So in order to simplify a rational expression what we want to do is cancel off anything that is in both the numerator and the denominator. Think about a fraction is you have 2 over 4, you can divide the top by 4, 2 in the bottom by 2 and you end up with one half. It's a similar concept writing it as simpler terms.
So how we do this is we just want to factor our numerator and our denominator. Numerator in this case we cannot cancel, so we are just left with x minus 1, and the denominator we want to factor out. It’s a trinomial we know we have x and x, our middle term is negative our last term is positive which tells us we have minus and minus and the only factors of 3 are 3 and 1.
So now we have x minus 1 in both the top and the bottom we can go ahead and cancel those out leaving us with 1 over x minus 3. So by factoring in common like factoring and cancelling like terms we are able to simplify this rational expression.
The other part of this question is asking about the domain. What values of x can we put to this expression and a common mistake that people do is they actually go to your simplified expression to find what your domain is. That’s not actually true what you have to do is go to your original statement before you cancel anything out.
So what we had is x² minus 4x plus 3 factors to this right here x minus 3 times x minus 1. This is where we have to look out to find out our domain. So this can’t be equal to 0 which tells our domain is everything but 1 and 3. These are almost equivalent statements except for the fact that we can’t plug in -1 into here, sorry we can’t plug 1 into here and we can’t plug 1 into here.
So whenever you are finding the domain for a rational expression you always want to go to the pre-cancelled form even though these are fair-