Simplifying rational expressions: opposite common binomial factors

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 now the one x value that would make this undefined is the x value that would make the denominator equal zero the x value that would make that equal to zero so when does that happen six minus x is equal to zero let's add X to both sides and we get six is equal to X so the domain of this function the domain of this function is equal to the set of all real numbers all real numbers except except six so X could be all real numbers except six because if X is six then you are dividing by zero and then this expression is undefined so we've got the we've stated the domain now let's do the simplifying the rational expression so let me rewrite it over here we have x squared minus 36 over six minus X now this might jump out at you immediately as it's that special type of binomials 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 six so this top expression right here can be factored as X plus 6 times X minus 6 all of that over six minus X now at first you might say well look you know 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 multiply let's multiply let's multiply by negative one and then by negative one again think of it that way so if I multiply by negative 1 times negative 1 obviously I'm just multiplying the numerator by one so I'm not in any way changing the numerator but what happens if we just multiply the X minus 6 by that first negative one what happens to that X minus 6 so let me rewrite the whole expression we have X plus 6 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 and I have a negative 1 times negative 1 and then it's all all of that is over 6 minus X now negative x plus 6 this is this 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 so 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 all you're left with this is negative 1 times X plus 6 and 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 of multiplying by negative 1 and negative 1 you should always be able to 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 but we're all done