Adding and subtracting polynomials of degree one | Algebra 1 (TX TEKS) | Khan Academy

Let's say that a is equal to 6 m - 4 N minus 7 p, and let's also say that b is equal to 7 m - 3 n + 5 P. What I want to do in this video is figure out what is a + b equal to, and I want to express that in terms of M's, n's, and P's. I want to use as few terms as possible. So why don't you pause this video and see if you can work through that on your own before we work through it together?

All right, now let's work through this together. So first, we have a, and I'm just going to rewrite it over here. So we have 6 M minus 4 N minus 7 p.

Then to that, we are going to add B right over here, which is 7 m - 3 n + 5 P. So what we can do is add the terms that are using the same variables. For example, I could add the 6 M to this 7 m. If I have six M's and then I add another 7 m, well, I'm going to have 13 M's here. So that's 13 m.

Next, I could think about adding you might want to say 4 N and 3 n. But since we're subtracting 4 N, we're subtracting 3 n here. You could view it as -4 n + -3 n, or you could say we're starting at -4 n and then we're subtracting three more n's. Well, -4 minus 3 is -7, so you're now going to have -7 n's. Or you could say we're subtracting 7 n.

Last but not least, we could say -7 p, and then we are going to add 5 P to that. So if you start at -7 p and then add 5 p, you're going to get to negative 2 p. Another way you could think about it is you have 5 p and we're subtracting 7 P from that, so you're now going to have negative -2 p.

And we're done! You can't combine any of these because this is in terms of M, this is in terms of N, and this is in terms of P.

Let's do another example here. In this one, let's do some subtraction. Let's imagine that we have the expression or we do have the expression 4x - z vs. 8 x - 4 y + 3 Z. See if you can do this subtraction. We're subtracting this expression from this expression over here. Pause this video and see if you can do that.

All right, now let's work through this together. So the way that I like to do this is essentially distribute this negative sign. You could view this as negative 1 times all of this, and now to remove the parentheses, I can just multiply -1 times each of those terms.

So let's do that. So this first part over here is just 4x - z, and now let's add, so plus, and I'm going to distribute this negative 1 onto each of these terms. So we have -1 * 8X, all right, -8X. Then we have -1 * -4 y; well, that's going to be positive 4 Y. Lastly, we have -1 * 3 Z; that would be -3 Z.

Now we can add terms that are dealing with the same variable. We can look at this 4X, and then we have -8x. So what's 4X plus -8X? Well, that's going to be -4X. Then we could go to, actually, let me go to Y next. Just my brain wants to go from X to Y to Z. I could have done Z first, but there's no y over here, and we just have a 4 Y over here. So I'll just rewrite that as + 4 Y.

Last but not least, we have a negative Z here, or we're subtracting Z, and now we're subtracting three more Z's right over here. So in total, we're subtracting a z and then subtracting three more Z's. We're going to subtract four Z's, so minus 4 Z.

And we are done!