### Video Transcript

Knowing that the length of a
rectangle is three π₯ minus four and its area is six π₯ to the fourth minus eight π₯
cubed plus nine π₯ squared minus nine π₯ minus four, express the width of the
rectangle as a polynomial in standard form.

Here we have a rectangle, its
length is equal to three π₯ minus four. We are solving for the width. And then our area is equal to this
polynomial. Length times width is equal to the
area. So if we are solving for the width,
which will be π, we need to divide both sides by πΏ. This means, in order to find the
width, we need to take the area and divide by the length.

So here we can solve by dividing,
and we can use something called long division. So we begin by looking at the first
terms. What do we multiply to three π₯ so
it looks like six π₯ to the fourth? We will need to multiply it by two
π₯ cubed, so we put that above the cubed term. And now we distribute. And now we subtract. So all of these cancel. So this means we need to bring down
the nine π₯ squared. And now we start over.

How do we get three π₯ to look like
nine π₯ squared? What do we multiply by? And that would be three π₯, we put
above the π₯ term. We can always add in a zero for
something that isnβt being used. So for here, thatβll be zero π₯
squared. We do not need to include it in our
final answer. So now we distribute the three
π₯. Now after weβve distributed, we
need another term, so we bring down the negative nine π₯. And now we subtract. So we have negative nine π₯ minus
negative 12π₯, so itβs really plus 12π₯. So we get three π₯, and we start
the process over again.

How do we get three π₯ to look like
three π₯? What do we multiply by? And that would be one. So we distribute, bring down the
minus four, and subtract. And we get zero. So this means our remainder is
zero.

So this means our width is two π₯
cubed plus three π₯ plus one.