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# Dividing polynomials: long division

Sal divides (x^2-3x+2) by (x-2) and then checks the solution. Created by Sal Khan and Monterey Institute for Technology and Education.

Video transcript

Divide x squared minus 3x
plus 2 divided by x minus 2. So we're going to
divide this into that. And we can do this really
the same way that you first learned long division. So we have x minus
2 being divided into x squared minus 3x plus 2. Another way we
could have written the same exact expression is
x squared minus 3x plus 2, all of that over x minus 2. That, that, and that are
all equivalent expressions. Now, to do this type
of long division-- we can call it algebraic
long division-- you want to look at the highest
degree term on the x minus 2 and the highest degree term on
the x squared minus 3x plus 2. And here's the x, and
here's the x squared. x goes into x squared
how many times? Or x squared divided
by x is what? Well, that's just equal to x. So x goes into x
squared x times. And I'm going to write
it in this column right here above all of the x terms. And then we want to
multiply x times x minus 2. That gives us-- x
times x is x squared. x times negative
2 is negative 2x. And just like you first
learned in long division, you want to subtract
this from that. But that's completely the
same as adding the opposite, or multiplying each of
these terms by negative 1 and then adding. So let's multiply
that times negative 1. And negative 2x times
negative 1 is positive 2x. And now let's add. x squared minus x
squared-- those cancel out. Negative 3x plus 2x--
that is negative x. And then we can bring
down this 2 over here. So it's negative x plus 2 left
over, when we only go x times. So then we say, can x minus
2 go into negative x plus 2? Well, x goes into negative
x negative one times. You can look at it right here. Negative x divided
by x is negative 1. These guys cancel out. Those guys cancel out. So negative 1 times x minus
2-- you have negative 1 times x, which is negative x. Negative 1 times
negative 2 is positive 2. And we want to subtract
this from that, just like you do
in long division. But that's the same thing
as adding the opposite, or multiplying each of
these terms by negative 1 and then adding. So negative x times
negative 1 is positive x. Positive 2 times
negative 1 is negative 2. These guys cancel
out, add up to 0. These guys add up to 0. We have no remainder. So we got this as being
equal to x minus 1. And we can verify it. If we multiply x minus 1 times
x minus 2, we should get this. So let's actually do that. So let's multiply x
minus 1 times x minus 2. So let's multiply negative
2 times negative 1. That gives us positive 2. Negative 2 times x--
that's negative 2x. Let's multiply x
times negative 1. That is negative x. And then x times x is x squared. And then add all the like terms. x squared, negative 2x minus
x-- that's negative 3x. And then 2 plus
nothing is just 2. And so we got that
polynomial again.