Subtracting rational expressions: factored denominators | High School Math | Khan Academy
Pause this video and see if you can subtract this magenta rational expression from this yellow one. All right, now let's do this together. The first thing that jumps out at you is that you realize that these don't have the same denominator, and you would like them to have the same denominator.
You might say, well, let me rewrite them so that they have a common denominator. A common denominator that will work will be one that is divisible by each of these denominators, so it has all of the factors of each of these denominators. Lucky for us, each of these denominators are already factored.
Let me just write the common denominator. I'll start rewriting the yellow expression. So you have the yellow expression—actually, let me just make it clear; I'm going to write both the yellow one, and then you're going to subtract the magenta one. Whoops, I'm saying yellow but drawing in magenta.
So you have the yellow expression, which I'm about to rewrite. Actually, I'm going to make a longer line. So the yellow expression minus the magenta one—minus the magenta one right over there. Now, as I mentioned, we want to have a denominator that has all the common factors.
The common denominator has to be divisible by both this yellow denominator and this magenta one. So it's got to have the z plus 8 in it, it's got to have the 9z minus 5 in it, and it's also got to have both of these well. We already accounted for the 9z minus 5. So it has to be divisible by z plus 6.
Notice, just by multiplying the denominator by z plus 6, we're now divisible by both of these factors. Both of these factors, because 9z minus 5 was a factor common to both of them. If you were just dealing with numbers when you were just adding or subtracting fractions, it works the exact same way.
All right, so what will the numerator become? Well, we multiply the denominator times z plus 6, so we have to do the same thing to the numerator. It's going to be negative z to the third times z plus 6. Now let's focus over here.
We want the same denominator, so we could write this as z plus 8 times z plus 6 times 9z minus 5. And these are equivalent; I've just changed the order that we multiply, and that doesn't change their value. If we multiplied the—so we had a 3 on top before, and if we multiply the denominator times z plus 8, we also have to multiply the numerator times z plus 8.
So there you go. This is going to be equal to—this is going to be equal to, actually, I'll just make a big line right over here. This is all going to be equal to—we have our probably don't need that much space. Let me see, maybe that may be about that much. So I'm gonna have the same denominator.
I'll just write it in a neutral color now: z plus eight times nine z minus five times z plus 6. So over here, just in this blue color, we want to distribute this negative z to the third.
Negative z to the third times z is negative z to the fourth. Negative z to the third times 6 is minus 6 z to the third. Now, this negative sign right over here, actually instead of saying negative z—negative of this entire thing, we could just say plus the negative of this.
Or another way of thinking about it, you could view this as negative 3 times z plus 8. So we could just distribute that. So let's do it. Negative 3 times z is negative 3 z, and negative 3 times 8 is negative 24. There you go; we are done.
We found a common denominator, and once you have a common denominator, you can just subtract or add the numerators. Instead of viewing this as minus this entire thing, I viewed it as adding and then having a negative 3 in the numerator, distributing that, and then these—I can't simplify it any further.
Sometimes you'll do one of these types of exercises, and you might have two second-degree terms or two first-degree terms or two constants or something like that, and then you might want to add or subtract them to simplify it. But here, these are all have different degrees, so I can't simplify it any further.
And so we are all done.