Complex Fractions to Simplify (taught every year)

Added: 2026-05-13

[00:00:00]In this video, I want to go over the three most popular types of simplifying complex fractions that you can expect to see on your homework, on your test, or even in your teacher's lesson. And the cool thing about these three problems is there is a common element between all of them. And guess what that common element is? It's going to be the common denominator. Now, you might say, "Oh, I know what the common denominator is." Or you might be like, "This is confusing."

[00:00:21]But don't worry. In each of these examples, I am going to show you in different ways how we can identify as well as use the common denominator to help simplify our complex fraction. So, let's get into it because these problems are going to show up on a test on a quiz and you need to know how to be able to simplify them with ease. So, in this first example, you can see I have two rational expressions being added in our numerator and then I have another rational expression here simpl in the denominator. Now the one thing that sticks out to me is in the denominator I have these two binomial expressions and in the denominator here I have a tromial. Now immediately I tell my students when I am teaching complex fractions that ding ding ding like that should like immediately bring your attention to I need to factor this. Now more often than not, okay, it's not all the time and it's not guaranteed but more often than not this fraction is a product of these two other denominators.

[00:01:17]Okay, so just keep that in the back of your h mind. But you also want to make sure you validate it. How are we going to validate it? We're going to factor this expression and I want to see what two numbers multiplied give me a positive four but add give me ne5. And guess what ladies and gentlemen? Yes, it is indeed the product of my other two denominators. Now that's very very important. Okay. So I am going to strike through this denominator and I'm just going to kind of represent my complex fraction and these are my denominators.

[00:01:43]Now when simplifying complex fractions, what I want to do is get rid of my denominators. You can see I have 1 2 3 and four or sorry this is my fourth denominator. Way too many denominators. All right we want to get rid of our denominators. To get rid of our denominators we need to find something that our denominators are going to evenly divide into. Now it gets a little bit more difficult with polomials but understand a polomial is going to divide into itself. Right? So my LCD in this case which is my least common denominator is simply going to be the product of x -1 * an x - 4. Right?

[00:02:18]Because this divides into that. X -1 divides in that. X -4 divides in that.

[00:02:24]So what I'm going to do is I'm simply going to multiply every single expression multiplied by my LCD. So I'm just going to write it like this. X -1 * X - 4. All right. But I'm going to multiply this times this expression.

[00:02:37]Multiply by this expression as well as multiply it by this denominator. Now when I do that to simplify a little bit of space, actually, you know what? For this first one, let's go ahead and write out. So, what I'll have here is a 2 over x -1.

[00:02:50]And then we're going to have x -1 * a x - 4. And that's going to be plus. And then I'll have a 3 over x - 4. And then again, x -1 * x - 4. And then my big denominator here, I have six. Again, I'm going to write it in its expanded form.

[00:03:08]x -1 * x - 4. And then again, multiply it there. Okay? So, this is really important. You got to make sure whatever you do in the numerator, you do in the denominator, right? That's going to produce an equivalent fraction. But also, you got to remember if you have terms or expressions, sorry, if you have expressions separated by addition or subtraction, you have to distribute that, right? We're playing the distributive property across here.

[00:03:30]So now, let's just go ahead and simplify things. So in this case, you can see my xus 1's are going to divide out or divide to one. All right? Over here, my xus 4s are going to divide out. And here everything is going to divide out.

[00:03:43]Here's the thing I want you to pay attention to, guys. By multiplying everything by my LCD, notice what happened to all my denominators.

[00:03:49]Bye-bye. They went away. Now, I still have this one big denominator, and that's okay. We'll try to see if we can simplify this. But at this point, right now, we have eliminated that denominator. Eliminated these little denominators. So now, let's see what we have here. I have distributive property, right? So I have to simplify that and I have to multiply this one. And then over here, I'm just going to have a six. So, uh, when multiplying this, I get a 2x - 8 and this is a positive 3. So, plus a 3x minus 3. And that's going to be all over a six. Now, I can combine like terms, right? Variables with variables and numbers with numbers. Now, again, when you're adding or combining variables, remember, you're just going to combine the coefficients. So, 2x + 3x is going to be a 5xgative8 minus 3 is going to be a negative 11.

[00:04:36]And that's divided by six. Now again depending on the instruction depending on what the um what your teacher is asking for we can also talk about the restricted values. Those would be the values from the original equation as well as to your final equation that will make the denominator equal to zero. Now there's no values that are going to make my denominator equal to zero for my final result. But if I look back at the original problem I recognize I cannot have my denominator equal to one. I cannot have it equal to four. And one and four would not work over here as well. So what I'll do is say oops why did I write it like that?

[00:05:08]I'll put a let me actually erase that and I'll write this as x cannot equal a positive 1 and a positive4 those are what we call our excluded values okay so simplistic one very common if you see two binomials and then you see a tromial as your third denominator look for that to be your common denominator that tromial in factored form but it's not always the case so make sure you check yourself okay all right let's go and take a look at another one in this case I I have a x^2 / x^2 - 25 / a x over a x - 5. Okay, again ladies and gentlemen, I see a binomial in this denominator. But here it's a quadratic. It's not a tromial, but it is a quadratic. And I immediately recognize I can simplify this, right? I can rewrite that as an x - 5 times a x + 5. Now, in this case, they don't share, right? Because there's not a third denominator here. I have a fraction simply being divided by another fraction. But that brings up something else to me. When I have a fraction divided by a fraction, right, like this or divided by a fraction, what does that tell us? When we have a fraction divided by a fraction, we can simply rewrite that as a product of the reciprocal.

[00:06:23]Okay, so if I have this fraction right here and that is being divided by this fraction. So again, I can rewrite it like this. x^2 / a x - 5 * x + 5 that's being divided by a x overx - 5 I'm sorry an x over that's being divided by a x - 5.

[00:06:47]I can simply go ahead and rewrite this as the product. So I can rewrite this as an x^2 all over x - 5 times an x + 5.

[00:06:57]and I'm going to multiply that by a x - 5 all over an x. Okay.

[00:07:03]Now, we could do the LCD in this case as well, which the LCD would be an x - 5 * x + 5. But I like understanding this when you have a fraction divide by another fraction because writing it into this form is really not that bad. Now, remember though, when I have an x squar, that is equivalent to an x x. Sometimes students get that confused and even when I'm doing like my dividing out the terms, I'll put a I'll I'll cross out the squared in the numerator, but students still get kind of confused with that. So for the purpose of this instruction on the video, I'm rewriting x^2 as x * x. The reason why this is important is because in my numerator I have x * x* a x - 5, right? They're multiplied by each other over there.

[00:07:46]Since I'm multiplying this expression, we're multiplying in our numerator and we're multiplying in our denominator.

[00:07:51]That's important because when you have terms that are separated by multiplication and they're exactly the same in the numerator as well as the denominator, the division property is applied. What that means is they will divide out to one over here. Again, these expressions are separated by multiplication. So my x - 5s are going to simply divide out leaving me with a final answer here of only one x divided by an x plus 5. Now again in this case we can see that x cannot equal a neg5.

[00:08:23]That's going to make my final result uh undefined right but again we just can't find our restrictions at the end right and that's a very common thing that will happen on a test especially like a multiple choice that will be an answer choice this you'll do all the work correctly but they'll have the one restriction as our simplified result.

[00:08:43]Don't take the bait. Okay. We have to remember we also have to look at our restrictions from our original equation.

[00:08:48]And if you look, not only does five make our denominator equal to zero of this denominator, but also a neg 5. So that is also going to be a restricted value.

[00:08:58]So not only is this a simplified result, but x cannot equal neg 5 as well as x cannot equal a positive 5. Okay? So it's very very important. And again, that's why I'm covering these problems, guys, because these are the ones that are the most common that you're not going to see, right? They're not the easiest.

[00:09:12]They're not the most difficult, but these are the most common that you are going to see on a test, a quiz, in your homework, and everything else. Um, that's why I wanted to cover them. So, let's go over another example over here.

[00:09:25]And this one's going to have four denominator or four expressions. I love that. And okay, so again, we recognize here this quadratic. And again, the whole idea, guys, to get rid of all these denominators. I want to be able to divide them into something. Now, more often than not, when you have a quadratic, that is going to be a product of your other denominators. Okay? Not always the case, but more often than not, and again, when I look at this and I go to simplify it, I say, "Hey, yeah, I can simplify that to x -2 * x + 2."

[00:10:00]This is easy because now this tells me exactly what my LCD is, right? Because I know my LCD has to include a x -2 times an x plus2, right? because a polomial has to be able to divide into itself. But then also my other denominators all divide into that. So we're good. Now all I simply need to do is multiply every single expression times a x - 2 * an x + 2. Now I'll save my Oops, I ran out of room here. I didn't run out of room. Yeah. Okay. So let's multiply it over here. So I'll do a x - 2 times an x plus two. Okay. So we're multiply here, here, here, and there. Okay. Now, to save myself a little bit of work, I'm just going to not rewrite the whole expression again. Let's just kind of hopefully you guys can follow me. When I multiply this expression times over here, the x -2's will divide out. That's going to leave me with a 3 * an x + 2.

[00:11:00]Over here, my x -2 and my x+2's divide out. So, that's just going to leave me with a -6.

[00:11:06]Uh, over here, my x+2's divide out. So, that'll leave me with a 3 * an x -2. And then over here, my x -2 will divide out.

[00:11:14]That's going to leave me with a 1 * a x plus2. Now, again, you don't really need to include the one, but I'll just do it for some accounting purposes. Okay? And ladies and gentlemen, the cool thing about, you know, complex fractions is true with, you know, factoring quadratics. It's true with um fractions.

[00:11:30]Just the more practice you get of going through these motions, the easier and the easier it is to not only identify patterns, but to be able to work with them without making mistakes. Okay? So, I know sometimes it takes a little bit slow of, you know, getting through understanding the motions, understanding what you're looking for. But I'm covering three problems in this exa in this video. I have multiple multiple examples of me working through problems as well as ones that you can practice on your own. That is how you're going to get better at this process. Okay? So, if you need more practice with simplifying complex fractions, check out those resources um in this description. Now, we just need to apply our good old distributive property here. So, I'll take the 3 * the x, 3 * 2, 3 * x, 3 * -2, and then x + 2. We'll just leave it as that, right? Okay. So, now what I'm going to have is a 3x + 6 - 6 and a 3x - 6 + 8 x + 2. combine our like terms, that's going to go to zero. So, I'm left with a 3x. And then over here, that's going to become a negative -4. That's gonna become a positive 4x. So, I have a 4x minus 4. Okay. Now, again, some students will make the mistakes. They'll divide out these x's, right? They see an x in the numerator and denominator, divide them out. But again, when I talked about the distrib or the I'm sorry, division property, that only works when you have your terms separated by multiplication.

[00:12:52]These terms here are not separated by multiplication. So you cannot apply the division property. So this is going to be our simplified result. You could if you wanted to or if it was on a multiple choice test or your teacher was saying like they want everything in factored form, you could also write it in that form as well. The important thing here though is I do want to tackle what x cannot equal. We recognize x cannot equal a one. A good common thing sometimes they'll put like xals 4 as a restriction. No, four does not make the denominator zero. It's in the denominator, but it does not make the denominator zero. The only value that makes the denominator zero is the number one. But again, this is important. We cannot forget our original problem. What were the other values that made our denominator equal to zero? -2 and positive2. Right? It shows up in all of these. So, we also have to restrict not only our simplified result.

[00:13:45]Why am I writing it like that? We cannot only take in our simplified result at the end, but we also have to include. So x cannot include a positive 1, ag -2, and a positive2. So there you go, ladies and gentlemen.

[00:14:01]Those are your most popular simplifying complex fractions that you absolutely need to know. If you want to see some challenging ones, then check out the next video I have for you here. Or if you just need some more practice or you if you just need some more practice, go and check out my description or check out the video where I have for starting here for simplifying complex fractions.

[00:14:20]They'll give you a good review as well.