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CCSS.Math:

simplify the rational expression and state the domain so once again we have a a trinomial over trinomial to see if we can simplify we need to factor both of them and that's also going to help us figure out the domain the domain is essentially figuring out all of the valid X's that we can put into this expression and not get something that's undefined so let's factor the numerator and the denominator so let's start with the numerator let's start with the numerator there and since we have a 2 out front since we have a 2 out front factoring by grouping will probably be the best way to go so let's just rewrite it here I'm just working on the numerator right now - x squared plus 13x plus 20 so we need to find two numbers a and B that if I multiply them a times B needs to be equal to let me write it over here on the right a times B needs to be equal to 2 times 20 so it has to be equal to positive 40 and then a plus B has to be equal to 13 and the numbers that jump out at me immediately are five and eight alright five times eight is forty five plus eight is 13 so we can break this 13 X into a 5x and an 8x and so we can rewrite this as 2x squared and it will break up the 13 X into and I'm going to write the 8x first I'm going to write 8x plus 5x and the reason why I wrote the 8x first is because the eight shares common factors with the two so maybe we can factor out a 2x here it'll simplify a little bit five shares factors with the 20 so let's see where this goes and then we finally have a plus 20 here and now we can group them that's the whole point of factoring by grouping so you group these first two characters right here let's factor out a 2x so this would become 2x times well 2x squared divided by 2x is just going to be X 8x divided by 2x is going to be plus 4 and then let's factor let's factor out these or let's group these two characters and if we factor out a 5 what do we get we get plus 5 Plus 5 times X + 4 5 x divided by 5 is X 20 divided by 5 is 4 and we have an x + 4 in both cases so we can factor that out right we have x + 4 x 2 terms we could undistribute it so this thing this thing over here will be x + 4 times x let me do that same color - x + 5 - x + 5 and we factored this numerator expression right there now let's do the same thing with the denominator expression I'll do that in a different let me see I want to run out of color so the denominator right over here let's do the same exercise with it so we have 2x squared + 17 x + 30 let's look for an A and a B when I multiply them I get 2 times 30 which is 60 and a plus a B when I add them I get 17 and once again let's see 5 and 12 seem to work so let's split this up let's split this up into 2 x squared so we're going to split up the 17 X into a into a what was it 12 X + a 5 X right that adds up to 17 X and when you multiply 12 times 5 you get 60 and then plus 30 plus 30 and then on this first group right here this first group right here we can factor out a 2x so if you factor out a 2x you get 2x times X plus 6 and in that second group in that second group we can factor out a 5 so you get plus 5 times X plus 6 and now we can factor out an X plus 6 and we get we get x + 6 x times 2x plus 5 2x plus 5 so we've now factored the numerator the denominator let's rewrite both of these expressions or write this entire rational expression with the numerator and the denominator factored so the numerator so this is going to be equal to X plus 4 times 2x plus 5 we figure that out right there and then the denominator the denominator is X plus 6 times 2x plus 5 now you might already might already jump out at you that you have a 2x plus 5 of the numerator in the denominator we can cancel them out and we will cancel them out but before we do that let's work on the second part of this question state the domain so what are the valid x-values that we could put in here or I get some a more interesting question what are the x-values that will make this this rational expression undefined what's the x-values that will make the denominator equal to 0 and when will the denominator equal to 0 well either when X plus 6 is equal to 0 or when 2x plus 5 is equal to 0 and we could just solve for X here subtract 6 from both sides you get X is equal to negative 6 and if you subtract 5 from both sides you get 2x is equal to negative 5 divide both sides by 2 you get X is equal to negative 5 halves so if X so we could say that the domain let me write this over here the domain is all real numbers all real numbers other than or except except X is equal to negative 6 and X is equal to negative 5 halves and the reason why we have to exclude those is those would make this denominator either way you're right it's going to make the denominator equal to zero and it would make the entire rational expression undefined so we've stated the domain now let's just simplify the rational expression we've already said that X cannot be equal to negative 5 halves or negative 6 so let's just divide the numerator in the denominator by 2x plus 5 or just look at the 2x plus 5 we know that 2x plus five won't be 0 because x won't be equal to negative 5 halves so we can cancel those out and the simplified rational expression is just X plus 4 over X +6