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# Adding rational expression: unlike denominators

CCSS.Math:

## Video transcript

pause the video and try to add these two rational expressions okay I'm assuming you've had a go at it now we can work through this together so the first thing that you might have hit when you try to do it is you realize that they have different denominators and it's hard to add fractions when they have different denominators you need to rewrite them so that you have a common denominator and the easiest way to get a common denominator is you can just multiply the two denominators especially in a case like this where they don't seem to share any factors the both of these are about as factors you can get and they don't share anything in common and so let's set let's set up a common denominator so this is going to be equal to it's going to be equal to something let's see it's going to be equal to something over our common denominator let's make it let's make it 2 X let me just in another color so we're going to make it 2 X minus 3 times 3 X plus 1 times 3 X plus 1 and then plus plus something else over 2 X minus 3 2 X minus 3 times 3 X plus 1 times 3 X plus 1 and so to go from 2 X to go from just a 2 X minus 3 here in the denominator to a 2 X 3 times 3 X plus 1 we multiply the denominator by 3 X plus 1 so if we do that to the denominator we don't want to change the value of the rational expression we'd have to we'd also have to do that to the numerator so the original numerator was 5x in that blue color so the original numerator was 5x and now we're going to multiply it by the 3x plus 1 so times 3x plus 1 notice I didn't change the value of this expression I multiplied it by 3 X plus 1 over 3x plus 1 which is 1 as long as as long as 3x plus 1 does not equal 0 so let's do the same thing over here over here over here I'm a denominator of 3x plus 1 I multiplied it by 2x minus 3 so I would take my numerator which is negative 4x squared and I would also multiply it by 2x minus 3 2x minus 3 let me put parentheses around this so it doesn't look like I'm subtracting 4x squared and so then I can rewrite all of this business as being equal to well in the numerator in the numerator I'm going to have 5x times 3x which is 15 x squared 5x times 1 which is plus 5x and then over here let me do some green let's see I could do 4x times 2 negative 4x times 2x which would be negative 8x squared and then negative 4x times negative 3 which is plus 12x squared did I do that right negative oh let me be very careful negative negative fly and my spider-sense could tell that I did something shady and in fact if you want to pause the video could see try to figure out what I just did that's wrong so negative 4x squared times 2x is negative 8x to the third power negative 8x to the third power and then negative 4x squared plus times negative 3 is 12x squared and then our entire denominator our entire denominator we had a common denominator now so we were able to just add everything is 2x minus 3 2x minus 3 times 3x plus 1 times 3x plus 1 and let's see how can we simplify this so this is all going to be equal to let me draw make sure we recognize it's a rational expression and so let's see we can look at we can we our highest degree term here is the negative 8x to the 3rd so it's negative 8 negative 8x to the 3rd power and then we have a 15 x squared and we also have a 12 x squared we could add those two together to get a 27 x squared so we've already taken care of this we've taken creator let me do it in that green color so we've taken care of this taking care of those two and we're just left with a 5x so plus 5x and then all of that is over 2x minus 3 times 3x plus 1x plus 1 and we are and we are all done it doesn't seem like there's any easy way to simplify this further you could factor out an X onto the numerator but that's not going to cancel out with anything in the denominator and it looks like we are all done