Subtracting rational expressions: unlike denominators

right over here we have one rational expression being subtracted from another rational expression I encourage you to pause the video and see what this would result in so actually do the subtraction alright now let's do this together and if we're subtracting two rational expressions we'd like to have them have the same denominator and they clearly don't have the same denominator and so we need to find a common denominator and a common denominator is one that is going to be divisible by either of these and then we can multiply them by by an appropriate expression or number so that it becomes the common denominator so the easiest common denominator I can think of especially because these factors these two expressions have no factors in common would just be their product so this is going to be equal to so we could just multiply these two so this is going to be C let me do let me do this one right over here in magenta so this is going to be equal to the common denominator if I say if I wanna just multiply those two denominators for this one I'll have my 8x plus 7 and now I'm going to multiply it by 3x plus 1 I'm multiplying it by the other denominator and I had negative 5x in the numerator but if I'm going to multiply the denominator by 3x plus 1 and I don't want to change the value of the expression I'll have to multiply the numerator by 3x plus 1 as well notice 3x plus 1 divided by 3x plus 1 is just 1 and you'd be left with what we started with and from that we are going to subtract all of this now there's a couple of ways you could think of the subtraction I could just write a minus sign right over here and do the same thing that I just did for the first for the first term or another way to think about it and actually for this particular case I like thinking about it better this way is to just add the negative of this and so if I just multiplied negative 1 times this expression I'd get negative 6x third over 3x plus 1 add more terms up here the numerator and I would have to be careful to distribute that negative sign but here I only have one term so I just made it negative and so I could say this is going to be plus and then we do this in a new color this in green our common denominator we already established is the product of our two denominators so it is going to be 8 X plus 7 times 3 X plus 1 now if we multiply the denominator here was 3x plus 1 we're multiplying it by 8 X plus 7 so that means we have to multiply the numerator by 8 X plus 7 as well 8 X plus 7 times negative 6 X to the third power notice 8 X plus 7 divided by 8 X plus 7 is 1 if you were to do that you would get back to your original expression right over here the negative 6 X to the third over 3x plus 1 and now we're ready to add this is all going to be equal to I'll write the denominator in white so we have our common denominator 8 X plus 7 times 3x plus 1 now in the magenta I would want to distribute the negative 5 X so negative 5 x times positive 3x is negative 15 x squared and then negative 5 X times 1 is minus 5x and then in the green I would have let's see I'll distribute the negative 6 X to the third power so negative 6 X to the 3rd times positive 8 X is going to be negative 48 X to the fourth power and the negative 6x 3rd times positive 7 it's going to be negative 42 negative 42 X to the third and I think I'm done because there's no more there's you know I only have one fourth degree term 1 third degree term one second degree term one first degree term and that's it I there's no more simplification here some of you might want to just write it write it in descending degree order so if you could write it as negative 48 X to the fourth minus 42 X to the third minus 15 x squared minus 5x all of that over all of that over 8 X plus 7 times 3x plus one but either way we are all done and it looks like up here yeah there's no nothing to factor out these two are divisible by five these are divisible by six but even if I were to factor that out nothing over here down here no five or six to factor out yes it looks like we are all done