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Adding rational expression: unlike denominators

Video transcript
- [Voiceover] 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 tried to do it, is you realized 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 case like this where they don't seem to share any factors. Both of these are about as factor as you can get and they don't share anything in common. And so 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 2x, I'm going to do this in another color. So we're going to make it 2x-3 times 3x+1 times 3x+1 and then plus plus something else over 2x-3 2x-3 times 3x+1. Times 3x+1. And so to go from 2x, to go from just a 2x-3 here the denominator to a (2x-3)(3x+1) we multiply the denominator by 3x+1. So if we do that to the denominator, we don't want to change the value of the rational expression. We'd also have to do that to the numerator. So the original numerator was 5x. I'll do that in blue color. So the original numerator was 5x and now we're going to multiply it by the 3x+1, so times 3x+1. Notice I didn't change the value of this expression. I multiplied it by 3x+1 over 3x+1, which is 1 as long as 3x+1 does not equal zero. So let's do the same thing over here. Over here I have a denominator of 3x+1, I multiplied it by 2x-3, so I would take my numerator, which is -4x² and I would also multiply it by 2x-3. 2x-3. Let me put parentheses around this so it doesn't look like I'm subtracting 4x². 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 15x² 5x times 1, which is + 5x, and then over here, let me do this in green, let's see, I could do -4x times 2x which would be -8x² and then -4x times -3 which is +12x². Did I do that right? Negative... Oh, let me be very careful. - 4, my spider sense could tell that I did something shady. In fact, if you want to pause the video you could see, try to figure out what I just did that's wrong. So -4x² times 2x is -8x to the third power. - 8x³ and then -4x² times -3 is 12x² and then our entire denominator our entire denominator we have a common denominator now so we were able to just add everything is 2x-3 2x-3 times 3x+1 times 3x+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, our highest degree term here is the -8x³ so it's -8 - 8x³ and then we have a 15x² and we also have a 12x². We could add those two together to get a 27x² so we've already taken care of this, we've taken care, let me do it in that green color, so we've taken of this, we've taken care of those two and we're just left with a 5x, so + 5x and then all of that is over 2x-3 times 3x+1 3x+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 out of the numerator but that's not going to cancel out with anything in the denominator and it looks like we are all done.