Michael wants to mulch his
garden. His garden is 𝑥 squared plus 18𝑥
plus 81 feet squared, and one bag of mulch covers 𝑥 squared minus 81 feet
squared. Write, in its simplest form, a
rational function that describes the number of bags of mulch Michael needs to
Now in this question, there are two
key bits of information. One is actually the area of his
garden, which is 𝑥 squared plus 18𝑥 plus 81, and the other is the amount of area
that actually one bag of mulch can cover, which is 𝑥 squared minus 81.
So in this question, what we’re
looking to do is actually describe the number of bags of mulch Michael needs to
buy. Well, let’s think of it like
this. The number of bags needed is gonna
be equal to the total area, so the total area of the garden, divided by the area
covered by one bag.
So therefore, if we substitute in
the values we’ve got, we can say that the number of bags is gonna be equal to 𝑥
squared plus 18𝑥 plus 81 over 𝑥 squared minus 81. Now whenever we’re actually looking
to simplify something like this, so an algebraic fraction, what you need to do is
actually see whether the numerator or denominator can actually be factored.
Well, actually both the numerator
and denominator can be factored. So we’ll start with the
numerator. If we factor the numerator, we’re
gonna get 𝑥 plus nine multiplied by 𝑥 plus nine. So now if you actually take a look
at the expression to see how we did this, we’ve got 𝑥 squared plus 18𝑥 plus
81. And what we need is to actually
find a pair of factors who add together to give us positive 18 but multiply together
to give us positive 81.
Well, actually before we do that,
what we can do is actually set up our two parentheses, and that’s because actually
we know that it’s quadratic. So we’re gonna have a pair of
parentheses. And actually the beginning of each
of our parentheses is going to be an 𝑥, because we’ve got an 𝑥 squared term.
Well, actually the two factors
we’re gonna have is positive nine and positive nine. That’s because nine multiplied by
nine gives us 81 and nine add nine gives us 18. So there we have it fully
factored. We’re gonna get 𝑥 plus nine
multiplied by 𝑥 plus nine.
And just remember, you can actually
find these factors with trial and error, so we did that. I actually just found straight away
nine and nine because it’s quite straightforward for this one because you can see
that actually both of them are gonna be positive. That’s cause we’ve got positive 18
and positive 81. And we can actually see that nine
add nine gives us 18 and nine multiplied by nine gives us 81. Okay, great! Now let’s move on to the
So now if we actually factor the
denominator, what we’re gonna get is 𝑥 plus nine multiplied by 𝑥 minus nine. And we actually get this because 𝑥
squared minus 81 is a special form of quadratic because actually it’s a difference
of two squares. So difference of two squares, for
instance, will give us say- let’s say we’ve got 𝑎 squared minus 𝑏 squared. It’s gonna be equal to 𝑎 minus 𝑏
multiplied by 𝑎 plus 𝑏. And that’s because actually we’ve
got two squared terms. So here we’ve got an 𝑥 squared, so
that’s a square term, and then 81; well that’s a square number. And then there’s got to be minus in
So in order to actually show why
the difference of two squares works, and this is the form we’ll get for any
difference of two squares, what I’m gonna do is actually show how we’ll expand the
parentheses of our factored answer to the denominator.
First, we have 𝑥 multiplied by 𝑥,
which gives us 𝑥 squared. Now we have 𝑥 multiplied by
negative nine, which gives us a negative nine 𝑥. Then we have nine multiplied by 𝑥,
which gives us positive nine 𝑥. And then finally, we have positive
nine multiplied by negative nine, which gives us negative 81.
And now if we actually simplify
this, we’ll just be left with 𝑥 squared minus 81. That’s because if we have negative
nine 𝑥 add nine 𝑥, these cancel out, and then we’re left with what we wanted,
which was the 𝑥 squared minus 81.
Okay, great! So we’ve actually fully factored
the numerator and denominator. One way you can check that you’ve
actually done this correctly is to have a look at the numerator and denominator
because now they should be actually at least one shared factor, cause in a question
like this that’s the reason they’ll get you to actually factor the numerator and the
denominator, to find a shared factor so that you can simplify.
So in our question, we can see that
actually we have a shared factor cause we have 𝑥 plus nine in both the numerator
and denominator. So therefore, what we can do is
actually divide the numerator and the denominator by 𝑥 plus nine. So these will cancel each other
out. Okay, so therefore, we can say
that, in its simplest form, the rational function that describes the number of bags
of mulch Michael needs to buy is 𝑥 plus nine over 𝑥 minus nine.