In this video, let's go and work on one where we have multiplication as well as subtraction.
Now, you might look at a problem like this and say, yeah, that looks like way too complicated.
I don't even want to try this problem. But let's just break it down step by step,
because if you look at a problem like this, if you just kind of follow things like one
step at a time, it's actually not that difficult. And this is actually a pretty straightforward
problem, as long as you know how to multiply rational expressions, and you know how to
subtract rational expressions, and you know a little bit of factoring, OK?
So the first thing I want you to recognize is we know from the order of operations, right,
way back in the day when we learned this, that we're always going to multiply before we go ahead
and subtract. So what I'm going to do is I'm just going to focus right now on multiplying fractions.
And if you forgot how to multiply rational expressions, what I want you to do is just
kind of think of numbers. All right, so when we're multiplying fractions with numbers, remember,
we just multiply a numerator times a numerator, a denominator times a denominator. We don't care
about the common denominator. That's why a lot of students love multiplying fractions.
So 3 times 2 is going to equal 6. 4 times 3 is equal to 12. Now, the one thing I want you
to remember, right, and this is kind of like, again, another arch nemesis for so many students,
when we're multiplying fractions or when we're doing any fraction operation, always look to
simplify. What is something in the numerator and the denominator that I can divide into both of
them that I can reduce this fraction? And hopefully you recognize I can divide a 6 in the numerator
and denominator. So therefore, this can be a simplified version of just 1 half.
Now, why do I bring that up? Because sometimes you also want to look into simplifying before
you multiply, right? Now, these are pretty small numbers, but if I had, like, really big numbers,
you'd probably want to simplify them first before doing the multiplication.
Otherwise, you're going to have even bigger numbers, right? A bigger number times another
big number is going to get even bigger. So you always want to look to simplify first,
as well as simplify at the end. And in this example, before I go ahead and multiply these
out, I have these polynomials. It looks pretty confusing in my opinion. So what I want to do is
say, all right, is there anything I can simplify? Now, when we're dealing with fractions, right,
we're looking into, you know, simplifying and reducing them. When we're looking into polynomials,
we're going to look to simplifying them by factoring. And I can't really factor this,
this, this. But here, I recognize the difference of two squares. I know I can factor an x squared
minus 36 as an x minus 6 times an x plus 6. And then over here, I recognize I have a 4,
right, and an 8. Now, the important thing you got to recognize is every single expression here,
I'm going to put these in parentheses so you can see what's going on. Between these terms,
right, that are in parentheses, everything is technically separated by multiplication. You
can put a multiplication in the numerator and denominator. Like, if you multiply straight
across, all these terms are separated by multiplication, right? So that's important
because when terms are separated by multiplication, you can reduce them. I can now divide out the x
plus 6s in the numerator and the denominator, and I can reduce a 4 over 8 into a 1 half,
all right? So now, when I go ahead and multiply this out, I'm going to be left with a 1 times
nothing else, right? Those are, sorry, 1 times x. Okay, that's going to be good. So I'll have
a 1 times x. That got divided out. That'd be 4, yeah, 1. So that's going to be 1. So it's going
to be an x. And then in the denominator over here, I'm just going to have an x minus 6 times 2.
And therefore, that's going to be minus a 3 times an x minus 6. Now, it's really, really,
really, really, really important that you recognize I put this by x minus 6 in parentheses.
And the reason why I did that is because when I want to simplify these with parentheses,
what I want to be able to do here is I need to make sure I get common denominators, right?
Multiplying fractions, you just multiply straight across. You don't really care what the denominators
are. In this example, we got to make sure we have common denominators before we can combine them.
That means adding or subtracting them. So here, I have an x minus 6. Here, I have a 2 times x minus
6. So to get these to be common denominators, what I'm going to need to do is multiply by 2,
all right? Now, what you can see I have is going to be an x minus a 6 in this case. And therefore,
I can keep that all over my common denominator. Now, again, you recognize here, I technically,
I technically can go ahead and rewrite this as a 1 times an x minus 6. Now, you can see my x minus
6s are going to divide out. And I'm just going to be left with a final answer of 1 half. So I took
this whole crazy expression, and I reduced it all the way down to a 1 half. Now, if we're going to
include our excluded values, just because all the xs got simplified out does not mean we don't have
any more excluded values. Our excluded values are still going to be x cannot equal a 6 or a negative
6 because those would make my denominator 0 for going back from my original equation.
So make sure you go ahead and do that when you need to go and create your excluded values.
Now, this was me simplifying a rational expression with multiplication and subtraction. In the next
example, I'll simplify a rational expression with division and addition. I'll see you in the next
video.
