Definition and Domain of a Rational Expression - Problem 4

In simplifying rational expression what we are looking for is things that we can cancel and in order to do that what we first typically have to do is to factor on your numerator and your denominator. So we are going to go ahead and do that with this problem behind me.

So our numerator factors, the last term is positive and the middle term is negative which tells me I have to be x minus and x minus and in order to get a negative 5 we just need to break that down into a 2 and a 3.

The denominator is a difference of squares, we are use to having our x terms first but that doesn’t really matter we are just sort of, which is a habit this will factor the same exact way just being 2 minus x and 2 plus x.

So we don’t have anything that any two terms that are identical as this, but what we do have is a x and a 2 in the numerator and a couple of x’s and 2’s in the denominator. What we are looking for is opposites, things that can cancel to -1, things will cancel to -1 and as long as every single term has the opposite sign. So what we have here is the x minus 2, one of them is negative and we are looking for one negative as well. Here they are both positive that’s not going to cancel. So what we have is x minus 2 over 2 minus x, these are actually going to cancel to -1.

If you want to prove it yourself take a second and factor out the -1 from one of these two terms you’ll see that what that happens after you factor that out is you get the same thing. So what we’ve done Is we’ve cancelled these out to -1 leaving us with -1 over times x minus 3 over 2 plus x.

If you wanted to you can distribute this -1 either into the numerator or the denominator I don’t really think it's necessary but if you want to you can go ahead and distribute that through. So we have simplified our expression.

The other thing that we are asked is to do is find the domain. The domain is any value of x that we can plug into this expression and to do that we have to look at our original expression we can’t look at our simplified version. So go back to our original which are these two polynomials and the first thing that we did was factor it down.

So what I want to look at is the factored form before I cancel these out and to figure out what can make my denominator 0. X is 2 here, I get denominator 0, if x is -2 here I get denominator 0. We can’t have that pop in so then our domain is everything not equal to + or -2.

So simplifying a rational expression looking for opposites and then a little bit about the domain as well.