Video Transcript
Use polynomial long division to
find the quotient 𝑞 of 𝑥 and the remainder 𝑟 of 𝑥 for 𝑝 of 𝑥 divided by d of
𝑥, where 𝑝 of 𝑥 is equal to 𝑥 of the seventh power plus 𝑥 to the sixth power
plus 𝑥 to the fourth power plus 𝑥 squared plus 𝑥 plus one and d of 𝑥 is equal to
𝑥 cubed plus 𝑥 plus one.
The question wants us to use
polynomial long division. We’re given our polynomial 𝑝 of 𝑥
and our divisor d of 𝑥. We need to find the quotient 𝑞 of
𝑥 and the remainder 𝑟 of 𝑥 when we divide 𝑝 of 𝑥 by d of 𝑥. Before we start answering this
question, there’s a couple of things we should check. For example, we should check that
both our polynomials 𝑝 of 𝑥 and d of 𝑥 are written in descending exponents of
𝑥. In this case, this is true, so we
can just carry on with our long division. We’ll set up our long division. We have our divisor d of 𝑥 is
dividing our polynomial 𝑝 of 𝑥.
Before we start doing our long
division, there’s one more thing we can check. If we look at our polynomial 𝑝 of
𝑥, we can see there’s no term for 𝑥 to the fifth power and there’s no term for 𝑥
cubed. In regular long division, when this
happens, we have a digit of zero in this place. However, because this is a
polynomial, we just didn’t write these terms in. There’s a few different ways we
could tackle this. We could just leave that as it is,
or we could add terms zero 𝑥 to the fifth power and zero 𝑥 cubed. And both of these methods work, and
you can use them if you prefer. However, in this video, we’re just
going to leave a gap where these terms are to keep our columns aligned.
Now, let’s move on to our long
division. The leading term in 𝑝 of 𝑥 is 𝑥
to the seventh power. We need to divide this by 𝑥
cubed. And of course, 𝑥 to the seventh
power divided by 𝑥 cubed is 𝑥 to the fourth power. We’ll write this in our quotient,
and we’ll write this in the column for 𝑥 to the fourth power terms. The next thing we need to do is
multiply our divisor by the term in our quotient 𝑥 to the fourth power. Multiplying these together, we get
𝑥 to the fourth power times 𝑥 cubed plus 𝑥 plus one. And if we distribute this and
simplify, we get 𝑥 to the seventh power plus 𝑥 to the fifth power plus 𝑥 to the
fourth power.
We now want to subtract this from
our polynomial 𝑝 of 𝑥. And remember, we want to keep each
term in its respective column. We’ll start with 𝑥 to seventh
power. Then we need to add 𝑥 to the fifth
power. Finally, we add a term for 𝑥 to
the fourth power. Now, we can just subtract this term
by term. First, we get 𝑥 to the seventh
power minus 𝑥 to the seventh power. This is equal to zero. You can write this term of zero in
if you prefer. However, this term will always give
us zero. So we’ll just leave this blank.
Next, we have 𝑥 to the sixth power
minus zero. Of course, this is just equal to 𝑥
to the sixth power. Next, we have zero minus 𝑥 to the
fifth power. This is negative 𝑥 to the fifth
power. Then in our next column, we have 𝑥
to the fourth power minus 𝑥 to the fourth power. This is equal to zero. We’ll leave this bank. And remember, we need to bring down
the rest of our terms. It’s worth pointing out some people
prefer to leave these terms at the top until they’re needed. But we’re going to always bring
these terms down.
We’re now ready to find the next
term in our quotient. We need to divide 𝑥 to the sixth
power by 𝑥 cubed. And 𝑥 to the sixth power divided
by 𝑥 cubed is just equal to 𝑥 cubed. And remember, we write this in our
column for 𝑥 cubed terms. The next step in our long division
will be to multiply 𝑥 cubed by our divisor 𝑥 cubed plus 𝑥 plus one. This gives us 𝑥 cubed times 𝑥
cubed plus 𝑥 plus one. And if we distribute this and
simplify, we get 𝑥 to the sixth power plus 𝑥 to the fourth power plus 𝑥
cubed.
We now need to subtract this from
our polynomial. Remember, it’s important that we
write each term in the correct column. We can then subtract this term by
term. In our first column, we get 𝑥 to
the sixth power minus 𝑥 to the sixth power, which is zero. In our second column, we get
negative 𝑥 to the fifth power minus zero, which is just equal to negative 𝑥 to the
fifth power. In our next column, we get zero
minus 𝑥 to the fourth power, which is negative 𝑥 to the fourth power. We get a simpler story in our next
column. We have zero minus 𝑥 cubed, which
is negative 𝑥 cubed. Then once again, we bring down the
remaining terms.
Once again, we need to find the
next term in our quotient. We need to divide negative 𝑥 to
the fifth power by 𝑥 cubed. Of course, if we do this, we get
negative 𝑥 squared. Once again, we need to multiply the
newly added term to our quotient by our divisor. Distributing this and simplifying,
we get negative 𝑥 to the fifth power minus 𝑥 cubed minus 𝑥 squared. We then need to subtract this from
our polynomial. Remember, we want to write each
term in its correct column. We evaluate the subtraction term by
term. This time, we get negative 𝑥 to
the fourth power plus two 𝑥 squared. And then we bring the rest of our
terms down.
And once again, we need to find the
next term in 𝑞 of 𝑥. We need to divide negative 𝑥 to
the fourth power by 𝑥 cubed. Doing this, we get negative 𝑥. Once again, we multiply negative 𝑥
by our divisor. And if we evaluate this, we get
negative 𝑥 to the fourth power minus 𝑥 squared minus 𝑥. Now, we need to subtract this from
our polynomial. Evaluating the subtraction and
bringing down our term of one, we get three 𝑥 squared plus two 𝑥 plus one. And now, if we were to try and find
the next term in our quotient, we would get three 𝑥 squared divided by 𝑥
cubed. This is three over 𝑥.
This is not a polynomial. This tells us we’re done. We can see that the polynomial
we’re left with has a lower degree than our divisor. So we’ve found our quotient 𝑞 of
𝑥 and our remainder 𝑟 of 𝑥. And this gives us our final
answer. We were able to show our quotient
𝑞 of 𝑥 is equal to 𝑥 to the fourth power plus 𝑥 cubed minus 𝑥 squared minus 𝑥
and our remainder 𝑟 of 𝑥 is equal to three 𝑥 squared plus two 𝑥 plus one.
