Video: Simplifying Algebraic Fractions Using Factorization

Michael wants to mulch his garden. His garden is (𝑥² + 18𝑥 + 81) ft² and one bag of mulch covers (𝑥² − 81) ft². Write, in its simplest form, a rational function that describes the number of bags of mulch Michael needs to buy.

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Video Transcript

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 buy.

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 denominator.

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 between.

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.

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