Adding and subtracting rational expressions
- [Voiceover] So 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, 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 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, so 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 want to just multiply those two denominators. For this one, I'll have my 8x plus seven and now I'm going to multiply by 3x plus one. 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 one and I don't want to change the value of the expression, we'll have to multiply the numerator by 3x plus one, as well. Notice, 3x plus one divided by 3x plus one is just one and you'll 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 can think of this subtraction. I could just write a minus sign right over here and do the same thing that I just did 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. So if I just multiply negative one times this expression, I'd get negative 6x third over 3x plus one. If I had more terms up here, in the numerator, I would have to be careful to distribute that negative sign, but here, I only have one term. So I just made a negative and so I can say this is going to be plus and, let me do this, in a new color, do this in green. Our common denominator, we already established, is the product of our two denominators so it is going to be 8x plus seven times 3x plus one. Now if we multiply the denominator here was 3x plus one, we're multiplying it by 8x plus seven. So that mean's we have to multiply the numerator by 8x plus seven as well. 8x plus seven times negative 6x to the third power. Notice, 8x plus seven divided by 8x plus seven is one. If you were to do that, you would get back to your original expression right over here, the negative 6x to the third over 3x plus one. 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, 8x plus seven times 3x plus one. Now, in the magenta, I would want to distribute the negative 5x, so negative 5x times positive 3x is negative 15x squared. And then, negative 5x times one is minus 5x. And then, in the green, I would have, let's see, I'll distribute the negative 6x to the third power, so negative 6x to the third times positive 8x is going to be, negative 48x to the fourth power. And then negative 6x to the third times positive seven is going to be negative 42, negative 42x 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, one third-degree term, one second-degree term, one first-degree term, and that's it. There's no more simplification here. Some of you might want to just write it in descending degree order, so you could write it as negative 48x to the fourth minus 42x to the third minus 15x squared minus 5x. All of that over 8x plus seven times 3x plus one. But either way, we are all done. I mean, 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. Yeah, so it looks like we are all done.