

00:02:29.268 --> 00:02:31.534
of these two terms right over here,

00:02:31.534 --> 00:02:34.434
and so it definitely fit the pattern.

00:02:34.434 --> 00:02:35.034
So, there you have it.

00:02:35.034 --> 00:02:37.601
We factored the first expression,

00:02:37.601 --> 00:02:39.971
and now let's try to factor
the second expression.

00:02:39.971 --> 00:02:41.235
And, immediately when you see this one,

00:02:41.235 --> 00:02:44.001
it looks like it's a
difference of squares,

00:02:44.001 --> 00:02:45.698
so this one right over there,

00:02:45.698 --> 00:02:48.215
looks like a difference of squares to me.

00:02:48.215 --> 00:02:53.215
This, we can rewrite as 2x squared,

00:02:53.934 --> 00:02:58.934
minus, minus, look at that
nice color, minus 3 squared,

00:03:00.401 --> 00:03:02.434
so minus 3 squared.

00:03:02.434 --> 00:03:04.864
This is a difference of squares,
we've seen multiple times,

00:03:04.864 --> 00:03:06.281
how to factor difference of squares.

00:03:06.281 --> 00:03:08.367
If this, again, looks foreign to you,

00:03:08.367 --> 00:03:09.907
I encourage you to watch those videos,

00:03:09.907 --> 00:03:12.468
and we explain how that
works, and why it works.

00:03:12.468 --> 00:03:13.301
What is is going to be,

00:03:13.301 --> 00:03:14.235
when you have something in the form

00:03:14.235 --> 00:03:16.368
A squared, minus B squared,

00:03:16.368 --> 00:03:20.655
it's going to be equal to
A plus B, times A minus B.

00:03:20.655 --> 00:03:22.801
So, this is going to be equal to,

00:03:22.801 --> 00:03:25.168
let me just put the two
binomials right over here,

00:03:25.168 --> 00:03:30.168
so this is going to be A
plus B, times A minus B.

00:03:30.334 --> 00:03:35.334
So, this is going to be 2x
plus 3, times 2x minus 3.

00:03:37.544 --> 00:03:41.274
So, 2x plus 3, times 2x minus 3.

00:03:41.284 --> 00:03:44.668
And so, what is their
common binomial factor?

00:03:44.668 --> 00:03:48.200
Well, they both involve,
when you factor them out,

00:03:48.200 --> 00:03:52.069
they both have a binomial
factor of 2x plus 3.

00:03:52.069 --> 00:03:54.328
This one right over here,
we could rewrite if we want.

00:03:54.328 --> 00:03:58.484
We could rewrite it as 2x
plus 3, times 2x plus 3.

00:03:58.484 --> 00:04:01.351
That might've been somewhat
obvious to you already.

00:04:01.351 --> 00:04:06.351
So, 2x, 2x, then you have plus 3, plus 3,

00:04:07.468 --> 00:04:08.767
these two are equivalent.

00:04:08.767 --> 00:04:13.767
And so you see, we see, that
we share in both of these,

00:04:15.098 --> 00:04:18.735
we share at least one,
or we share exactly one,

00:04:18.735 --> 00:04:22.234
2x plus 3, so that's the
binomial factor that they share:

00:04:22.234 --> 00:04:24.234
2x plus 3.


Factoring perfect squares: common factor

WEBVTT
Kind: captions
Language: en

00:00:00.363 --> 00:00:01.361
- [Voiceover] So let's say that we've got

00:00:01.361 --> 00:00:05.367
the polynomial 16x to the third

00:00:05.367 --> 00:00:09.556
plus 24x squared

00:00:09.556 --> 00:00:11.576
plus nine x.

00:00:11.576 --> 00:00:13.097
Now what I'd like you to do is pause

00:00:13.097 --> 00:00:14.246
the video and see if you could

00:00:14.246 --> 00:00:17.346
factor this polynomial completely.

00:00:18.526 --> 00:00:19.847
Now let's work through it together.

00:00:19.847 --> 00:00:21.288
So the first thing that you might notice

00:00:21.288 --> 00:00:24.224
is that all of the
terms are divisible by x

00:00:24.224 --> 00:00:27.544
so we can actually factor out an x.

00:00:27.544 --> 00:00:28.925
So let's do that, and
actually, if we look at

00:00:28.925 --> 00:00:31.642
these coefficients, it
looks like, let's see,

00:00:31.642 --> 00:00:32.745
it looks like they don't have any

00:00:32.745 --> 00:00:34.708
common factors other than one.

00:00:34.708 --> 00:00:37.598
So it looks like the
largest monomial that we can

00:00:37.598 --> 00:00:39.722
factor out is just going to be an x.

00:00:39.722 --> 00:00:41.719
So let's do that. Let's factor out an x.

00:00:41.719 --> 00:00:43.843
So then this is gonna be x times.

00:00:43.843 --> 00:00:46.560
When you factor out an
x from 16x to the third

00:00:46.560 --> 00:00:50.046
you're gonna be left with 16x squared,

00:00:50.046 --> 00:00:55.046
and then plus 24x and then plus nine.

00:00:55.370 --> 00:00:57.549
Now this is starting to look interesting

00:00:57.549 --> 00:00:58.640
so let me just rewrite it.

00:00:58.640 --> 00:01:00.706
It's gonna be x times.

00:01:00.706 --> 00:01:01.647
This part over here looks interesting

00:01:01.647 --> 00:01:04.259
because when I see the 16x squared,

00:01:04.259 --> 00:01:06.348
this looks like a perfect square.

00:01:06.348 --> 00:01:09.879
Let me write it out. 16x squared.

00:01:09.879 --> 00:01:13.883
That's the same thing as four x squared,

00:01:13.883 --> 00:01:17.098
and then we have a nine over there,

00:01:17.098 --> 00:01:18.643
which is clearly a perfect square.

00:01:18.643 --> 00:01:22.322
That is three squared. Three squared.

00:01:22.322 --> 00:01:25.445
And when we look at this 24x, we see

00:01:25.445 --> 00:01:29.300
that it is four times three times two,

00:01:29.300 --> 00:01:32.086
and so we can write it as,
let me write it this way.

00:01:32.086 --> 00:01:35.991
So this is going to be plus two times

00:01:35.991 --> 00:01:39.277
four times three x.

00:01:39.277 --> 00:01:44.277
So let me make it, so two times four

00:01:44.872 --> 00:01:49.330
times three times three x.

00:01:49.330 --> 00:01:51.455
Now why did I take the trouble,

00:01:51.455 --> 00:01:55.355
why did I take the trouble of
writing everything like this?

00:01:55.355 --> 00:01:56.795
Because we see that it fits the

00:01:56.795 --> 00:01:59.175
pattern for a perfect square.

00:01:59.175 --> 00:02:00.057
What do I mean by that?

00:02:00.057 --> 00:02:02.193
Well in previous videos, we saw that

00:02:02.193 --> 00:02:03.807
if you have something of the form

00:02:03.807 --> 00:02:07.150
Ax plus B, and you were to square it,

00:02:07.150 --> 00:02:11.204
you're going to get Ax squared

00:02:11.204 --> 00:02:15.033
plus two ABx

00:02:15.033 --> 00:02:19.346
plus B squared, and we have
that form right over here.

00:02:19.346 --> 00:02:22.432
This is the Ax squared.
Let me do the same color.

00:02:22.432 --> 00:02:25.866
The Ax squared. Ax squared.

00:02:25.866 --> 00:02:30.731
We have the B squared.
You have the B squared.

00:02:30.731 --> 00:02:34.522
And then you have the two ABx.

00:02:34.522 --> 00:02:38.144
Two ABx right over there.

00:02:38.144 --> 00:02:42.578
So this section, this entire
section, we can rewrite

00:02:42.578 --> 00:02:44.762
as being we know it, what A and B are.

00:02:44.762 --> 00:02:49.696
A is four and B is three
so this is going to be

00:02:49.696 --> 00:02:54.349
Ax, so four x plus

00:02:54.349 --> 00:02:59.111
four x plus b, which we know to be three.

00:02:59.111 --> 00:03:02.455
That whole thing, that whole
thing squared, and now we

00:03:02.455 --> 00:03:06.274
can't forget this x out front
so we have that x out front.

00:03:06.274 --> 00:03:08.119
And we're done. We have
just factored this.

00:03:08.119 --> 00:03:09.919
We have just factored this completely.

00:03:09.919 --> 00:03:12.113
We could write it as x
times four x plus three

00:03:12.113 --> 00:03:13.832
and then write out and then say times

00:03:13.832 --> 00:03:15.713
four x plus three or we could just write

00:03:15.713 --> 00:03:20.055
x times the quantity four
x plus three squared.

00:03:20.055 --> 00:03:21.888
And so we've just factored it completely,

00:03:21.888 --> 00:03:23.293
and the key realizations here is well,

00:03:23.293 --> 00:03:26.056
one: what could I factor
out from all of these terms?

00:03:26.056 --> 00:03:27.936
I could factor out an x
from all of those terms,

00:03:27.936 --> 00:03:30.328
and then to realize what we had left over

00:03:30.328 --> 00:03:33.834
was a perfect square,
really using this pattern

00:03:33.834 --> 00:03:37.858
that we were able to
see in previous videos.


Identifying perfect square form

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:04.370
- [Voiceover] We wanna figure
out what AX plus B squared is,

00:00:04.370 --> 00:00:05.671
and I encourage you to pause the video

00:00:05.671 --> 00:00:06.864
and figure out what that is

00:00:06.864 --> 00:00:09.302
in terms of capital-A and capital-B.

00:00:09.302 --> 00:00:11.172
So let's work through it.

00:00:11.172 --> 00:00:14.144
This is the same thing as multiplying

00:00:14.144 --> 00:00:18.625
AX plus B times AX plus B.

00:00:18.625 --> 00:00:20.443
So let me fill that in.

00:00:21.052 --> 00:00:24.825
This is AX there, another AX there.

00:00:24.825 --> 00:00:25.869
I just wrote it in that order

00:00:25.869 --> 00:00:28.453
to make the color switching
a little bit easier.

00:00:28.586 --> 00:00:31.518
So AX plus B times AX plus B.

00:00:31.732 --> 00:00:33.404
Well, what's that going to be equal to?

00:00:33.404 --> 00:00:36.746
Well, if you take this AX
and you multiply it times

00:00:36.746 --> 00:00:40.793
that AX, you're going to get AX squared.

00:00:41.403 --> 00:00:44.532
AX, the entire thing squared.

00:00:44.746 --> 00:00:48.647
And then if you take, if you take this AX

00:00:48.647 --> 00:00:51.561
and then multiply it times this B,

00:00:51.561 --> 00:00:55.734
you're going to get ABX.

00:00:56.366 --> 00:00:59.002
Then if you take this
B and you multiply it

00:00:59.002 --> 00:01:02.770
times this AX, you're
going to get another ABX.

00:01:03.273 --> 00:01:05.462
ABX.

00:01:05.700 --> 00:01:07.976
And then last but not
least, if you take this B

00:01:07.976 --> 00:01:11.047
and multiply it times the other B,

00:01:11.110 --> 00:01:14.367
it's going to be plus B squared.

00:01:14.570 --> 00:01:16.341
And so what are you left with?

00:01:16.671 --> 00:01:19.550
Well, you're going to be left with A,

00:01:19.550 --> 00:01:20.897
I'll write it like this,

00:01:20.897 --> 00:01:25.854
AX squared, we actually if we want,

00:01:25.854 --> 00:01:28.640
well, I'll write it in a
different way in a second,

00:01:28.640 --> 00:01:32.541
and then you have plus, plus two,

00:01:32.541 --> 00:01:34.781
that's a slightly different color.

00:01:34.781 --> 00:01:36.604
I'm gonna do that other color.

00:01:36.604 --> 00:01:40.800
Plus two ABX,

00:01:40.800 --> 00:01:43.639
and then finally plus B squared.

00:01:43.639 --> 00:01:45.717
Plus B squared.

00:01:45.717 --> 00:01:47.296
Now, I said I could write it
in a slightly different way,

00:01:47.296 --> 00:01:50.813
what I could do is just
rewrite-out AX squared

00:01:50.813 --> 00:01:53.019
as being the same thing.

00:01:53.019 --> 00:01:58.019
This is the same thing
as A-squared X-squared,

00:01:58.278 --> 00:02:01.041
and then I can write out
everything else the same way.

00:02:01.041 --> 00:02:05.528
Plus two ABX,

00:02:05.528 --> 00:02:06.985
and then plus B squared.

00:02:06.985 --> 00:02:09.829
Now, why did I, what's
interesting about doing this?

00:02:09.829 --> 00:02:11.768
Well, now we can see the pattern

00:02:11.768 --> 00:02:15.343
for the square of any binomial

00:02:15.343 --> 00:02:17.282
or binomial like this,

00:02:17.282 --> 00:02:20.660
so for example, if someone
were to walk up to you

00:02:20.660 --> 00:02:25.525
and say "alright, I have
a trinomial of the form,"

00:02:25.525 --> 00:02:28.716
let's say they have a
trinomial of the form

00:02:28.716 --> 00:02:32.408
25X squared

00:02:32.408 --> 00:02:36.761
plus 20X plus 4,

00:02:36.761 --> 00:02:38.909
and they were to tell you to factor this,

00:02:38.909 --> 00:02:40.302
well, actually, let's just do that.

00:02:40.302 --> 00:02:41.370
Why don't you pause the video and see

00:02:41.370 --> 00:02:44.087
if you could factor this as
the product of two binomials.

00:02:44.877 --> 00:02:46.261
Well, when you look at this,

00:02:46.261 --> 00:02:48.145
you'd say "well look, there's 25X squared,

00:02:48.145 --> 00:02:50.386
"that looks like a perfect square.

00:02:50.386 --> 00:02:55.099
"25X squared, that's the
same thing as five-squared

00:02:55.099 --> 00:02:57.038
"x-squared,"

00:02:57.038 --> 00:03:02.038
or you could write it as five X squared.

00:03:02.268 --> 00:03:05.280
This four here, that's a perfect square.

00:03:05.280 --> 00:03:08.374
That's the same thing as two squared.

00:03:08.519 --> 00:03:10.585
And let's see, 20, right over here,

00:03:10.585 --> 00:03:12.547
if we want it to fit this pattern,

00:03:12.547 --> 00:03:16.228
we would see that A is five and B is two,

00:03:16.228 --> 00:03:19.548
and so let's see, what
would be two times AB?

00:03:19.548 --> 00:03:23.019
Well, five times two AB would be 10,

00:03:23.019 --> 00:03:25.201
and then two times that would be 20.

00:03:25.201 --> 00:03:27.650
So this right over here is,

00:03:27.650 --> 00:03:32.650
that is plus two times five.

00:03:33.477 --> 00:03:37.291
Two times five times two

00:03:37.291 --> 00:03:40.200
times two X.

00:03:40.588 --> 00:03:43.984
Times two X, I'll do it in this color.

00:03:44.210 --> 00:03:46.062
Times two X.

00:03:46.149 --> 00:03:49.006
So you see that this completely
matches this pattern here

00:03:49.006 --> 00:03:53.579
where A is equal to five
and B is equal to two.

00:03:53.579 --> 00:03:56.620
Once again, this is AX,
the whole thing squared,

00:03:56.620 --> 00:04:01.514
then you have two times A times BX,

00:04:01.914 --> 00:04:03.232
you see that there,

00:04:03.296 --> 00:04:05.803
and then finally you have the B squared.

00:04:05.803 --> 00:04:08.195
So if you wanted to
factor this, you could say

00:04:08.195 --> 00:04:09.917
"well, this is just
going to the same thing

00:04:09.917 --> 00:04:11.995
"since we know what A and B are,

00:04:11.995 --> 00:04:14.920
"this is going to be five X plus two."

00:04:14.920 --> 00:04:19.845
Five X plus two.

00:04:19.845 --> 00:04:24.845
Five X plus two, whole thing squared.

00:04:25.577 --> 00:04:28.314
So the whole point of doing
this is to start recognizing

00:04:28.314 --> 00:04:30.577
when we actually have perfect squares,

00:04:30.577 --> 00:04:32.818
especially perfect squares
where the leading coefficient

00:04:32.818 --> 00:04:35.552
isn't one.


Factoring difference of squares: shared factors

WEBVTT
Kind: captions
Language: en

00:00:00.359 --> 00:00:02.106
- [Voiceover] We're told that
the quadratic expressions

00:00:02.106 --> 00:00:05.530
m squared minus 4m minus 45, and

00:00:05.530 --> 00:00:08.607
6m squared minus 150,

00:00:08.607 --> 00:00:11.620
share a common binomial factor.

00:00:11.620 --> 00:00:14.291
What binomial factor do they share?

00:00:14.291 --> 00:00:16.067
And like always pause the video and

00:00:16.067 --> 00:00:17.956
see if you can work through this.

00:00:17.956 --> 00:00:19.792
All right, now let's work
through this together and

00:00:19.792 --> 00:00:21.036
the way I am going to do this is

00:00:21.036 --> 00:00:23.264
I'm just going to try and
factor both of them into the

00:00:23.264 --> 00:00:25.511
product of binomials and
maybe some other things and

00:00:25.511 --> 00:00:28.142
see if we have any
common binomial factors.

00:00:28.142 --> 00:00:30.518
So first let's focus on

00:00:30.518 --> 00:00:33.659
m squared minus 4m minus 45.

00:00:33.659 --> 00:00:35.217
So let me write it over here,

00:00:35.217 --> 00:00:39.471
m squared minus 4m minus 45.

00:00:40.191 --> 00:00:43.531
So when you're factoring a
quadratic expression like this,

00:00:43.531 --> 00:00:46.255
where the coefficient
on the, in this case,

00:00:46.255 --> 00:00:49.054
m squared term, on the
second degree term is one,

00:00:49.054 --> 00:00:52.189
we could factor it as being equal to

00:00:52.189 --> 00:00:56.950
m plus a, times m plus b,

00:00:56.950 --> 00:00:59.316
where a plus b is going to be equal

00:00:59.316 --> 00:01:01.522
to this coefficient right over here,

00:01:01.522 --> 00:01:03.959
and a times b is going to be equal

00:01:03.959 --> 00:01:06.531
to this coefficient right over here.

00:01:06.531 --> 00:01:09.155
So let's be clear, so, a,

00:01:09.155 --> 00:01:11.110
let me see another color,

00:01:11.110 --> 00:01:13.833
so a plus b needs to
be equal to negative 4,

00:01:13.833 --> 00:01:18.633
a plus b needs to be equal to negative 4,

00:01:18.633 --> 00:01:22.660
and then a times b needs
to be equal to negative 45.

00:01:22.660 --> 00:01:27.660
A times b is equal to negative 45.

00:01:27.727 --> 00:01:30.303
Now I like to focus on the
a times b and think about,

00:01:30.303 --> 00:01:34.853
well, what could a and b
be to get to negative 45?

00:01:34.853 --> 00:01:36.461
Well if I'm taking the
product of two things and

00:01:36.461 --> 00:01:38.156
if the product is negative that means that

00:01:38.156 --> 00:01:40.374
they are going to have different signs and

00:01:40.374 --> 00:01:44.100
if when we add them we get a
negative number that means that

00:01:44.110 --> 00:01:47.093
the negative one

00:01:47.093 --> 00:01:50.890
has a larger magnitude.

00:01:50.890 --> 00:01:52.591
So let's think about this a little bit.

00:01:52.591 --> 00:01:54.900
So a times b is equal to negative 45.

00:01:54.900 --> 00:01:58.843
So this could be, let's
try some values out.

00:01:58.843 --> 00:02:02.966
So, 1 and 45, those are too far apart.

00:02:02.966 --> 00:02:03.856
Let's see.

00:02:03.856 --> 00:02:08.856
3 and 15, those still
seem pretty far apart.

00:02:08.990 --> 00:02:13.873
Let's see, it looks like
5 and 9 seem interesting.

00:02:13.873 --> 00:02:15.377
So if we say,

00:02:15.377 --> 00:02:18.159
if we say 5 times,

00:02:18.159 --> 00:02:22.971
if we were to say, 5 times negative 9,

00:02:22.975 --> 00:02:26.265
that indeed is equal to negative 45,

00:02:26.265 --> 00:02:30.247
and 5 plus negative 9

00:02:30.247 --> 00:02:33.087
is indeed equal to negative 4.

00:02:33.087 --> 00:02:35.387
So a could be equal to 5 and

00:02:35.387 --> 00:02:38.062
b could be equal to negative 9.

00:02:38.062 --> 00:02:41.062
And so if we were to factor
this, this is going to be

00:02:41.062 --> 00:02:43.483
m plus 5,

00:02:43.483 --> 00:02:46.132
times m, I could say m plus negative 9,

00:02:46.132 --> 00:02:48.837
but I'll just write m minus 9.

00:02:48.837 --> 00:02:52.387
So just like that I've
been able to factor this

00:02:52.387 --> 00:02:54.129
first quadratic expression right

00:02:54.129 --> 00:02:56.904
over there as a product of two binomials.

00:02:56.904 --> 00:02:59.911
So now let's try to factor the
other quadratic expression.

00:02:59.911 --> 00:03:04.911
Let's try to factor 6m squared minus 150.

00:03:05.430 --> 00:03:08.441
And let's see, the first
thing I might want to do is,

00:03:08.441 --> 00:03:12.405
both 6m squared and 150,

00:03:12.405 --> 00:03:14.807
they're both divisible by 6.

00:03:14.807 --> 00:03:16.737
So let me write it this
way, I could write it as,

00:03:16.737 --> 00:03:21.318
6m squared minus 6 times,

00:03:21.318 --> 00:03:26.005
let's see, 6 goes into 150, 25 times.

00:03:26.375 --> 00:03:27.593
So all I did is I rewrote this

00:03:27.593 --> 00:03:30.890
and really I just wrote 150 as 6 times 25.

00:03:30.890 --> 00:03:34.372
And now you can clearly see
that we can factor out a 6.

00:03:34.372 --> 00:03:36.717
You can view this as undistributing the 6.

00:03:36.717 --> 00:03:38.777
So this is the same thing as

00:03:38.777 --> 00:03:43.205
6 times m squared minus 25,

00:03:43.205 --> 00:03:45.691
which we recognize this is
a difference of squares.

00:03:45.691 --> 00:03:47.915
So it's all going to be 6 times,

00:03:47.915 --> 00:03:52.915
m plus 5, times m minus 5.

00:03:53.542 --> 00:03:56.615
And so we've factored this out
as a product of binomials and

00:03:56.615 --> 00:04:00.253
a constant factor here, 6,

00:04:00.253 --> 00:04:03.721
and so, what is their shared, common

00:04:03.721 --> 00:04:06.578
or what is their common
binomial factor that they share?

00:04:06.578 --> 00:04:11.309
Well you see when we factored
it out, they both have

00:04:11.309 --> 00:04:13.979
an m plus 5.

00:04:13.979 --> 00:04:16.457
So m plus 5

00:04:16.457 --> 00:04:21.107
is the binomial factor that they share.


Factoring difference of squares: analyzing factorization

WEBVTT
Kind: captions
Language: en

00:00:00.466 --> 00:00:03.322
- [Voiceover] Moussa and Fatu
were each asked to factor

00:00:03.322 --> 00:00:04.900
the quadratic expression

00:00:04.900 --> 00:00:09.300
16 x-squared minus 64.

00:00:09.300 --> 00:00:11.947
Their responses are shown below.

00:00:11.947 --> 00:00:13.886
So Moussa factored it this way.

00:00:13.886 --> 00:00:15.999
Fatu factored it this way.

00:00:15.999 --> 00:00:18.204
Which student wrote an
expression that is equivalent

00:00:18.204 --> 00:00:20.944
to 16 x-squared minus 64?

00:00:20.944 --> 00:00:23.405
So I encourage you to pause
the video and figure that out.

00:00:23.405 --> 00:00:26.215
Which student wrote an
expression that is equivalent to

00:00:26.215 --> 00:00:30.324
our original one, 16 x-squared minus 64?

00:00:30.324 --> 00:00:32.658
Well let's work through it
together, so let's see if first

00:00:32.658 --> 00:00:35.920
we can factor this out
somehow to get what Moussa got

00:00:35.920 --> 00:00:38.717
and it looks like Moussa
first factored out a 16

00:00:38.717 --> 00:00:40.819
and then he was left with
a difference of squares.

00:00:40.819 --> 00:00:42.398
So let's see if we can do that.

00:00:42.398 --> 00:00:45.102
So, we can write our original expression.

00:00:45.102 --> 00:00:48.562
16 x-squared minus 64,
we can write that as

00:00:48.562 --> 00:00:51.557
16 times x-squared

00:00:51.557 --> 00:00:55.133
minus 16 times four.

00:00:55.133 --> 00:00:57.083
And when you write it like
that, it's very clear that you

00:00:57.083 --> 00:00:58.882
can factor out a 16.

00:00:58.882 --> 00:01:02.214
So this is going to be equal to 16 times

00:01:02.214 --> 00:01:04.640
what you have left over is x-squared

00:01:04.640 --> 00:01:07.972
minus four and then x-squared minus four,

00:01:07.972 --> 00:01:10.696
that's a difference of
squares right over there.

00:01:10.696 --> 00:01:13.598
So, that part we can factor as,

00:01:13.598 --> 00:01:16.611
so we have our original 16 and then...

00:01:17.987 --> 00:01:20.878
this part right over here, we can write as

00:01:20.878 --> 00:01:23.502
x plus two times x minus two.

00:01:23.502 --> 00:01:26.787
x plus two times x minus two.

00:01:26.787 --> 00:01:30.056
If what I just did in this
last step, going from x-squared

00:01:30.056 --> 00:01:33.991
minus four to x plus two times x minus two

00:01:33.991 --> 00:01:35.988
doesn't make any sense, I
encourage you to watch some of

00:01:35.988 --> 00:01:38.147
the introductory videos on factoring

00:01:38.147 --> 00:01:39.505
and difference of squares.

00:01:39.505 --> 00:01:42.222
But the basic idea, I have a form here of

00:01:42.222 --> 00:01:45.379
a-squared minus b-squared, so
it's going to have the form

00:01:45.379 --> 00:01:48.525
of a plus b times a minus b

00:01:48.525 --> 00:01:51.265
and in this case it's
x-squared minus two squared.

00:01:51.265 --> 00:01:54.539
So it's going to be x plus
two times x minus two.

00:01:54.539 --> 00:01:56.861
So that's exactly what Moussa got.

00:01:56.861 --> 00:01:59.821
So this one, so Moussa,

00:01:59.821 --> 00:02:01.434
did get an expression
that is equivalent to

00:02:01.434 --> 00:02:03.861
16 x-squared minus 64.

00:02:03.861 --> 00:02:05.393
Now let's think about Fatu.

00:02:05.393 --> 00:02:08.737
So Fatu didn't factor
out a 16 from the get-go.

00:02:08.737 --> 00:02:11.625
It looks like he just
immediately recognized that

00:02:11.625 --> 00:02:14.295
our original expression is
itself a difference of squares

00:02:14.295 --> 00:02:16.315
even if we don't factor out a 16,

00:02:16.315 --> 00:02:17.859
and so let's re-write it.

00:02:17.859 --> 00:02:21.783
So our original expression, we could write

00:02:21.783 --> 00:02:25.419
as, so instead of writing,
well I'm just going to

00:02:25.419 --> 00:02:27.160
write it like this, this
is our original expression.

00:02:27.160 --> 00:02:30.271
16 x-squared minus 64.

00:02:30.271 --> 00:02:32.837
That's the same thing as, 16
x-squared is the same thing as

00:02:32.837 --> 00:02:36.053
four x, the whole thing squared

00:02:36.053 --> 00:02:38.850
and then minus eight squared.

00:02:38.850 --> 00:02:40.801
So when you write it like
this, it's clear that this is a

00:02:40.801 --> 00:02:42.751
difference of squares,
so this is going to be

00:02:42.751 --> 00:02:46.129
four x plus eight times
four x minus eight.

00:02:46.129 --> 00:02:49.310
Four x plus eight times
four x minus eight.

00:02:49.310 --> 00:02:51.761
Once again, if this last
step that I did doesn't make

00:02:51.761 --> 00:02:54.431
a lot of sense I encourage
you to watch the video

00:02:54.431 --> 00:02:56.451
on factoring difference
of squares where we go

00:02:56.451 --> 00:02:58.575
a lot more into the intuition of it.

00:02:58.575 --> 00:03:00.595
But when you see it this
way you realize that Fatu

00:03:00.595 --> 00:03:03.312
also got an expression
that is equivalent to

00:03:03.312 --> 00:03:07.207
16 x-squared minus 64, so they both did.


Factoring difference of squares: missing values

WEBVTT
Kind: captions
Language: en

00:00:00.423 --> 00:00:02.060
- [Voiceover] The polynomial expression,

00:00:02.060 --> 00:00:07.060
36 y to the third minus
100y can be factored as

00:00:07.494 --> 00:00:10.373
4y times all of this,

00:00:10.373 --> 00:00:13.519
My plus g, times My minus g,

00:00:13.519 --> 00:00:16.016
where M and g are integers.

00:00:16.016 --> 00:00:18.419
Sally wrote that g could be equal to 3.

00:00:18.419 --> 00:00:21.821
Brandon wrote that g could be equal to 10.

00:00:21.821 --> 00:00:24.317
Which student is correct?

00:00:24.317 --> 00:00:26.325
Now, when you look at this,
it seems really daunting,

00:00:26.325 --> 00:00:27.951
all this M's and g's here,

00:00:27.951 --> 00:00:30.516
but we just need to realize
that they're factoring out,

00:00:30.516 --> 00:00:33.419
first they factored out a
4y from the 36y to the third

00:00:33.419 --> 00:00:35.857
minus 100y, and it looks
like whatever's left

00:00:35.857 --> 00:00:38.260
is a difference of squares,
which they then factor

00:00:38.260 --> 00:00:39.386
even further.

00:00:39.386 --> 00:00:41.720
So I encourage you to pause
the video and just factor

00:00:41.720 --> 00:00:42.986
this out as much as you can.

00:00:42.986 --> 00:00:44.820
First factoring out a 4y,

00:00:44.820 --> 00:00:46.782
and then we can think
about what g is going to be

00:00:46.782 --> 00:00:50.985
equal to, or whether Sally
or Brandon is correct.

00:00:50.985 --> 00:00:53.725
So, then now let's work
through this together.

00:00:53.725 --> 00:00:55.722
So, if we look at,

00:00:55.722 --> 00:00:57.521
if we look at this
expression right over here,

00:00:57.521 --> 00:01:00.087
we want to factor out a 4y,

00:01:00.087 --> 00:01:04.510
so 36y to the third

00:01:04.510 --> 00:01:07.250
minus 100y,

00:01:07.250 --> 00:01:08.887
that's the same thing as,

00:01:08.887 --> 00:01:12.289
36y to the third is the same thing as 4y

00:01:12.289 --> 00:01:14.356
times, let's see,

00:01:14.356 --> 00:01:19.127
4y times 9y squared,

00:01:19.127 --> 00:01:20.428
right?

00:01:20.428 --> 00:01:22.587
Because 4 times 9 is 36,

00:01:22.587 --> 00:01:25.455
and y times y squared is y to the third,

00:01:25.455 --> 00:01:27.788
so all I did to get the
9y squared is I divided

00:01:27.788 --> 00:01:31.887
36 by 4 to get the 9, and
I divided y to the third

00:01:31.887 --> 00:01:34.580
by the y to get y squared,

00:01:34.580 --> 00:01:37.494
so if you factor out a 4y,
you're left with 9y squared

00:01:37.494 --> 00:01:39.991
for that first term, and then,

00:01:39.991 --> 00:01:42.173
for the second term, let's see,

00:01:42.173 --> 00:01:43.857
if we, we're going to subtract,

00:01:43.857 --> 00:01:46.724
if we factor out a 4y again,

00:01:46.724 --> 00:01:49.290
what's left over?

00:01:49.290 --> 00:01:52.448
100 divided by 4 is 25,

00:01:52.448 --> 00:01:57.448
and then y divided by y

00:01:57.626 --> 00:02:00.993
is just 1, so we're left with a 25 here.

00:02:00.993 --> 00:02:03.454
So just to be clear what's going on,

00:02:03.454 --> 00:02:05.393
this 36y to the third,

00:02:05.393 --> 00:02:09.190
I just rewrote it as 4y times 9y squared.

00:02:09.190 --> 00:02:11.523
One way to think about it
is I wrote it with the 4y

00:02:11.523 --> 00:02:14.553
factored out, and then,

00:02:14.553 --> 00:02:16.725
the 100y, right over here,

00:02:16.725 --> 00:02:19.627
I wrote it with the 4y factored out,

00:02:19.627 --> 00:02:22.053
so it's 4y times 25,

00:02:22.053 --> 00:02:24.294
and now it's very clear
that we can factor out 4y

00:02:24.294 --> 00:02:26.291
from this entire thing,

00:02:26.291 --> 00:02:28.160
so we can factor out,

00:02:28.160 --> 00:02:31.179
you can think of it as
undistributing the 4y,

00:02:31.179 --> 00:02:33.559
so this is going to be equal to

00:02:33.559 --> 00:02:38.319
4y and what is left over?

00:02:38.319 --> 00:02:40.560
Well, if you factor out a
4y out of this first term,

00:02:40.560 --> 00:02:43.381
you're going to have a 9y squared,

00:02:43.381 --> 00:02:48.381
9y squared, and then minus 25,

00:02:48.989 --> 00:02:53.424
and then we're going to
be left with minus 25,

00:02:53.424 --> 00:02:54.387
and when we write it like this,

00:02:54.387 --> 00:02:56.187
we see what we have in parentheses here.

00:02:56.187 --> 00:03:00.157
This is a difference of squares, and,

00:03:00.157 --> 00:03:02.120
we could skip a step, but
let me just rewrite it,

00:03:02.120 --> 00:03:04.360
so we could rewrite it as, literally,

00:03:04.360 --> 00:03:05.788
a difference of squares.

00:03:05.788 --> 00:03:10.225
9y squared, that is the same thing as,

00:03:10.225 --> 00:03:14.660
that is the same thing as

00:03:14.660 --> 00:03:17.619
3y, that whole thing to the 2nd power,

00:03:17.619 --> 00:03:21.057
3 squared is 9, y squared is y squared,

00:03:21.057 --> 00:03:24.018
and then we have

00:03:24.018 --> 00:03:28.325
minus 25, we can rewrite as 5 squared,

00:03:28.325 --> 00:03:30.392
so you see, we have a
difference of squares,

00:03:30.392 --> 00:03:32.052
and we've seen this
pattern multiple times.

00:03:32.052 --> 00:03:33.724
If this is the first time you're
seeing it, I encourage you

00:03:33.724 --> 00:03:35.186
to watch the videos on Khan Academy

00:03:35.186 --> 00:03:39.552
on difference of squares, but
we know anything of the form,

00:03:39.552 --> 00:03:44.552
anything of the form a
squared minus b squared,

00:03:44.887 --> 00:03:48.347
let me do it in that color,

00:03:48.347 --> 00:03:53.347
minus b squared can be
factored as being equal to,

00:03:53.387 --> 00:03:55.825
this is equal to, if I were
to write it as a product

00:03:55.825 --> 00:03:58.054
of two binomials,

00:03:58.054 --> 00:03:59.958
this is going to be equal to

00:03:59.958 --> 00:04:04.958
a plus b

00:04:06.448 --> 00:04:10.524
times a minus b,

00:04:10.524 --> 00:04:12.288
and you can verify that that works,

00:04:12.288 --> 00:04:13.925
if you've never seen this before,

00:04:13.925 --> 00:04:15.922
or you can watch those videos for review.

00:04:15.922 --> 00:04:18.790
So this right over here
can be rewritten as

00:04:18.790 --> 00:04:21.228
4y, which we factored
out at the beginning,

00:04:21.228 --> 00:04:24.850
it's going to be times the
product of two binomials

00:04:24.850 --> 00:04:26.522
for this part right over here,

00:04:26.522 --> 00:04:28.751
and so in this case, a is 3y,

00:04:28.751 --> 00:04:31.921
so it's going to be 3y
plus 5, times 3y minus 5,

00:04:31.921 --> 00:04:33.651
so let me write that down.

00:04:33.651 --> 00:04:38.492
So 3y plus 5

00:04:38.492 --> 00:04:42.660
times 3y minus 5.

00:04:42.660 --> 00:04:47.660
3y minus 5.

00:04:49.556 --> 00:04:51.785
So now that we factored
this, let's go back to what

00:04:51.785 --> 00:04:56.151
they originally told
us, so then we have 4y,

00:04:56.151 --> 00:05:00.853
so this 4y corresponds to
that 4y, right over there,

00:05:00.853 --> 00:05:04.684
and then you have My plus g,

00:05:04.684 --> 00:05:06.855
and then you have My minus g,

00:05:06.855 --> 00:05:09.792
so you could view the My,

00:05:09.792 --> 00:05:11.592
the My's right over there,

00:05:11.592 --> 00:05:14.286
that's the 3y right over there,

00:05:14.286 --> 00:05:17.223
so we could say that M is equal to 3,

00:05:17.223 --> 00:05:19.591
M is equal to 3,

00:05:19.591 --> 00:05:21.658
and then we do plus 5 and minus 5,

00:05:21.658 --> 00:05:24.990
plus g and minus g, so,

00:05:24.990 --> 00:05:28.148
g, if we're pattern
matching right over here,

00:05:28.148 --> 00:05:30.957
g is going to be equal to 5.

00:05:30.957 --> 00:05:33.651
So g is equal to 5.

00:05:33.651 --> 00:05:35.520
So what's interesting about this problem

00:05:35.520 --> 00:05:38.455
is that neither one of them

00:05:38.455 --> 00:05:42.117
are correct, so, I could write,

00:05:42.117 --> 00:05:44.184
neither is correct.

00:05:44.184 --> 00:05:47.260
G is equal to 5.

00:05:47.260 --> 00:05:48.706
That was a tricky one.


Emancipation Proclamation

WEBVTT
Kind: captions
Language: en

00:00:00.146 --> 00:00:01.752
- [Voiceover] So again, we've
been talking a lot about

00:00:01.752 --> 00:00:06.059
the early stages of the Civil
War which were about slavery

00:00:06.059 --> 00:00:08.486
and the spread of slavery,

00:00:08.486 --> 00:00:11.227
and I think in popular culture,
when we think about things

00:00:11.227 --> 00:00:14.669
like Abraham Lincoln and
the Civil War and slavery,

00:00:14.669 --> 00:00:17.922
the term Emancipation
Proclamation, or the idea

00:00:17.922 --> 00:00:19.722
of the Emancipation
Proclamation comes to mind

00:00:19.722 --> 00:00:23.586
and I think a lot of folks
imagine it to be this really

00:00:23.586 --> 00:00:26.590
amazing speech that at
some point, you know,

00:00:26.590 --> 00:00:28.028
Lincoln makes this proclamation,

00:00:28.028 --> 00:00:30.294
and after that point,
slavery is just gone from

00:00:30.294 --> 00:00:31.215
the United States,

00:00:31.215 --> 00:00:33.094
is that what it was?

00:00:33.094 --> 00:00:35.074
- [Voiceover] No, it's
actually a lot more complicated

00:00:35.074 --> 00:00:35.692
than that.

00:00:35.692 --> 00:00:38.488
So, the Emancipation Proclamation itself

00:00:38.488 --> 00:00:40.038
was not a speech.

00:00:40.038 --> 00:00:42.971
It was actually closer to a decree.

00:00:42.971 --> 00:00:47.659
Even though Lincoln didn't
originally come in to

00:00:47.659 --> 00:00:51.068
his presidency intending to end slavery,

00:00:51.068 --> 00:00:54.401
it became pretty clear in the
first year of the war that--

00:00:54.401 --> 00:00:55.526
- [Voiceover] He was anti-slavery.

00:00:55.526 --> 00:00:56.654
He just didn't want it to spread.

00:00:56.654 --> 00:00:58.260
He didn't think he could end it.

00:00:58.260 --> 00:00:59.467
- [Voiceover] Right, so he

00:00:59.467 --> 00:01:02.544
specifically says in his
inaugural address that he has

00:01:02.544 --> 00:01:06.616
no intention of ending slavery
and that if the slave states

00:01:06.616 --> 00:01:09.261
returned to the Union,
everything will just go back

00:01:09.261 --> 00:01:13.075
to the status quo, but it
very quickly becomes apparent

00:01:13.075 --> 00:01:17.071
in the Civil War that ending
slavery is going to have

00:01:17.071 --> 00:01:19.785
to be a goal of the North.

00:01:19.785 --> 00:01:24.643
For one thing, the slaves
provide so much of the production

00:01:24.643 --> 00:01:28.864
for the South that interfering
with that production

00:01:28.864 --> 00:01:32.714
is really important for
the North to secure victory

00:01:32.714 --> 00:01:36.016
but it also is important
on a moral ground.

00:01:36.016 --> 00:01:39.273
Almost as soon as the war begins

00:01:39.273 --> 00:01:41.953
and slave men and women
in the South start going

00:01:41.953 --> 00:01:45.864
to Union lines hoping for
the army to protect them,

00:01:45.864 --> 00:01:48.031
they want to be part of the war effort,

00:01:48.031 --> 00:01:50.588
they interpret it as a war to end slavery

00:01:50.588 --> 00:01:53.885
long before Abraham Lincoln
interprets it as a war

00:01:53.885 --> 00:01:54.896
to end slavery.

00:01:54.896 --> 00:01:59.887
So it becomes clear by
the summer of 1862 that

00:01:59.887 --> 00:02:02.302
the Union is going to
have to deal with slavery,

00:02:02.302 --> 00:02:03.162
one way or another.

00:02:03.162 --> 00:02:04.737
- [Voiceover] Now, once
again, the reason why it's

00:02:04.737 --> 00:02:06.737
strategically important is
that this war is not going

00:02:06.737 --> 00:02:08.732
as well as the North would have hoped

00:02:08.732 --> 00:02:09.712
or as Lincoln would have hoped.

00:02:09.712 --> 00:02:12.123
It's lasting longer,

00:02:12.123 --> 00:02:14.574
the first years as we mentioned
in the previous video,

00:02:14.574 --> 00:02:18.576
the South was winning and it
was going on the offensive,

00:02:18.576 --> 00:02:20.393
and then also on the moral argument,

00:02:20.393 --> 00:02:24.339
it gave the North something
more to fight for.

00:02:24.339 --> 00:02:28.916
- [Voiceover] Yes, and even
in some pockets of the South,

00:02:28.916 --> 00:02:32.349
individual Union generals were even

00:02:32.349 --> 00:02:35.181
making their own small
Emancipation Proclamation.

00:02:35.181 --> 00:02:38.279
So it became clear that
the North was going to need

00:02:38.279 --> 00:02:43.279
an overall stance on whether
this was a war to end slavery.

00:02:43.512 --> 00:02:46.504
Lincoln decides, yes,
this is going to be a war

00:02:46.504 --> 00:02:48.173
to end slavery.

00:02:48.173 --> 00:02:49.726
- [Voiceover] And so, does
he announce it, you know,

00:02:49.726 --> 00:02:52.882
just all of a sudden, or,
is he kind of waiting for

00:02:52.882 --> 00:02:54.448
the right time?

00:02:54.448 --> 00:02:56.487
- [Voiceover] He's definitely
waiting for the right time.

00:02:56.487 --> 00:03:01.487
In the summer of 1862,
he makes the first draft

00:03:01.629 --> 00:03:04.684
of the Emancipation Proclamation,

00:03:04.684 --> 00:03:07.805
and even in the summer,
he shows it to the members

00:03:07.805 --> 00:03:10.266
of his cabinets, and they're shocked.

00:03:10.266 --> 00:03:13.759
Most of them are, themselves,
strongly abolitionist

00:03:13.759 --> 00:03:17.825
but they are terrified that
this document is going to be

00:03:17.825 --> 00:03:22.615
so revolutionary at a time
when the North is already

00:03:22.615 --> 00:03:26.316
struggling, that it's going to
force the border states out,

00:03:26.316 --> 00:03:29.475
that many of the whites in the North

00:03:29.475 --> 00:03:32.547
who don't care about the fate of slaves

00:03:32.547 --> 00:03:35.109
are going to leave the war effort,

00:03:35.109 --> 00:03:40.109
so they advise Lincoln
not to just proclaim

00:03:40.268 --> 00:03:43.309
the Emancipation Proclamation,

00:03:43.309 --> 00:03:46.716
but rather to wait for a big Union victory

00:03:46.716 --> 00:03:50.702
to kind of put a stamp on the document.

00:03:50.702 --> 00:03:52.075
- [Voiceover] And just to
make sure we understand,

00:03:52.075 --> 00:03:54.804
I mean, the fear of his advisors was

00:03:54.804 --> 00:03:56.769
not all the Union states were free states.

00:03:56.769 --> 00:03:59.717
You had states like Missouri and Kentucky

00:03:59.717 --> 00:04:03.753
and Maryland that were

00:04:03.753 --> 00:04:06.516
slave states but still part of the Union

00:04:06.516 --> 00:04:08.859
and so the fear is, if you emancipate,

00:04:08.859 --> 00:04:11.407
or if you make this
proclamation, all of a sudden,

00:04:11.407 --> 00:04:12.703
those states would leave.

00:04:12.703 --> 00:04:16.053
- [Voiceover] Right, and those
are very important states.

00:04:16.053 --> 00:04:19.476
They have a very large white population,

00:04:19.476 --> 00:04:22.109
relative to their enslaved population,

00:04:22.109 --> 00:04:24.949
which means that if they go to the South,

00:04:24.949 --> 00:04:27.717
all of those fighters
will go for the South,

00:04:27.717 --> 00:04:30.320
and it's also where a lot
of the South's industry is

00:04:30.320 --> 00:04:34.486
so it would mean that a
tremendous amount of man power

00:04:34.486 --> 00:04:38.012
and also industrial power
would join the South,

00:04:38.012 --> 00:04:41.397
so Lincoln and his
cabinet are very concerned

00:04:41.397 --> 00:04:44.041
about alienating these states.

00:04:44.041 --> 00:04:46.380
They want to make sure that their power

00:04:46.380 --> 00:04:48.277
doesn't go with the South.

00:04:48.277 --> 00:04:49.415
- [Voiceover] Does the
Emancipation Proclamation

00:04:49.415 --> 00:04:50.717
end slavery in those states,

00:04:50.717 --> 00:04:52.922
or the entire United States,
or just in the South,

00:04:52.922 --> 00:04:54.497
or just in the Confederate states?

00:04:54.497 --> 00:04:59.497
- [Voiceover] So this is a
strategy of military necessity,

00:04:59.504 --> 00:05:01.070
as Lincoln says.

00:05:01.070 --> 00:05:04.597
He specifically exempts
the border states from the

00:05:04.597 --> 00:05:06.180
Emancipation Proclamation.

00:05:06.180 --> 00:05:06.842
- [Voiceover] He exempts them?

00:05:06.842 --> 00:05:07.640
- [Voiceover] He exempts them.

00:05:07.640 --> 00:05:09.797
- [Voiceover] He says, this
does not apply to them.

00:05:09.797 --> 00:05:13.337
So, it's not a pure moral
grounds that slavery is evil

00:05:13.337 --> 00:05:15.126
and needs to be gone from the Earth.

00:05:15.126 --> 00:05:19.087
He's saying, in the states
that are trying to secede,

00:05:19.087 --> 00:05:20.438
he's banning slavery.

00:05:20.438 --> 00:05:23.402
- [Voiceover] Right, and,
he almost makes it like

00:05:23.402 --> 00:05:27.818
a form of enticement for
the South to come back.

00:05:27.818 --> 00:05:29.663
He says,

00:05:29.663 --> 00:05:32.438
here is my plan for emancipation.

00:05:32.438 --> 00:05:36.271
I am going to put it into effect

00:05:36.271 --> 00:05:39.688
on the first day of January, 1863.

00:05:39.688 --> 00:05:43.644
He puts it out in

00:05:43.644 --> 00:05:46.307
September of 1862

00:05:46.307 --> 00:05:49.906
just after the Union victory at Antietam,

00:05:49.906 --> 00:05:54.844
so this big Union victory gives him the...

00:05:54.844 --> 00:05:56.289
- [Voiceover] Confidence, or--

00:05:56.289 --> 00:05:56.946
- [Voiceover] The confidence.

00:05:56.946 --> 00:05:58.423
- [Voiceover] the backdrop, the context.

00:05:58.423 --> 00:05:59.454
- [Voiceover] The context

00:05:59.454 --> 00:06:00.811
- [Voiceover] to do this.
- [Voiceover] in which he

00:06:00.811 --> 00:06:02.894
can put the Emancipation
Proclamation out there,

00:06:02.894 --> 00:06:04.154
but he says,

00:06:04.154 --> 00:06:08.470
if the states that are in
rebellion come back into the Union

00:06:08.470 --> 00:06:12.067
before January 1st, then
it will be null and void.

00:06:12.067 --> 00:06:13.748
- [Voiceover] I see, so he's
still trying to get them back

00:06:13.748 --> 00:06:15.650
in the fold, and he's
willing to accept slavery,

00:06:15.650 --> 00:06:16.971
at least for the time being,

00:06:16.971 --> 00:06:19.797
if he can end this war that's
turning out far bloodier

00:06:19.797 --> 00:06:22.334
and far longer-lasting
than frankly anyone,

00:06:22.334 --> 00:06:23.702
especially folks in the North,

00:06:23.702 --> 00:06:24.707
had expected it to be,

00:06:24.707 --> 00:06:26.873
so didn't immediately
say, hey, no more slavery,

00:06:26.873 --> 00:06:29.299
which might have made the
South fight even harder.

00:06:29.299 --> 00:06:32.226
It's like, hey, well, I'm
going to make this effective

00:06:32.226 --> 00:06:35.290
in January 1863, which
would be another year,

00:06:35.290 --> 00:06:38.089
so you have a couple months to come back,

00:06:38.089 --> 00:06:39.723
come back to the mother ship so to speak.

00:06:39.723 --> 00:06:42.343
- [Voiceover] Right, and the
South does not take him up

00:06:42.343 --> 00:06:47.343
on this offer, so when
January 1st, 1863 rolls around

00:06:47.715 --> 00:06:50.842
he signs the Emancipation Proclamation.

00:06:50.842 --> 00:06:53.282
- [Voiceover] And, we
have some of the text of

00:06:53.282 --> 00:06:55.966
the Proclamation here, and
what I think is interesting,

00:06:55.966 --> 00:06:58.446
and we have this right
on the Khan Academy site,

00:06:58.446 --> 00:07:00.559
in one of the articles, is that,

00:07:00.559 --> 00:07:03.633
it doesn't have the
beauty of the Declaration

00:07:03.633 --> 00:07:06.650
of Independence or even the Constitution.

00:07:06.650 --> 00:07:09.537
It actually reads kind
of like a legal document.

00:07:09.537 --> 00:07:10.532
I mean, if I read this,

00:07:10.532 --> 00:07:13.176
"That on the first day of
January, in the year of our Lord,

00:07:13.176 --> 00:07:15.985
"one thousand eight
hundred and sixty-three,

00:07:15.985 --> 00:07:18.632
"all persons held as slaves
within any States or designated

00:07:18.632 --> 00:07:19.459
"part of a State,"

00:07:19.459 --> 00:07:21.076
so it's kind of this legalese,

00:07:21.076 --> 00:07:24.036
"the people whereof shall
then be in rebellion against

00:07:24.036 --> 00:07:26.369
"the United States, shall
be then, thenceforth

00:07:26.369 --> 00:07:27.792
"and forever free,

00:07:27.792 --> 00:07:29.253
"and the Executive Government
of the United States,

00:07:29.253 --> 00:07:31.330
"including the military and
naval authority thereof,

00:07:31.330 --> 00:07:33.868
"will recognize and maintain
the freedom of such persons,

00:07:33.868 --> 00:07:36.892
"and will do no act or acts
to repress such persons,

00:07:36.892 --> 00:07:39.159
"or any of them in efforts,

00:07:39.159 --> 00:07:41.528
"in any efforts they make
for their actual freedom,"

00:07:41.528 --> 00:07:42.352
which is a,

00:07:42.352 --> 00:07:44.362
in spirit a really nice thing to say,

00:07:44.362 --> 00:07:45.993
but it reads like a legal document

00:07:45.993 --> 00:07:49.036
and I could keep going and I
encourage the folks listening

00:07:49.036 --> 00:07:52.034
to read it, and it literally
lists the various states

00:07:52.034 --> 00:07:54.672
and as you mentioned, it's
much more of a decree.

00:07:54.672 --> 00:07:56.437
It's not this beautiful speech.

00:07:56.437 --> 00:07:58.566
- [Voiceover] Right, and we
know Lincoln for his beautiful

00:07:58.566 --> 00:08:01.695
speeches, the Gettysburg
Address is perhaps the most

00:08:01.695 --> 00:08:04.243
famous speech in American history,

00:08:04.243 --> 00:08:07.660
and so, it's a little bit
strange that, arguably,

00:08:07.660 --> 00:08:10.449
the most important thing
that Lincoln ever does

00:08:10.449 --> 00:08:15.066
is arguably the ugliest
piece of writing that Lincoln

00:08:15.066 --> 00:08:17.250
ever put out there.

00:08:17.250 --> 00:08:19.315
It's because he's a lawyer,

00:08:19.315 --> 00:08:23.783
and we forget sometimes
about Lincoln's background

00:08:23.783 --> 00:08:26.473
as a lawyer, but he practiced for decades,

00:08:26.473 --> 00:08:29.676
and so, what he's trying
to do here is really create

00:08:29.676 --> 00:08:33.819
an ironclad legal document that
no one can turn around later

00:08:33.819 --> 00:08:35.689
and say, "Well, here's a loophole."

00:08:35.689 --> 00:08:37.181
- [Voiceover] Yup, although
it has its moments.

00:08:37.181 --> 00:08:38.003
I mean, right here,

00:08:38.003 --> 00:08:41.893
"And upon this act, sincerely
believed to be an act

00:08:41.893 --> 00:08:44.914
"of justice, warranted
by the Constitution,

00:08:44.914 --> 00:08:47.482
"upon military necessity,
I invoke the considerate

00:08:47.482 --> 00:08:49.139
"judgment of mankind,

00:08:49.139 --> 00:08:51.024
"and the gracious favor of Almighty God,"

00:08:51.024 --> 00:08:52.540
so, I mean, it's kind of,

00:08:52.540 --> 00:08:54.059
yeah, it's still kind of a legal,

00:08:54.059 --> 00:08:57.076
yeah, it's not moving to the passions

00:08:57.076 --> 00:08:59.882
or to the moral as much as, kind of,

00:08:59.882 --> 00:09:02.065
and he even also talks about,

00:09:02.065 --> 00:09:03.438
well, it is what it is.

00:09:03.438 --> 00:09:06.926
We'll let other people
judge it on their own.

00:09:06.926 --> 00:09:09.183
- [Voiceover] But, you know,
what's really important

00:09:09.183 --> 00:09:12.683
about the Emancipation Proclamation is

00:09:12.683 --> 00:09:16.216
from that point forward,
even though he's particularly

00:09:16.216 --> 00:09:20.512
putting this on the
southern states where any

00:09:20.512 --> 00:09:24.243
jurisdiction he has is really
just what can be enforced

00:09:24.243 --> 00:09:26.153
by the army,

00:09:26.153 --> 00:09:29.695
it turns the Union army
into an army of liberation,

00:09:29.695 --> 00:09:32.510
that as they go throughout the South,

00:09:32.510 --> 00:09:35.616
enslaved people will flock to them

00:09:35.616 --> 00:09:39.393
and want to contribute to the war effort

00:09:39.393 --> 00:09:42.510
and shortly after this, Lincoln will

00:09:42.510 --> 00:09:43.553
make it possible,

00:09:43.553 --> 00:09:47.733
one part of the Emancipation
Population is saying that

00:09:47.733 --> 00:09:51.824
African Americans can join
the Union army as soldiers,

00:09:51.824 --> 00:09:54.134
and so, they really,

00:09:54.134 --> 00:09:55.729
from this point forward,

00:09:55.729 --> 00:09:58.818
can play an active role,

00:09:58.818 --> 00:10:02.915
direct and recognized
role in ending slavery

00:10:02.915 --> 00:10:07.065
and this is really the moment when slavery

00:10:07.065 --> 00:10:09.310
truly starts to crumble.


Early phases of Civil War and Antietam

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.002
- [Voiceover] All right, Kim.

00:00:01.002 --> 00:00:03.836
So where we left off, Lincoln gets elected

00:00:03.836 --> 00:00:05.287
in November of 1860.

00:00:05.287 --> 00:00:08.668
He's not inaugurated
until we get into 1861.

00:00:08.668 --> 00:00:11.201
Shortly after his inauguration,
you have the whole situation

00:00:11.201 --> 00:00:14.651
at Fort Sumter, which is really
the start of the Civil War.

00:00:14.651 --> 00:00:16.120
We don't have the first major battle

00:00:16.120 --> 00:00:17.690
until we get to Bull Run.

00:00:17.690 --> 00:00:21.130
And the overall theaters of war,

00:00:21.130 --> 00:00:23.136
we have this corridor
here in the Northeast

00:00:23.136 --> 00:00:24.582
in Virginia and Maryland.

00:00:24.582 --> 00:00:27.802
And you also have it in the
west along the Mississippi.

00:00:27.802 --> 00:00:29.941
And the North, the strategy is

00:00:29.941 --> 00:00:31.341
well let's use our industrial base.

00:00:31.341 --> 00:00:32.887
Let's use our larger population.

00:00:32.887 --> 00:00:34.529
Let's use our navy to see

00:00:35.633 --> 00:00:37.147
if we can essentially blockade the South.

00:00:37.147 --> 00:00:38.715
Well the South says, "Hey
we have the home court.

00:00:38.715 --> 00:00:39.873
"We have better leadership.

00:00:39.873 --> 00:00:42.611
"We just need to outlast the North."

00:00:42.611 --> 00:00:44.273
And so what happens as we get started,

00:00:44.273 --> 00:00:48.030
we talked about Bull Run
being the first major battle.

00:00:48.030 --> 00:00:49.943
Who kind of

00:00:51.352 --> 00:00:54.032
comes out better in some
of these first engagements?

00:00:54.032 --> 00:00:56.198
- [Voiceover] Well I think
it's a surprise to everyone

00:00:56.198 --> 00:01:00.149
when the South does much
better in the first year

00:01:00.149 --> 00:01:01.581
of the war than the North.

00:01:01.581 --> 00:01:06.305
Knowing the major advantages
that the North has

00:01:06.305 --> 00:01:10.284
in industrial power, in railroads,

00:01:10.284 --> 00:01:13.252
and just in the sheer number of people,

00:01:14.385 --> 00:01:18.374
it's very surprising that the leadership

00:01:18.374 --> 00:01:21.493
in the South does such an incredible job

00:01:21.493 --> 00:01:24.147
of really blocking the North's advances.

00:01:24.147 --> 00:01:27.330
The North is attempting to take Richmond,

00:01:27.330 --> 00:01:30.719
and Lee repeatedly keeps

00:01:30.719 --> 00:01:34.887
General McClellan from
getting to Richmond.

00:01:34.887 --> 00:01:36.600
- [Voiceover] Right and
then Lee actually goes

00:01:36.600 --> 00:01:38.732
on the offensive to some degree.

00:01:38.732 --> 00:01:40.489
The South essentially wins in Bull Run,

00:01:40.489 --> 00:01:41.762
and they have a series of victories

00:01:41.762 --> 00:01:43.923
as you mentioned in year one.

00:01:43.933 --> 00:01:48.838
- [Voiceover] Right, so one
problem that the North has

00:01:48.838 --> 00:01:52.034
is that Lincoln's generals

00:01:52.034 --> 00:01:54.629
are just not nearly as skilled.

00:01:54.629 --> 00:01:57.265
George B. McClellan
that we've talked about,

00:01:57.265 --> 00:02:01.687
his idea of the South's power is perhaps

00:02:01.687 --> 00:02:04.518
considerably greater than
the South's actual power.

00:02:04.518 --> 00:02:08.371
As he is forever telling
Lincoln, "I need more troops.

00:02:08.372 --> 00:02:09.882
"I need more supplies.

00:02:09.882 --> 00:02:12.319
"Send me more things."

00:02:12.320 --> 00:02:15.200
He loves parading his army,

00:02:15.200 --> 00:02:16.655
but I think he was actually

00:02:16.655 --> 00:02:18.689
a little too close to the troops himself.

00:02:18.689 --> 00:02:21.184
He was really afraid to lose anyone,

00:02:21.184 --> 00:02:23.304
which made him very popular with the army,

00:02:23.304 --> 00:02:27.227
but drove Lincoln crazy
because the North comes out

00:02:27.227 --> 00:02:30.189
with this really strong numeric

00:02:30.189 --> 00:02:32.206
and industrial advantage.

00:02:32.206 --> 00:02:36.134
And as McClellan delays,
it gives the South time

00:02:36.134 --> 00:02:38.636
to build things up over and over again.

00:02:38.636 --> 00:02:41.259
In fact Lincoln, who we
often think of as being sort

00:02:41.259 --> 00:02:45.787
of this great-grandfatherly,
sweet character,

00:02:45.787 --> 00:02:49.637
who has so many words of
wisdom, his letters to McClellan

00:02:49.637 --> 00:02:51.434
are downright snarky.

00:02:51.434 --> 00:02:53.702
He says to McClellan,

00:02:53.702 --> 00:02:55.399
"If you're not using the army,

00:02:55.399 --> 00:02:57.413
"could I borrow it?"
(laughter)

00:02:58.473 --> 00:03:00.421
- [Voiceover] And is that
what historians believe too?

00:03:00.421 --> 00:03:03.356
It looks like Lincoln felt
that the reason why year one

00:03:03.356 --> 00:03:04.475
went in favor of the South.

00:03:04.475 --> 00:03:06.903
And what we talked
about in previous years,

00:03:06.903 --> 00:03:08.699
everyone thought this was
gonna be a fast engagement.

00:03:08.699 --> 00:03:10.482
The North had all of these advantages.

00:03:11.621 --> 00:03:13.293
Lincoln believed that maybe McClellan

00:03:13.293 --> 00:03:14.670
wasn't being aggressive enough.

00:03:14.670 --> 00:03:16.250
And do historians
- [Voiceover] Yes absolutely.

00:03:16.250 --> 00:03:17.251
- [Voiceover] believe that too?

00:03:17.251 --> 00:03:19.122
- [Voiceover] Yeah, no
I think that's true.

00:03:19.122 --> 00:03:21.009
It's really borne out by the numbers,

00:03:21.009 --> 00:03:24.907
that in many cases where
McClellan thought he was facing

00:03:24.907 --> 00:03:27.713
just thousands of troops,
he was really only facing

00:03:27.713 --> 00:03:29.315
a fraction of that.

00:03:29.315 --> 00:03:31.521
- [Voiceover] And so that
made him be a little bit more

00:03:31.521 --> 00:03:32.518
cautious?
- [Voiceover] He was very

00:03:32.518 --> 00:03:33.964
- [Voiceover] cautious.

00:03:33.964 --> 00:03:35.403
- [Voiceover] At what
point is the turning point,

00:03:35.403 --> 00:03:37.818
at least in these early
stages of the Civil War,

00:03:37.818 --> 00:03:39.363
as we have here in this timeline?

00:03:39.363 --> 00:03:43.858
We go from April 1861
to roughly April 1865.

00:03:43.858 --> 00:03:45.238
The first year,

00:03:46.403 --> 00:03:47.676
I can draw that.

00:03:47.676 --> 00:03:49.496
So the first year would be

00:03:50.701 --> 00:03:53.125
roughly this.

00:03:54.761 --> 00:03:57.457
We've had several battles after Bull Run,

00:03:57.457 --> 00:03:58.732
but then we get to Antietam.

00:03:58.732 --> 00:04:01.673
- [Voiceover] Right so Lee,
since he's done so well

00:04:01.673 --> 00:04:03.630
in Virginia, he decides that he's

00:04:03.630 --> 00:04:05.128
gonna take the army to the North.

00:04:05.128 --> 00:04:07.326
This is the first time that he heads up

00:04:07.326 --> 00:04:10.156
into the border state of Maryland,

00:04:10.156 --> 00:04:14.563
and he meets at Antietam
Creek with McClellan.

00:04:14.563 --> 00:04:16.155
- [Voiceover] And this goes
back to the naming conventions

00:04:16.155 --> 00:04:17.154
between the North and South.

00:04:17.154 --> 00:04:18.902
It's called Antietam,
that's the body of water,

00:04:18.902 --> 00:04:21.048
which the North does.

00:04:21.048 --> 00:04:22.915
This is Antietam right there.

00:04:22.915 --> 00:04:24.512
- [Voiceover] Right well
the South refers to it

00:04:24.512 --> 00:04:27.367
as the nearby town which
is Sharpsburg, Maryland.

00:04:27.367 --> 00:04:29.078
And once again,
- [Voiceover] I see.

00:04:29.078 --> 00:04:30.108
- [Voiceover] this is a big deal.

00:04:30.108 --> 00:04:32.366
This is the South invading the North now,

00:04:32.366 --> 00:04:33.569
taking the offensive.

00:04:33.569 --> 00:04:36.897
- [Voiceover] Right and this is

00:04:36.897 --> 00:04:39.473
the bloodiest day

00:04:39.473 --> 00:04:41.336
in American history when--

00:04:41.336 --> 00:04:42.586
- [Voiceover] Let me make sure

00:04:42.586 --> 00:04:43.756
I digested what you just said.

00:04:43.756 --> 00:04:45.496
The bloodiest day?

00:04:46.639 --> 00:04:49.938
I imagine things like
Pearl Harbor and D-Day.

00:04:49.938 --> 00:04:53.720
- [Voiceover] Right,
so 4,000 Americans died

00:04:53.720 --> 00:04:57.209
on a single day, September 17th, 1862

00:04:57.209 --> 00:04:59.851
when these two armies meet at Antietam.

00:04:59.851 --> 00:05:03.837
And on no other day in American history

00:05:03.837 --> 00:05:06.007
have so many Americans died,

00:05:06.007 --> 00:05:08.407
not even on September 11th

00:05:08.407 --> 00:05:10.759
did that many Americans die.

00:05:10.759 --> 00:05:13.712
- [Voiceover] And was
this a surprise to folks?

00:05:13.712 --> 00:05:15.466
- [Voiceover] Yeah, well I think

00:05:17.001 --> 00:05:20.964
one of the truisms, perhaps
about military strategy

00:05:20.964 --> 00:05:23.604
in general, is that
people are always planning

00:05:23.604 --> 00:05:25.324
for the last war.

00:05:25.324 --> 00:05:27.933
They're not planning for the
next war, and so they learn

00:05:27.933 --> 00:05:30.759
from their mistakes,
but what they don't know

00:05:30.759 --> 00:05:33.071
how to do always is anticipate

00:05:33.071 --> 00:05:34.960
what's going to be new about this war.

00:05:34.960 --> 00:05:38.483
And there were so many new inventions

00:05:38.483 --> 00:05:41.198
during this time period that
really made the Civil War

00:05:41.204 --> 00:05:43.419
an incredibly deadly war.

00:05:43.419 --> 00:05:44.497
- [Voiceover] Yeah and you can see.

00:05:44.497 --> 00:05:45.507
These are pictures.

00:05:45.507 --> 00:05:46.768
These are Antietam right here?

00:05:46.768 --> 00:05:47.744
- [Voiceover] This is Antietam,

00:05:47.744 --> 00:05:48.755
yes.
- [Voiceover] And this looks

00:05:48.755 --> 00:05:50.143
- [Voiceover] like Lincoln and McClellan.

00:05:50.143 --> 00:05:51.756
- [Voiceover] Right, meeting at Antietam.

00:05:51.756 --> 00:05:52.768
- [Voiceover] Which is incredibly bloody.

00:05:52.768 --> 00:05:56.864
And you talk about new
technologies or new weapons,

00:05:56.864 --> 00:05:59.919
this rifle here looks like one of 'em.

00:05:59.919 --> 00:06:01.108
- [Voiceover] (chuckles) Yes.

00:06:01.108 --> 00:06:04.192
So this is a war where
there's a transition

00:06:04.192 --> 00:06:07.007
from the musket to the rifle,

00:06:07.007 --> 00:06:09.490
and what's different about a rifle

00:06:09.490 --> 00:06:12.517
is that inside the barrel of a rifle,

00:06:12.517 --> 00:06:14.762
there is a

00:06:14.770 --> 00:06:16.616
sort of spiral-shaped groove.

00:06:16.616 --> 00:06:20.447
And the spiral-shaped
groove makes the rifle

00:06:20.447 --> 00:06:23.323
much more accurate at a
much farther distance.

00:06:23.323 --> 00:06:25.055
It's sort of the difference

00:06:25.055 --> 00:06:28.453
between just hurling a
football end over end

00:06:28.453 --> 00:06:32.272
and throwing a spiral,
so you can hit a target

00:06:32.272 --> 00:06:33.638
at 600 yards,

00:06:33.638 --> 00:06:36.115
which is much greater.
- [Voiceover] Was unheard of

00:06:36.115 --> 00:06:38.527
- [Voiceover] for a musket or
hard, very hard with a musket.

00:06:38.527 --> 00:06:40.042
So it gets the bullet spinning,

00:06:40.042 --> 00:06:41.592
which keeps it on its trajectory better.

00:06:41.592 --> 00:06:42.958
- [Voiceover] Exactly.

00:06:42.958 --> 00:06:47.414
So we have much more accurate technology

00:06:47.414 --> 00:06:50.479
and old military strategy.

00:06:50.482 --> 00:06:52.912
If you see paintings of,

00:06:52.912 --> 00:06:55.149
for example the Napoleonic Wars,

00:06:55.149 --> 00:06:58.689
just involved a whole
bunch of soldiers lining up

00:06:58.689 --> 00:06:59.914
and going toward each other.

00:06:59.914 --> 00:07:01.621
Well when you've got soldiers in a line

00:07:01.621 --> 00:07:03.902
and very accurate weapons--

00:07:03.902 --> 00:07:05.068
- [Voiceover] I never got why that ever

00:07:05.068 --> 00:07:06.397
made sense though.
(Kim laughs)

00:07:06.397 --> 00:07:07.667
- [Voiceover] I'm not sure I do

00:07:07.667 --> 00:07:08.850
either to be perfectly
- [Voiceover] Yeah I don't

00:07:08.850 --> 00:07:09.844
- [Voiceover] honest.
- [Voiceover] consider myself

00:07:09.844 --> 00:07:10.969
- [Voiceover] a great military strategist,

00:07:10.969 --> 00:07:13.387
but wearing these
(Kim laughs)

00:07:13.387 --> 00:07:16.528
bright uniforms and marching in step

00:07:16.528 --> 00:07:18.021
in these kind of--
- [Voiceover] Yeah it does

00:07:18.021 --> 00:07:20.093
- [Voiceover] seem to make
you a very good target.

00:07:20.093 --> 00:07:23.239
- [Voiceover] Yes, yes, but
anyway, you have the rifle now,

00:07:23.239 --> 00:07:24.256
much more

00:07:25.262 --> 00:07:26.983
accurate

00:07:26.983 --> 00:07:29.476
and you end up with scenes like this,

00:07:29.476 --> 00:07:31.500
but what was the outcome of Antietam?

00:07:32.571 --> 00:07:36.247
- [Voiceover] There are two
very major outcomes of Antietam,

00:07:36.247 --> 00:07:37.264
I would say.

00:07:37.264 --> 00:07:40.173
One, on the negative side for the North,

00:07:40.173 --> 00:07:44.039
is this is a battle that
is widely photographed

00:07:44.039 --> 00:07:45.314
as you can see.

00:07:45.314 --> 00:07:48.374
Mathew Brady, who was the leading

00:07:48.374 --> 00:07:51.487
photography studio owner of his time,

00:07:51.487 --> 00:07:53.243
and it's one t.

00:07:53.243 --> 00:07:54.275
- [Voiceover] Mathew, okay.

00:07:54.275 --> 00:07:55.744
- [Voiceover] Yeah, very
important, only one t.

00:07:55.744 --> 00:07:56.744
- [Voiceover] Yes, a
(Kim laughs)

00:07:56.744 --> 00:07:59.134
nontraditional spelling of Matthew.

00:08:00.301 --> 00:08:03.076
- [Voiceover] He sends
out his photographer

00:08:03.076 --> 00:08:05.540
that works for him,
named Alexander Gardner,

00:08:05.540 --> 00:08:08.526
and they have roving
photographers for the first time.

00:08:08.526 --> 00:08:10.084
They have wagons.

00:08:10.084 --> 00:08:11.084
And they take--
- [Voiceover] 'Cause

00:08:11.084 --> 00:08:13.991
- [Voiceover] photography's
just becoming a used technology

00:08:13.991 --> 00:08:16.503
at this time.
- [Voiceover] Right and so

00:08:16.511 --> 00:08:20.092
- [Voiceover] they have
Alexander Gardner photograph

00:08:20.092 --> 00:08:22.797
the battlefields at Antietam.

00:08:22.797 --> 00:08:24.525
And as you can see.

00:08:24.525 --> 00:08:25.655
- [Voiceover] Like a E-R?

00:08:25.655 --> 00:08:26.663
- [Voiceover] That's right, yes.

00:08:26.663 --> 00:08:28.249
- [Voiceover] Gardner.

00:08:28.249 --> 00:08:30.185
- [Voiceover] As you can see,

00:08:30.190 --> 00:08:33.344
this is just about as far away

00:08:33.344 --> 00:08:35.979
from the kind of heroic paintings

00:08:35.979 --> 00:08:37.511
of what battles looked like

00:08:37.511 --> 00:08:40.404
that people had been used to
seeing up until this point.

00:08:40.404 --> 00:08:43.809
These don't look like the sort of heroes

00:08:43.809 --> 00:08:46.562
of the Revolutionary War,
like George Washington.

00:08:46.562 --> 00:08:47.655
This is,

00:08:47.655 --> 00:08:48.909
this is gruesome.
- [Voiceover] Gruesome.

00:08:48.909 --> 00:08:50.382
- [Voiceover] And this
is really fascinating

00:08:50.382 --> 00:08:52.454
'cause we take it for granted
in today's day and age

00:08:52.454 --> 00:08:56.702
is that the effect of media
on people's perception

00:08:56.702 --> 00:08:57.782
of things like war.

00:08:57.782 --> 00:09:01.642
Before the camera, before photographs,

00:09:01.642 --> 00:09:05.396
if I'm a civilian, I just hear
about these great stories,

00:09:05.396 --> 00:09:07.855
and I see these paintings
that look very valiant

00:09:07.855 --> 00:09:09.625
and very heroic.

00:09:09.625 --> 00:09:12.412
But now with photographs,
you see the grim reality

00:09:12.412 --> 00:09:14.722
of war, people just shot in their tracks.

00:09:14.722 --> 00:09:18.484
And young men just kind of

00:09:18.484 --> 00:09:19.770
just piled up.

00:09:20.741 --> 00:09:22.912
It's just very dark.

00:09:24.282 --> 00:09:27.505
- [Voiceover] Yeah and
it's a real PR problem

00:09:27.505 --> 00:09:29.248
for the North

00:09:29.248 --> 00:09:32.988
because this is before we
can really put photography

00:09:32.988 --> 00:09:34.024
in newspapers.

00:09:34.024 --> 00:09:35.997
They don't have that technology yet,

00:09:35.997 --> 00:09:38.965
but these photographs were put on display

00:09:38.965 --> 00:09:40.488
in Brady's studios.

00:09:40.488 --> 00:09:43.772
He had one in Washington,
D.C. and one in New York City,

00:09:43.772 --> 00:09:46.569
and people would go and
look at these photographs,

00:09:46.569 --> 00:09:48.103
and it was very shocking to them.

00:09:48.103 --> 00:09:50.682
It was a level of detail
that they had never seen.

00:09:50.682 --> 00:09:53.536
In some cases, they
could make out the faces

00:09:53.536 --> 00:09:56.195
of the individual men who
were dead on the battlefield.

00:09:56.195 --> 00:09:59.921
And that just seemed beyond

00:09:59.921 --> 00:10:01.979
what was imaginable to these people.

00:10:01.979 --> 00:10:05.928
To think of some poor person
going into one of these studios

00:10:05.928 --> 00:10:09.603
and seeing their son dead
there on the battlefield

00:10:09.603 --> 00:10:13.809
made it very difficult for
the North to keep up morale.

00:10:13.809 --> 00:10:15.503
- [Voiceover] Fascinating.