What's up guys? In this video, what I want to do is cover one of the most common mistakes that I see
students make when we're trying to subtract rational expressions. Now,
most students get the idea of these denominators are not the same, so therefore the first thing
we need to do is find the least common denominator. Now, that is one thing where students make a lot
of mistakes in their own right, but that's not the most common mistake that I want to get to
in this video. So, to find the common denominator, a lot of times the quickest, fastest, easiest way
is just to multiply the denominators. But before you do that, you always want to make sure that
you can factor them. And now with an x here, I can't simply factor anything, so I'm just going
to leave that as that. But over here, I have an x squared minus a 3x, and hopefully you recognize
that they both share an x. Basically, I can factor out an x. Now, if you're not seeing this,
let's just maybe rewrite the x squared as an x times x, so you can see that they have an x in
common. Okay, so what I want you to see here is instead of writing x squared as like, you know,
x squared, I'm writing it as an x times x. So now what I want you to see is they both have,
both of these terms have an x in common. So I can factor that out, or another way would the
operation we're basically doing is dividing that term out. So when I do that, what I'm going to
simply do is say, all right, I'm going to put that x over here, and that's going to leave me now with
an x minus a 3. And again, we can always go back and check our work when we are looking into, did
we factor this correctly? That's always a great question to like, ask yourself, like, did I do
that correctly? And to do that, all you simply need to do is just multiply whatever term you
factored out back through your parentheses, right, via the distributed property. And you notice that
x times x gives you back an x squared, right, which is what, and x times negative 3x gives you
that negative 3x. So we're good there. So a lot of times when I have this, what I like to do is
just go ahead and rewrite it in the factor form just below it. Now, again, this is not the chemical
mistake that we are looking for. We're still not there yet. But again, that is something really
important for I want you to recognize here. So what we have here is on this side, we have an x
times x minus 3. And then over here, we just have the x. So if we want to get a common denominator,
what you recognize here is, well, this already has the x and x minus 3. This already has the x.
What else do I need to obtain on this right-hand side of this fraction? So therefore, it's going
to be the same denominator. And hopefully, you can just realize that, oh, well, if this has an x
minus 3, this doesn't. That means I need to multiply this by an x minus 3. Now, just remember,
whatever you do in the denominator, you also have to make sure you do in the numerator. That's what
are we going to be calling our equivalent fractions. So now, we're going to get into
the mistake. This is where students are going to make the mistake. And where that mistake comes
through is they just go ahead and apply their operation in the numerator. See, the cool thing
over here is I didn't have to do anything, right? This is basically just preserved as a 1 over x
times x minus 3. But over here, I need to simplify this. And what I'm doing here is I'm typically
going to multiply the 2 times x, right, and then the 2 times negative 3, following that
distributive property. But the problem here is this is actually going to be what we're not going
to want to do is multiply this by positive. We're actually going to want to multiply this by a
negative 2. So what I like to do is whenever I see subtraction, I don't want to do subtraction.
I want to do addition. So what I'm going to do, though, to make sure I'm preserving the
operation correctly, I'm going to change the subtraction to an addition of a negative. So
what that is doing, again, like adding a negative is the same thing as subtracting, right? If you
think about it this way, if I had like 5 minus 4 is negative 3. And the same thing if I said a 1,
5 minus 4. Did I say a 5 minus 4? 1 minus 4 is a negative 3. Same thing if I said a 1 plus a
negative 4. That's also going to equal to a negative 3, right? So these are going to be
those equivalent equations, okay? Or equivalent expressions to represent that. So I prefer to add
a negative. So whenever I see subtraction, it might even be helpful just to go ahead and make
that addition and then put the negative up top. Because I'm telling you, this is where students
make the mistake, especially if they're taking a test or taking a quiz and they're moving fastly
and they kind of forget about things. They will forget to bring this negative. Now, again,
you don't have to do this. But if you've made this mistake before, you know exactly where you
made it. You just weren't paying attention to the details. So now that I have this as a negative,
I can now apply distributive property. And again, remember, I changed it to a positive,
so it's going to be plus. And now I have a negative 2x plus a 6, all right? So you're not
going to get the right answer if you don't do this correctly. And then again, that's going to be all
over our common denominator of x times an x minus a 3. So now in this case, now we notice that our
denominators are exactly the same. So now all we need to do is apply our operation to our numerator.
And the only thing I can really combine here is the 1 and the 6. So that's going to give me a
final answer of negative 2x plus 6 all over x times x minus 3. And sorry, that's a 1 plus 6 is
going to be 7. And then also we want to make sure we include our excluded values. Those are going to
be the values that are going to make my denominator equal to 0. And you can see if I were to plug
those numbers in for x in the denominator, they would make them 0. That's why they are excluded
from the domain of this expression. Now, there's something that was really important that I
mentioned in this video that I want you to always do first when combining rational expressions.
If you're not sure you caught it, then go ahead and check out the next video where I will explain
why we always want to do this first before we simplify rational expressions.
