 So what else can we do with a polynomial? So remember, there's two useful ideas to keep in mind. First, arithmetic is bookkeeping. And second, algebra is generalized arithmetic. And so this leads to the idea of a polynomial. A polynomial is a generalization of the concept of a whole number. And once we have whole numbers, we can do other things like form fractions. And this leads us to the idea of a rational expression. A rational expression x over y is an expression where x and y are polynomials. And if polynomials are the generalizations of the whole numbers, then rational expressions are the generalizations of fractions. And this means that everything we did with fractions, we can do with rational expressions. For example, one of the things we can do with fractions is we can simplify them. Well, let's try and simplify a rational expression. It helps to remember what we can do with fractions. So remember, for n not equal to 0, the fraction a over b is equal to na over nb. Now this rule for fractions only really applies if we're dealing with real numbers. If our expressions are polynomials, we have a radically different rule which looks like this. For n not equal to 0, the rational expression x over y is the same as nx over ny. And again, we can multiply numerator and denominator by the same thing. Or if we have a common factor in numerator and denominator, we can remove it. And while I'm not a great fan of bumper sticker mathematics, mathematics that you can reduce to a short, quick, memorable phrase, here's a useful one, if in doubt, factor out. And what that means is that a useful thing to do if you're not sure what to do next is try factoring. Maybe something will appear. In this case, our numerator doesn't look too bad. That's just a number. We're good with numbers. But our denominator is a more complicated expression. So let's see if we can factor it. 3x plus 9, we stare at this for a moment and we realize that both terms have a factor of 3. So we can write 3x as 3 times x. Well, it's already written that way. But we can also write 9 as 3 times 3. And now, since both of our terms have a common factor of 3, we can remove it. Equals means replaceable. So 3x plus 9 can be replaced with 3 times x plus 3. And we take a look at our 12 again. We have a factor of 3 in the denominator. And we see that 12 also has a factor of 3. So we'll rewrite 12 as 3 times 4. But now there's a common factor of 3 in the numerator and the denominator. So we can drop that common factor and what we have left for the numerator, x plus 3 in the denominator. What about adding two rational expressions? 5 over 3x plus 7 over 3x. So again, it helps to remember what we did when we had fractions. Our fraction a over b plus c over b, as long as the denominator was the same, we could add them by adding the numerators and keeping the same denominator. Again, the radical difference made by switching to polynomials is, well, nothing really. If we're adding two rational expressions, x over y plus z over y, as long as our denominators are the same, we can add the numerators, x plus z, and keep the same denominator. So here we're adding two rational expressions with the same denominator, so we'll add the numerators and keep the denominator the same. And we can do a little cleanup. 5 plus 7 is 12. And strictly speaking, we could leave the answer in this form 12 over 3x, but it's nice to remove any common factors if we can. So here we note that the denominator is 3 times x and the numerator is 3 times 4. And so now numerator and denominator have a common factor, and we can remove it to obtain our answer in simplified form. What about subtraction? Well, if we subtract two fractions with the same denominator, we subtract the numerators and keep the common denominator. And if I have two rational expressions, I'm going to subtract the numerator and keep the common denominator. So here is one important thing to keep in mind. When we're dealing with fractions, c is a number, and so we don't have any real difficulty subtracting a number. If we're dealing with rational expressions, it's possible that one or both numerators may be polynomials. And so parentheses are cheap. Make sure that we're subtracting the entire second polynomial. So our denominators are the same, so we can subtract the numerators x plus 4, making sure that everything's in parentheses. Now let's try and simplify that numerator, x minus 8 minus quantity x minus 4, that minus distributes, so we get x minus 8 minus x minus 4. We'll rearrange things a little bit and simplify. Getting a final value of minus 12 as our numerator and our denominator stays the same. Now we should try to factor to simplify this, but remember a factor only matters if it's a common factor. If it's not a common factor, we can't remove it anyway. So if we look at our denominator x plus 3, that doesn't factor. So we don't really care if or how the numerator will factor, and we can leave our final answer in this form.