### Video Transcript

Find the value of π that makes the
expression 30π₯ to the fifth power plus 57π₯ squared minus 48π₯ cubed minus 20π₯ to
the fourth power plus π divisible by five π₯ squared minus eight.

Now, for this expression to be
divisible by five π₯ squared minus eight, that tells us that five π₯ squared minus
eight is a factor of it. If this is the case, then when we
perform the division, we should have no remainder, or a remainder of zero. So, letβs perform the long
division. Now, if we look carefully at our
expression, we see that the terms appear to be in a bit of a funny order. They are usually written in
decreasing powers of π₯. And so, we might be tempted to
rearrange them and write it as 30π₯ to the fifth power minus 20π₯ to the fourth
power, and so on. Doing it this way, weβll make the
problem a little bit easier, but it is going to look a little bit strange.

Letβs begin as normal. We divide the term in our dividend
with highest power of π₯ by the term in the divisor, also with the highest power of
π₯. 30 divided by five is six. Then, when weβre dividing numbers
whose bases are the same, we subtract their exponents. So, π₯ to the fifth power divided
by π₯ squared is π₯ to the power of five minus two, which is π₯ cubed. This means that 30π₯ to the fifth
power divided by five π₯ squared is six π₯ cubed. And so, we write six π₯ cubed above
this term. We now multiply this value by each
term in our divisor. Six π₯ cubed times five π₯ squared
gives us 30π₯ to the fifth power. And then, when we multiply six π₯
cubed by negative eight, we get negative 48π₯ cubed.

Now weβre going to line that up
directly underneath the π₯ cubed terms. And weβre going to add in another
term; weβre going to add in zero π₯ to the fourth power. This isnβt entirely necessary, but
it can make it a little bit easier to follow what happens next. Our next step is to divide each of
these three terms by the corresponding terms above. 30π₯ to the fifth power minus 30π₯
to the fifth power is zero. Negative 20π₯ to the fourth power
minus zero π₯ to the fourth power is negative 20π₯ to the fourth power. And negative 48π₯ cubed minus
negative 48π₯ cubed is zero.

Next, we bring down 57π₯
squared. And weβre now going to divide
negative 20π₯ to the fourth power by five π₯ squared. Negative 20 divided by five is
negative four, and π₯ to the fourth power divided by π₯ squared is just π₯
squared. And so, when we do this division,
we get negative four π₯ squared. And this is the next term in our
quotient. Remember, each time we find a term
in our quotient, we multiply it by each part of the divisor. So, weβre going to work out
negative four π₯ squared times five π₯ squared, which is negative 20π₯ to the fourth
power.

When we multiply negative eight by
negative four π₯ squared, we get 32π₯ squared. So, we can line this up directly
under 57π₯ squared. And then, we subtract each of these
terms from the terms immediately above them. Negative 20π₯ to the fourth power
minus negative 20π₯ to the fourth power is zero. Then, 57π₯ squared minus 32π₯
squared is 25π₯ squared. And weβre nearly there. We bring down the final term;
thatβs the π.

Now, donβt worry too much that we
donβt yet know what π is. Remember, weβre looking for a
remainder of zero. Weβre going to divide 25π₯ squared
by five squared. Well, 25 divided by five is five,
and π₯ squared divided by π₯ squared is one. So, the final term in our quotient
is five. We take that five and we multiply
it by five π₯ squared and negative eight. And that gives us 25π₯ squared
minus 40.

Now, we know that since weβre
subtracting the final two terms, weβre going to get a remainder of zero. So, 25π₯ squared plus π minus 25π₯
squared minus 40 has to give us zero. Well, 25π₯ squared minus 25π₯
squared is also zero. And so to ensure that our remainder
is zero, and thus our expression is divisible by five π₯ squared minus eight, we can
say that π minus negative 40 itself must be equal to zero. π minus negative 40 is, of course,
π plus 40.

And so letβs subtract 40 from both
sides to solve for π. That gives us π is equal to
negative 40. And so, the value of π that makes
our expression divisible by five π₯ squared minus eight is π equals negative
40. Note that at this point, we could
go ahead and multiply our quotient. Thatβs six π₯ cubed minus four π₯
squared plus five by five π₯ squared minus eight. If we had performed the calculation
correctly, we would end up with the expression 30π₯ to the fifth power plus 57π₯
squared minus 48π₯ cubed minus 20π₯ to the fourth power minus 40.