### Video Transcript

Knowing that the volume of a box is
12π₯ cubed plus 20π₯ squared minus 21π₯ minus 36, its length is two π₯ plus three,
and its width is three π₯ minus four, express the height of the box
algebraically.

We assume here that this box is in
the shape of a cuboid. So its volume will be equal to its
length multiplied by its width multiplied by its height. Weβve been given an algebraic
expression for the volume, 12π₯ cubed plus 20π₯ squared minus 21 π₯ minus 36, as
well as algebraic expressions for the length and width of the box, two π₯ plus three
and three π₯ minus four. So we can substitute each of these
expressions into the volume formula. To find the height of this box, we
need to divide both sides of this equation by the expressions two π₯ plus three and
three π₯ minus four, giving the quotient on the left of the screen.

Weβre going to need to simplify
this quotient using polynomial division. But before we do, letβs expand the
brackets in the denominator. Two π₯ multiplied by three π₯ gives
six π₯ squared. Two π₯ multiplied by negative four
gives negative eight π₯. Positive three multiplied by three
π₯ gives positive nine π₯. And positive three multiplied by
negative four gives negative 12. We can simplify by grouping the
like terms in the center of the expansion to give six π₯ squared plus π₯ minus
12. So the height of the box is equal
to the quotient 12π₯ cubed plus 20π₯ squared minus 21π₯ minus 36 over six π₯ squared
plus π₯ minus 12. And we use algebraic division to
simplify.

We set up our division with the
divisor on the outside and the expression weβre dividing into on the inside. We begin by looking at the highest
powers of these two expressions. And we ask ourselves, βWhat do we
need to multiply six π₯ squared by to give 12 π₯ cubed?β Well, we need to multiply six by
two to give 12 and π₯ squared by π₯ to give π₯ cubed. So overall, we must multiply by two
π₯.

We now multiply the full expression
six π₯ squared plus π₯ minus 12 by two π₯. Six π₯ squared multiplied by two π₯
gives 12π₯ cubed, as already discussed. π₯ multiplied by two π₯ gives two
π₯ squared. And negative 12 multiplied by two
π₯ gives negative 24π₯. Next, we subtract this expression
from the original expression. 12π₯ cubed minus 12π₯ cubed gives
zero. 20π₯ squared minus two π₯ squared
gives 18π₯ squared. And negative 21π₯ minus negative
24π₯ becomes negative 21π₯ plus 24π₯, which is positive three π₯. We also have negative 36 minus
zero. So, we still have negative 36.

We still have a remainder at this
point. So, we keep going. And again, we look at the highest
powers, asking ourselves, βWhat do we multiply six π₯ squared by to give 18π₯
squared?β We must multiply by three. Six multiplied by three is 18. So six π₯ squared multiplied by
three is 18π₯ squared. Now we multiply the full expression
by three. Six π₯ squared multiplied by three
gives 18π₯ squared. π₯ multiplied by three gives plus
three π₯. And negative 12 multiplied by three
gives negative 36.

As before, we subtract in
columns. 18π₯ squared minus 18π₯ squared
gives zero. Three π₯ minus three π₯ gives
zero. And negative 36 minus negative 36 β
thatβs negative 36 plus 36 β is also zero. So, we have no remainder. And weβve therefore reached the end
of our algebraic division. In total, we multiplied six π₯
squared plus π₯ minus 12 by two π₯ and then by positive three. So overall, weβve multiplied by two
π₯ plus three. This tells us that the answer to
the algebraic division 12π₯ cubed plus 20π₯ squared minus 21π₯ minus 36 divided by
six π₯ squared plus π₯ minus 12 is two π₯ plus three. And so this is the height of this
box expressed algebraically. The height is two π₯ plus
three.