Describing numerical relationships with polynomial identities | Algebra 2 | Khan Academy

What we're going to do in this video is use what we know about polynomials and how to manipulate them and what we've talked about of whether two polynomials are equal to each other for all values of the variable that they're written in. So whether we're dealing with a polynomial identity, and we're going to use those skills in order to prove some properties of relationships between numbers.

So if I were to list out some integers, I could go 0, I could go 1, I could go 2, 3, 4, 5. And if I were to list out the squares of these, if I were to create a sequence of integer squares, well, 0 squared would be 0, 1 squared would be 1, 2 squared is 4, 3 squared is 9, 4 squared is 16, 5 squared is 25. We could of course keep going in either case, but the first thing I want you to think about before we even write down a polynomial or try to construct one is look at this sequence of integer squares.

Do you see any pattern in terms of the difference between successive terms of this sequence of integer squares? All right, now let's think about this together. So to go from zero to one, you add one, and you go from one to four, you add three. To go from four to nine, you add five. To go from nine to sixteen, you add seven. It seems like a pattern here. As we go to successive terms of this sequence of integer squares, we're adding increasing odd numbers. So I'm guessing that if I add 9 here, which is the next odd number, I'm going to get to 25, and that indeed is the case, and you could test that out.

Well, what if I add 11, which would be the next odd number? What do I get to? I get to 36, which is the square of 6. But how can we feel good that this always is true, that this never breaks down? Well, one way to do it is to think a little bit more generally, and that's where our algebra and our knowledge of polynomials are going to be useful.

So let's say we go all the way, and we're just speaking generally now. So we have the number n, and then the next number after that is going to be n plus 1. And then, if we think about what the corresponding terms in the sequence of integer squares would be, well, that would be when we squared. When we get to n, we would get n squared, and when we get to n plus 1, we would have n plus 1 squared.

Let's see if we could think about what the difference between these two things are. The difference between 25 and 16 is 9. The difference between 16 and 9 is 7. So let's think about what the difference between n plus 1 squared and n squared is, and how do we write that as a polynomial? Well, it'll just be n plus 1 squared minus n squared.

Now let's see if we can rewrite this algebraically, manipulate this, so we can set up a polynomial identity that describes this pattern that we just saw. So what I'll do is I'm just going to expand out n plus 1 squared right over there. So that is going to be n squared plus 2n plus 1, and then we have this minus n squared here. So minus n squared, and so we see that n squared minus n squared cancels out.

We can rewrite everything we have here as n plus 1 squared minus n squared. So this is really the difference between successive terms in our sequence of integer squares. It's going to be equal to 2n plus 1 for any integer n. Well, for any integer n, what is 2n plus 1 going to be? Especially here, we're dealing with the positive integers.

Well, for any integer n, this is going to be an odd integer. If you take any integer, you multiply it by 2, this part is going to be even, but then you add 1 to that, you're going to get an odd integer. You can also see that this increases by 2 as n increases. So when you go from one odd integer, you add 2 to the next odd integer, you add 2 to the next odd integer, which is exactly what is described there.

So this is pretty neat! We've just used a little bit of algebra, a little bit of what we know about polynomial identities, to show that the difference between successive terms in this sequence of integer squares right over here is going to be increasing odd numbers.