Video Transcript
Find the value of 𝑘 that makes the
expression 30𝑥 to the fifth power plus 57𝑥 squared minus 48𝑥 cubed minus 20𝑥 to
the fourth power plus 𝑘 divisible by five 𝑥 squared minus eight.
Now, for this expression to be
divisible by five 𝑥 squared minus eight, that tells us that five 𝑥 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, let’s 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 𝑥. And so, we might be tempted to
rearrange them and write it as 30𝑥 to the fifth power minus 20𝑥 to the fourth
power, and so on. Doing it this way, we’ll make the
problem a little bit easier, but it is going to look a little bit strange.
Let’s begin as normal. We divide the term in our dividend
with highest power of 𝑥 by the term in the divisor, also with the highest power of
𝑥. 30 divided by five is six. Then, when we’re dividing numbers
whose bases are the same, we subtract their exponents. So, 𝑥 to the fifth power divided
by 𝑥 squared is 𝑥 to the power of five minus two, which is 𝑥 cubed. This means that 30𝑥 to the fifth
power divided by five 𝑥 squared is six 𝑥 cubed. And so, we write six 𝑥 cubed above
this term. We now multiply this value by each
term in our divisor. Six 𝑥 cubed times five 𝑥 squared
gives us 30𝑥 to the fifth power. And then, when we multiply six 𝑥
cubed by negative eight, we get negative 48𝑥 cubed.
Now we’re going to line that up
directly underneath the 𝑥 cubed terms. And we’re going to add in another
term; we’re going to add in zero 𝑥 to the fourth power. This isn’t 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. 30𝑥 to the fifth power minus 30𝑥
to the fifth power is zero. Negative 20𝑥 to the fourth power
minus zero 𝑥 to the fourth power is negative 20𝑥 to the fourth power. And negative 48𝑥 cubed minus
negative 48𝑥 cubed is zero.
Next, we bring down 57𝑥
squared. And we’re now going to divide
negative 20𝑥 to the fourth power by five 𝑥 squared. Negative 20 divided by five is
negative four, and 𝑥 to the fourth power divided by 𝑥 squared is just 𝑥
squared. And so, when we do this division,
we get negative four 𝑥 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, we’re going to work out
negative four 𝑥 squared times five 𝑥 squared, which is negative 20𝑥 to the fourth
power.
When we multiply negative eight by
negative four 𝑥 squared, we get 32𝑥 squared. So, we can line this up directly
under 57𝑥 squared. And then, we subtract each of these
terms from the terms immediately above them. Negative 20𝑥 to the fourth power
minus negative 20𝑥 to the fourth power is zero. Then, 57𝑥 squared minus 32𝑥
squared is 25𝑥 squared. And we’re nearly there. We bring down the final term;
that’s the 𝑘.
Now, don’t worry too much that we
don’t yet know what 𝑘 is. Remember, we’re looking for a
remainder of zero. We’re going to divide 25𝑥 squared
by five squared. Well, 25 divided by five is five,
and 𝑥 squared divided by 𝑥 squared is one. So, the final term in our quotient
is five. We take that five and we multiply
it by five 𝑥 squared and negative eight. And that gives us 25𝑥 squared
minus 40.
Now, we know that since we’re
subtracting the final two terms, we’re going to get a remainder of zero. So, 25𝑥 squared plus 𝑘 minus 25𝑥
squared minus 40 has to give us zero. Well, 25𝑥 squared minus 25𝑥
squared is also zero. And so to ensure that our remainder
is zero, and thus our expression is divisible by five 𝑥 squared minus eight, we can
say that 𝑘 minus negative 40 itself must be equal to zero. 𝑘 minus negative 40 is, of course,
𝑘 plus 40.
And so let’s subtract 40 from both
sides to solve for 𝑘. That gives us 𝑘 is equal to
negative 40. And so, the value of 𝑘 that makes
our expression divisible by five 𝑥 squared minus eight is 𝑘 equals negative
40. Note that at this point, we could
go ahead and multiply our quotient. That’s six 𝑥 cubed minus four 𝑥
squared plus five by five 𝑥 squared minus eight. If we had performed the calculation
correctly, we would end up with the expression 30𝑥 to the fifth power plus 57𝑥
squared minus 48𝑥 cubed minus 20𝑥 to the fourth power minus 40.
