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# Dividing polynomials with remainders: monomial divisors

Video transcript

Simplify the expression
18x to the fourth minus 3x squared plus 6x minus
4, all of that over 6x. So there's a couple ways
to think about them. They're all really equivalent. You can really just
view this up here as being the exact same thing
as 18x to the fourth over 6x plus negative 3x
squared over 6x, or you could say minus
3x squared over 6x, plus 6x over 6x,
minus 4 over 6x. Now, there's a couple of
ways to think about it. One is I just kind of decomposed
this numerator up here. If I just had a bunch
of stuff, a plus b plus c over d, that's clearly
equal to a/d plus b/d plus c/d. Or maybe not so clearly,
but hopefully that helps clarify it. Another way to think
about it is kind of like you're
distributing the division. If I divide a whole
expression by something, that's equivalent to
dividing each of the terms by that something. The other way to
think about it is that we're multiplying
this entire expression. So this is the same
thing as 1 over 6x times this entire thing, times
18x to the fourth minus 3x squared plus 6x minus 4. And so here, this would just
be the straight distributive property to get to this. Whatever seems to
make sense for you-- they're are all equivalent. They're all logical, good things
to do to simplify this thing. Now, once you have
it here, now we just have a bunch of monomials that
we're just dividing by 6x. And here, we could just
use exponent properties. This first one over here,
we can take the coefficients and divide them. 18 divided by 6 is 3. And then you have x
to the fourth divided by x to the-- well,
they don't tell us. But if it's just an x,
that's the same thing as x to the first power. So it's x to the fourth
divided by x to the first. That's going to be x to
the 4 minus 1 power, or x to the third power. Then we have this
coefficient over here, or these coefficients. We have negative 3 divided by 6. So I'm going to
do this part next. Negative 3 divided
by 6 is negative 1/2. And then you have x
squared divided by x. We already know that x is the
same thing as x to the first. So that's going to be x to the 2
minus 1 power, which is just 1. Or I could just leave
it as an x right there. Then we have these
coefficients, 6 divided by 6. Well, that's just 1. So I could just--
well, I'll write it. I could write a 1 here. And let me just write the 1
here, because we said 2 minus 1 is 1. And then x divided by x is x to
the first over x to the first. You could view it two ways--
anything divided by anything is just 1. Or you could view it as x to
the 1 divided by x to the 1 is going to be x
to the 1 minus 1, which is x to the 0,
which is also equal to 1. Either way, you knew how
to do this before you even learned that exponent property. Because x divided by x
is 1, and then assuming x does not equal to 0. And we kind of have to
assume x doesn't equal 0 in this whole thing. Otherwise, we would
be dividing by 0. And then finally,
we have 4 over 6x. And there's a couple of
ways to think about it. So the simplest way
is negative 4 over 6 is the same thing
as negative 2/3. Just simplified that fraction. And we're multiplying
that times 1/x. So we can view this 4 times 1/x. Another way to think
about it is you could have viewed this 4 as
being multiplied by x to the 0 power, and this being
x to the first power. And then when you tried to
simplify it using your exponent properties, you would have--
well, that would be x to the 0 minus 1 power, which is x
to the negative 1 power. So we could have written
an x to the negative 1 here, but x to the negative 1
is the exact same thing as 1 over x. And so let's just write our
answer completely simplified. So it's going to be 3x to the
third minus 1/2 x plus 1-- because this thing
right here is just 1-- and then minus 2 times
1 in the numerator over 3 times x in
the denominator. And we are done. Or we could write this. Depending on what you
consider more simplified, this last term right
here could also be written in minus 2/3
x to the negative 1. But if you don't want
a negative exponent, you could write it like that.