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# Subtracting rational expressions: factored denominators

CCSS.Math:

## Video transcript

pause this video and see if you can subtract this magenta rational expression from this yellow one alright now let's do this together and the first thing that jumps out at you is that you realize that these don't have the same denominator and you would like them to have the same denominator and so you might say well let me rewrite them so that they have a common denominator and a common denominator that will work will be one that is divisible by each of these denominators so it has all of the factors of each of these denominators I'm lucky for us each of these denominators are already factored so let me just write the common denominator I'll start rewriting the yellow expression so you have the yellow expression actually let me just make it clear I'm going to write both so yellow one and then you're going to subtract the magenta one whoops I'm saying yellow but drawing in magenta so you have the yellow expression which I'm about to rewrite actually want to make a longer line so the yellow expression - - the magenta one - the magenta one right over there now as I mentioned we want to have a denominator that has all the common denominator has to have be divisible by both this yellow denominator and at this magenta one so it's got to have the Z plus 8 in it it's got to have the 9z minus 5 in it and it's also got to have both of these well already we already accounted for the 9z minus 5 so it has to have be divisible by z plus 6z plus 6 notice just by multiplying the denominator by Z plus 6 we're now divisible by both of these factors and both of these factors because 9 Z minus 5 was a factor common to both of them and if you were just dealing with numbers when you were just adding and subtracting fractions it works the exact same way all right so what will the numerator become well we multiply the denominator times Z plus 6 so we have to do the same thing to the numerator it's going to be negative Z to the 3rd times Z plus 6 now let's focus over here we had well we want the same denominator so we could write this as Z plus 8z e + 8 x z + 6 x times Z plus 6 times 9 Z minus 5 and these are equivalent I've just changed the order that we multiply in and that doesn't change their value and if we multiplied the so we had a 3 on top before and if we multiplied the denominator times Z + 8 we also have to multiply the numerator times Z plus 8 Z + 8 so there you go and so this is going to be equal to this is going to be equal to actually I'll just make a big line right over here this is all going to be equal to we have our probably don't need that much space let me see maybe that may be about that much so I'm gonna have the same denominator I'll just write in a neutral color now Z plus 8 times 9 Z minus 5 times Z plus 6 so over here just in this blue color we want to distribute this negative Z to the 3rd negative Z to the 3rd times Z is negative Z to the 4th negative Z to the 3rd times 6 is minus 6 Z to the 3rd and now this negative sign right over here actually instead of saying negative Z negative of this whole entire thing we could just say plus the negative of this or another way of thinking about it you could view this as negative 3 times Z plus 8 so we could just distribute that so let's do it so negative 3 times Z is negative 3z and negative 3 times 8 is negative 24 and there you go we are we are done we found a common denominator and once you have a common denominator you can just subtract or add the numerators and instead of viewing this as minus this entire thing I viewed it as adding is and then having a negative 3 in the numerator distributing that and then these I can't simplify it any further sometimes you'll do one of these types of exercises and you might have to second degree terms or two first degree terms or two constants or something like that and then you might want to add or subtract them to simplify it but here these are all have different degrees so I can't simplify it any further and so we are all done