Name: 用多項式同位素描述數字關係｜代數2｜Khan Academy (Describing numerical relationships with polynomial identities | Algebra 2 | Khan Academy)
Uploaded: 2021-01-14T10:14:39.000Z
Duration: 4 min 34 s
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- [Instructor] 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

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.

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

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

if I were to create a sequence of integer squares,

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

But the first thing I want you to think about,

is look at this sequence of integer squares.

in terms of the difference between successive terms

All right, now let's think about this together.

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

which is the next odd number, I'm gonna get to 25,

which would be the next odd number, what do I get to?

But how can we feel good that this always is true,

and that's where our algebra and our knowledge

So we have the number n, and then with the next number

And then if we think about what the corresponding terms

in the sequence of integer squares would be,

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

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

The difference between 25 and 16 is nine.

between n plus one squared and n squared is.

And how do we write that as a polynomial?

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

So what I'll do is I'm just going to expand out

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

And then we have this minus n squared here,

And so we see that n squared minus n squared cancels out.

And so we can rewrite everything we have here as

So this is really the difference between successive terms

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

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,

that this increases by two as n increases.

which is exactly what is described there.