 A rational expression is basically a fraction with a polynomial numerator and denominator. This means that if we have to add, subtract, multiply, or divide rational expressions, we can use the same rules for adding, subtracting, multiplying, or dividing fractions with the appropriate changes. So with fractions, for b not equal to 0, a over b is the same as a divided by b. And so for y not equal to 0, x over y is the same as x divided by y. If we're trying to add or subtract fractions with the same denominator, we can combine them by adding or subtracting their numerators. And likewise, if we have rational expressions x over y plus or minus z over y, we can combine them to get x plus or minus z over y. If I have a fraction, I can multiply numerator and denominator by the same thing. And likewise, if I have a rational expression, I can multiply numerator and denominator by the same thing as long as it's not equal to 0. If I want to multiply two fractions, I get the product of the numerators as the new numerator and the product of the denominators as the new denominator. With rational expressions, we have the radically different rule, well, actually it's the same rule. And a useful thing to keep in mind, if I multiply a fraction by its denominator, I'll be able to remove that denominator and be left with just the numerator. And we can do the same thing with rational expressions. If we multiply a rational expression by its denominator, we get just the numerator. So let's consider this division. So first, remember that any division can be expressed as a fraction with the dividend as the numerator and the divisor as the denominator. But we like our numerator and denominators to be simple. When they are fractions like this, we get what's called a compound fraction. And even though it may feel like one of these, it's actually pretty easy to deal with. Now it is helpful, as with one of these things, to wrap things up properly. Since both of these have numerators and denominators, and the expression itself has a numerator and denominator, it'll be helpful to be able to distinguish between them. So the thing to remember is that our dividend is the numerator, so we'll throw parentheses around that. And our divisor is the denominator, so we'll throw parentheses around that. So remember that if I multiply a fraction by its denominator, I'll eliminate the denominator. So here I have a denominator of 3x, so I'll multiply the numerator by 3x. But remember we also have to multiply the denominator by 3x as well. And so our numerator simplifies. And our denominator, well remember, factored form is best. So until and unless we have to multiply things out, we'll leave the denominator as a product. But wait, there's more. We still have a denominator x here. So let's multiply numerator and denominator by x. And in the denominator fraction, multiplying by this denominator x, we'll get rid of it, leaving us with just the numerator. And in the numerator, we'll get 5 times x. Now we do want to simplify, so let's remove a common factor. So here x is a factor in the numerator, and here 3 times x, x is also a factor in the denominator, so we can remove it. And because factored form is best, we'll leave this in this form. Or if we take a more complicated expression, remember a division can be rewritten as a fraction. And again, it's helpful to remind ourselves that the dividend is the numerator. So let's throw parentheses around that. And the divisor is our denominator, so we'll use parentheses here too. We can get rid of this first denominator x squared minus 9 by multiplying numerator and denominator by it. And again, while that will take care of the denominator in the numerator fraction, in the denominator fraction, we'll just record that as 3 times x squared minus 9 over x squared minus 4x plus 3. Now we still have this denominator in the denominator, so we'll multiply by the denominator. But we also have to multiply the numerator by the same amount and simplify. Now we do want to simplify, and so we'll start with the hardest thing possible. Wait, no, we'll start with the easy things. So we notice that one of the factors in the denominator is a difference of squares. And so the denominator will factor easily. Now we do have this numerator that we might try to factor, but remember a factor only matters if it's a common factor. And so we can see if either x minus 3 or x plus 3 is a factor of the numerator. And here we might notice that our x term is subtracted, so we'll try x minus 3 first. Could it be possible to write x squared minus 4x plus 3 as x minus 3 times something? Well, if it is possible that something has to be x minus 1, but we have to check, and this is true, so here's our factorization. And now we can remove the common factors. And again, factored form is best, so unless we're bored and really need something to do, we'll go ahead and leave our final answer in this form. So let's take this division again. We can rewrite any division as a fraction where our dividend is the numerator and our divisor is the denominator. Here we only have one denominator that we want to get rid of, the c squared plus 8c plus 15. So we'll multiply numerator and denominator by that and simplify. And again, remember that factorization is the hardest, easy problem in mathematics. So we want to start out by trying to factor the easiest thing possible, and it looks like that will be, well, maybe 15c minus 5c squared. So that looks easy to factor, so we'll start with that. This factors as the other easy thing to factor is c squared minus 9, which is a difference of squares. So let's factor that. And again, a factor only matters if it's a common factor, so while we can try to factor c squared plus 8c plus 15 by taking random guesses, the only thing that really matters is whether c minus 3 or c plus 3 is a factor. So we'll see if either c minus 3 or c plus 3 is a factor of c squared plus 8c plus 15. Now since the c term is added, we'll check c plus 3 first. Is it true that c squared plus 8c plus 15 is c plus 3 times something? Well, if it is, that something has to be c plus 5, but we do have to check. Is this true? And we're in luck. It is, and so we have our factorization. And there's one more factorization we can do. Remember that a minus b is the same as minus b minus a. And so this 3 minus c can be rewritten as minus c minus 3. The other factors haven't changed, so we'll just copy them straight down. And now we can remove our common factors to get our final answer.