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Factoring - Rational Expressions


In this lesson, we will examine rational expressions. Now, what exactly are rational expressions? A rational expression is a ratio, a fraction, of two algebraic expressions. So, for example, x minus 3 over x plus 2. That would be a rational expression. Many of the rational expressions you will see on the test will be ones that can be simplified, and your job will be to simplify them.

To simplify a rational expression, we have to factor the expressions in the numerator and the denominator, and then cancel the common factors. So, for example, suppose we're asked to simplify this rational expression. Well, we have an ordinary quadratic in the numerator and another ordinary quadratic in the denominator. In fact, the one in the denominator is just the square of a difference pattern.

So when we factor, what we get is x minus 3 times x plus 7 in the numerator, x minus 3 squared in the denominator. We can cancel a factor of x minus 3 and we're left with x plus 7 over x minus 3. And that is a simplified form of the same expression. Here's another one. The expression on the top is a bit tricky, but notice that we can factor out a greatest common factor and then that just becomes a difference of two squares.

The denominator is just an ordinary quadratic, so we factor that in the ordinary way. We have an extra factor of x plus 2 that we can cancel. And this simplifies it. Two x squared time x minus 2, divided by x minus 7. Now, this is tricky.

We have a y squared plus 2x minus 8 over x minus 4. Well, this one, we can't do any factoring because we have two different variables in the numerator, but here's something else we can do. We can always split a fraction up by addition or subtraction in the numerator, so we can split this up into two fractions. It's sort of the opposite of finding the common denominator.

We split them up into fractions with the common denominator. So we get a y squared over x minus 4 and we get a 2 x minus 8 over x minus 4. Well, notice in that second fraction, we can factor out a 2 in the numerator, and we get 2 times x minus 4. Well, the x minus 4 and the x minus 4 cancel. So the first fraction, we can't do anything with that.

We can't simplify at all. But the second part just becomes plus 2. So, sometimes it's very useful to separate a fraction out by addition or subtraction in the numerator, and then simplify one piece of it. Here's a practice question. Pause the video and work on this.

So okay, if we're gonna do this a very straightforward way, we might cross multiply. We get 22 times x plus 15. I don't wanna have to figure out what 22 times 15 is. Then we'd have some very large quadratic that we'd have to solve.

Well, notice the following. Notice that we start with this and just factor the numerator of that. We get a factor of x plus 15. Cancel those x plus 15s and we're down to the equation 22 equals x minus 3. So x, of course, equals 25. Notice how deviously easy this question is if you realize you can simplify by cancelling.

Here's another practice question. Pause the video and work with this, and then we'll talk about it. Okay. So, obviously we can't factor here because we have a bunch of different variables. But let's start with that and I'm gonna say, let's group them a little bit differently.

Let's put the, the 3Q in front and let's get the P and R together, and then we'll separate out. So we have the numerator without P and R separate from the numerator with P and R. So it looks like that first one we're not going to be able to factor at all. But that second piece, we can factor out a 4 in the numerator, then cancel P minus R, and what we're left with is 3Q over P minus R plus 4.

And this equals the original expression. That's interesting. The original expression equals 19. So if we subtract 4 from both sides now, we get 3Q over P minus R equals 15. Well, now we'll just divide both sides by 3, divide that numerator and divide the other side by 3, and we get Q over P minus R equals 5.

And thus we have answered the question. In this lesson, we've learned how to simplify rational expressions by factoring the numerator and denominator and canceling the common factors. We also learned how to simplify by separating the numerator into two parts.

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