yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Simplifying rational expressions: two variables | High School Math | Khan Academy


3m read
·Nov 11, 2024

Let's see if we can simplify this expression, and like always, pause the video and have a go at it. Now, this one is interesting because it involves two variables, but it's really the same ideas that we've done when we factored things with one variable.

So, for example, up here in the numerator, well, I never like having a non-one coefficient on the second degree term. I mean, sometimes you have to, but it looks like every term here is divisible by five. So let's factor out a five first. The numerator I can rewrite as five times... five times, you factor out a five here, you get x; factor out a five here, you get plus 4. Actually, I'm going to rewrite it as 4YX, and you'll see in a second why I'm doing that.

Actually, I'll tell you why I'm doing that right now. Why I'm writing the Y there is that this way it seems to hit the pattern of how we're used to seeing quadratics. So, let's see. You have X plus 4YX; you can view the 4Y as a coefficient on the first-degree X term right over there, plus 4Y^2. And it's going to be over... over, now the denominator here; can we factor this out?

Well, let's just think about it. Do we know two numbers, or I guess you could say do we know two expressions, that when you multiply them you get -6Y^2 and then when you add them, you get -Y? That's actually why I liked writing it like this. So, actually, let me rewrite this. This is the same thing as negative YX, and so you can view the coefficient here as -1Y.

Now, we need to think of two numbers, or two expressions, A and B, that are equal to -6Y^2, and when I add them A + B, I get -Y. So, you can imagine both of them are going to be expressions that involve Y. Let's see if this was just a -1 and if this is just a -6. Well, we would do -3 and positive 2.

Let's see if we did -3Y and positive 2Y. That indeed is going to be equal to -6Y^2, and -3Y + 2Y does indeed equal -Y. So that's our A and B right over there. If it seems a little mysterious how I just all of a sudden got -2Y or -3Y and positive 2Y, let me write an analogous quadratic here that only has one variable.

If I were to write X^2 - X - 6 and I were to ask you to factor that out, you'd say, "Oh, okay, well, this is going to be -2; I have -3 * 2, which is -6, and if I add them, well, that's going to be -1." So you would say, "Well, that's going to be X - 3 and X + 2." The only difference between this and that is instead of having just a negative one here, you have a -1Y. Instead of having just a 6 here, you have a -6Y^2.

So you could just think of this instead of just -3 and positive 2 as -3Y and positive 2Y. Hopefully, that makes sense, and if it doesn't, I encourage you to kind of play around with this, multiply these out a little bit, and get a little bit more familiar with this.

But now that we know that it can be factored like this, let's rewrite this. This is going to be X - 3Y times X + 2Y. Nothing seems to simplify out just yet, but it looks like what we have in magenta here could be simplified further, and we're going to do a very similar exercise to what we did just now.

What two expressions, if I multiply them, I get 4Y^2, and if I add them, I get 4Y? It looks like 2Y would do the trick. So, it seems like we can rewrite the numerator. This is going to be, let me draw a little line here to make it clear that this is going to be equal to 5 times (X + 2Y) times (X + 2Y).

Once again, 2Y times 2Y is 4Y^2, and 2Y + 2Y is 4Y. That's all going to be over... that is all going to be over (X - 3Y)(X + 2Y). So now, I have a common factor (X + 2Y) in both the numerator and the denominator.

So I can cancel (X + 2Y) / (X + 2Y). Well, that's just going to be one if we assume that (X + 2Y) does not equal zero. That's actually an important constraint because once we cancel this out, you lose that information.

If you want this to be algebraically equivalent, we could say that (X + 2Y) cannot be equal to zero. Alternatively, you could say that X cannot be equal to -2Y. I just subtracted 2Y from both sides there. So what you're left with, and we can redistribute this five if we want to write it out in expanded form, we could rewrite it as the numerator would be 5X + 10Y, and the denominator is X - 3Y.

But once again, if we want it to be algebraically equivalent, we would have to say X cannot be equal to -2Y. Now this is algebraically equivalent to what we had up here, and you can argue that it's a little bit simpler.

More Articles

View All
15 Life-Changing Lessons We Learned in 2023
A man who does not reflect on the year that’s passed is destined to repeat it. With this year coming to a close, we make a priority of externalizing the most valuable insights we’ve drawn, and we’re about to share them with you. Here are 15 valuable lesso…
Cutting shapes into equal parts | Math | 3rd grade | Khan Academy
Is each piece equal to one-fourth of the area of the pie? So we have a pie, and it has one, two, three, four pieces. So it does have four pieces. So is one of those pieces equal to one-fourth of the pie? Well, let’s talk about what we mean when we have a…
Your Family Tree Explained
This is you, this is your family tree and this is your family tree explained. You have parents, and your parents have parents. These are your grandparents, who also have parents - your great grandparents. Keep adding parents, keep adding “greats.” For eve…
The Bullet Block Experiment
Alright, here is the setup: I have a rifle mounted vertically and we’re going to shoot a bullet into this block, right into the middle of it. So obviously the block is going to go flying into the air. But we’re going to do this again and instead of firin…
Cellular respiration | Energy and matter in biological systems | High school biology | Khan Academy
In this video, we’re going to talk about cellular respiration, which sounds like a very fancy thing, but it’s really just about the biochemical processes that can take things that we find in food and convert it into forms of energy that we can use to do t…
Why Most People Will Never Be Rich
Some people will never be rich, and no, it’s not about where you grew up, who your parents are, your gender, or the color of your skin. Let us explain. Welcome to alux.com. These 100 dots are meant to symbolize the world’s population. From a quality of l…