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 zero, I could go one, I could go two,

three, four, five.

And if I were to list out the squares of these,

if I were to create a sequence of integer squares,

well, zero squared would be zero,

one squared would be one,

two squared is four,

three squared is nine,

four squared is 16,

five squared is 25.

And we could, of course, keep going in either case.

But the first thing I want you to think about,

before you even write down a polynomial

or try to construct one,

is look at this sequence of integer squares.

And 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 to go from one to four, you add three.

To go from four to nine,

you add five.

To go from nine to 16, 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 nine here,

which is the next odd number, I'm gonna 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 six.

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 with the next number

after that is going to be n plus one.

And then if we think about what the corresponding terms

in the sequence of integer squares would be,

well, that would be, when we square it,

when we get to n, we would get n squared.

And when we get to n plus one,

we would have n plus one squared.

And let's see if we could think about what the difference

between these two things are.

The difference between 25 and 16 is nine.

Difference between 16 and nine is seven.

So let's think about what the difference

between n plus one squared and n squared is.

And how do we write that as a polynomial?

Well, it'll just be n plus one squared

minus n squared.

And 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 one squared right over there.

So that is going to be n squared plus two n plus one.

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.

And so we can rewrite everything we have here as

n plus one squared

minus n squared.

So this is really the difference between successive terms

in our sequence of integer squares

is going to be equal to two n plus one

for any integer n.

Well, for any integer n, what is two n plus one going to be?

And 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 two,

this part is going to be even.

But then you add one to that,

you're going to get an odd integer.

And you can also see

that this increases by two as n increases.

So when you go from one odd integer,

you go add two to the next odd integer.

You add two 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.