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Dividing rational expressions: unknown expression

Video transcript

we're told the following equation is true for all real values of Y for which the expression on the left is defined and D is a polynomial expression and they have this equation here what is D alright so essentially what they're saying is they don't want us to somehow solve this equation they're saying D is going to be some type of a polynomial expression they tell us that right over there D is a polynomial expression and if we if we figure out what D is this left-hand side of the expression is going to evaluate to 1 for all real values of Y for which the expression is defined so let's talk about how we would tackle that well the first thing that pops into my mind if I'm dividing by a fraction or a rational expression that's the same thing as multiplying by the reciprocal so let's let's just rewrite this on the left hand side so this is xx Y squared minus 80 over d x times 4 oh let me do the reciprocal you can be careful times the reciprocal of this if I divide by something the same thing as multiplying by the reciprocal so let me just swap the numerator denominator into numerators and denominators numerator and denominator alright Y to the 3rd plus 9y squared all of that over 4y squared minus 8y that's going to be equal to 1 now let's see if we can simplify all of this business on the left hand side a little bit so let's see over here I can divide both terms by xx so let me factor out xx because I think then it's going to end up being a difference of squares so if i factor out a 20 so this is going the same thing as 20 times y squared minus 4 and Y squared minus 4 we can rewrite as y plus 2 times y minus 2 it is a difference of squares so let me write that y plus 2 times y minus 2 all right this down here 4y squared minus 8y well it looks like we can factor out for why and so this is going to be the same thing as for Y times y minus 2 all right so let me cross that out so this is same thing as 4y times y minus 2 and I already see this y minus 2 here and this Y minus 2 here are going to cancel out and let's see up here both the both terms are divisible by Y squared so I can rewrite this as I don't know if this is actually going to be helpful because it's kind of if you factor let me just do it just in case so that's the same thing as y squared times y squared times y plus 9 all right and so we can rewrite all of these things if we were to multiply everything together we would end up getting in the numerator would get 20 times y plus 2 times y minus 2 times y squared times y plus 9 I'm just multiplying all the numerators and that's going to be over in the denominator I would have whatever the expression d is x 4y for y times y minus 2 and that's all going to be equal to 1 now let's think about it we can divide we can we have Y minus 2 divided by Y minus 2 so those cancel out let's see we have a we can divide the numerator or the denominator by Y so that would just become 1 and then that would just become Y to the first power and so what we'd be left with what would we be left with in the numerator is 20 times this Y times y plus 2 times y plus 9 over over for d-4d is equal to 1 now if we want to solve for D well we could just multiply both sides by D and 1 times e is going to be a DS you're going to have D equals something over here would be done so let's do that D times that let's multiply that times D let me be clear what I'm doing here let me draw little divider here to be to make it clear that to make it clear that that's happening on the other side of the page all right so D times this those cancel out and we're going to be left with 20 Y times actually let me I can simplify to even more 20 divided by 4 is 5 so the numerator is now just 1 so we have 5 times y times y plus 2 times y plus 9 is equal to D and we're done this is D this is the polynomial whoops that is that is the polynomial expression that we are looking for if you were to substitute this back in and then try to simplify it well you would end up with all of this over here and D would be this and so it would all just cancel out and you would be left with 1 for all real values Y for which the expression on the left is actually defined and you know we there are there are some values of Y for which the expression on the left is not defined if Y is equal to 0 this denominator is 0 and you're dividing by 0 well that's not defined and then when you make it when you divide when you multiply by the reciprocal if if this were to become 0 then that wouldn't be cool either and there's multiple ways to make this equal to 0 Y could be equal to 2 negative 9 that would also make this bottom zero so we could think about that if we wanted to but they're not asking us to do that they're saying for all of the real values for y for it's the expression is defined find the D that makes all of this business equal to 1 and we just did that