Adding and subtracting rational expressions is similar to adding fractions. When adding and subtracting rational expressions, we find a common denominator and then add the numerators. To find a common denominator, factor each first. This strategy is especially important when the denominators are trinomials.
Adding and subtracting rational expressions with variables is pretty much the exact same process as adding and subtraction of fractions okay? What we need is a least common denominator in order to combine everything, so the way that I find least common denominators is to factor out everything in the denominator to its most simplistic form, so by doing that on this example up here 4 is 2 times 2, 10 is 2 times 5 and 12 is 4 times 3 or 2 times 2 times 3. So basically looking at this you figure out what you need. Okay so I need for this I need 2 twos, a 5 and a 3 okay in order to accomodate every single denominator possible I need these 2 twos which occur in the 12 and the 4, the 5 that occurs in the 10 and the 3 that occurs in the 12 okay? So 2 times 2 is 4 times 5 is 20 times 3 is 60 so my least common denominator for all of this is 60.
Then what I need to do is make my denominator 60 for every single term so to get 60 from 4 I need to multiply this by 15 over 15, to get the 60 for the 10 I need to multiply this by 6 over 6 and from the 12, 5 over 5. Okay, I'm not going to finish this one up because hopeffuly you remember how to do that but basically what you end up with is 30-6+25 all over 60 okay.
Same idea for with rational expressions except instead of dealing with just numbers you're going to be dealing with variables and factors and things like that. Okay, so looking at this we have 12a squared and b and we have 6ab. We need to find the least common denominator. We have a 12 and a 6 so I know that my smallest number is going to be 12. I have an a squared and then a so I know I need at least 2 a's and then b and b, so my least common denominator for this is 12 a squared b which is this term right here so to make that out of this term we need to multiply this by 2a over 2a okay leaving me with 5. Once my denominators ars the same we can just subtraction as normal so we get 12 of over 12a squared b minus 2a okay. Common mistake that people frequently make when dealing with this is you multiply by your least common, you multiply to get your least common denominator and then typically there's a lot of simplification that can go on in the numerator and so in doing that people tend to leave off the denominator because instead of having to write this 12a squared b over and over and over again while we simplify this numerator it's easier just to let it leave it off. I understand that but you have to make sure you put that back in in the end okay but I said the fraction is one fourth I can't just drop the 4 off and say that's the same thing as 1. You really need that denominator there to round out your fraction. The same thing with this kind of problem okay it's okay in my book it may not be in your teacher's book to leave off that denominator while you work through the problem but in the end make sure you throw it back in.