Simplifying rational expressions
Simplify the rational expression and state the domain. Let's see if we can start with the domain part of the question, if we can start with stating the domain. Now, the domain is the set of all of the x values that you can legitimately input into this if you view this as a function, if you said this is f of x is equal to that. The domain is a set of all x values that you could input into this function and get something that is well-defined. The one x value that would make this undefined is the x value that would make the denominator equal 0-- the x value that would make that equal 0. So when does that happen? Six minus x is equal to 0. Let's add x to both sides. We get 6 is equal to x, so the domain of this function is equal to the set of all real numbers except 6. So, x could be all real numbers except 6, because if x is 6 then you're dividing by 0, and then this expression is undefined. We've stated the domain, now let's do the simplifying the rational expression. Let me rewrite it over here. We have x squared minus 36 over 6 minus x. Now, this might jump out at you immediately, as it's that special type of binomial. It's of the form a squared minus b squared, and we've seen this multiple times. This is equivalent to a plus b times a minus b. And in this case, a is x and b is 6. This top expression right here can be factored as x plus 6 times x minus 6, all of that over 6 minus x. Now, at first you might say, I have a x minus 6 and a 6 minus x. Those aren't quite equal, but what maybe will jump out at you is that these are the negatives of each other. Try it out. Let's multiply by negative 1 and then by negative 1 again. Think of it that way. If I multiply by negative 1 times negative 1, obviously, I'm just multiplying the numerator by 1, so I'm not in any way changing the numerator. What happens if we just multiply the x minus 6 by that first negative 1? What happens to that x minus 6? Let me rewrite the whole expression. We have x plus 6, and I'm going to distribute this negative 1. If I distribute the negative 1, I have negative 1 times x is negative x. Negative 1 times negative 6 is plus 6. And then I have a negative 1 out here. I have a negative 1 times negative 1, and all of that is over 6 minus x. Now, negative plus 6. This is the exact same thing as 6 minus x if you just rearrange the two terms. Negative x plus 6 is the same thing as 6 plus negative x, or 6 minus x. Now you could cancel them out. 6 minus x divided by 6 minus x, and all you're left with is a negative 1-- I'll write it out front-- times x plus 6. If you want, you can distribute it and you would get negative x minus 6. That's the simplified rational expression. In general, you don't have to go through this exercise, multiplying by a negative 1 and a negative 1. But you should always be able recognize that if you have a minus b over b minus a that that is equal to negative 1. Or think of it this way: a minus b is equal to the negative of b minus a. If you distribute this negative sign, you get negative b plus a, which is exactly what this is over here. We're all done.