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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

  • 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.

- [Instructor] What we're going to do in this video is use

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B1 中級

多項式の等式を用いて数値関係を記述する|代数学2|カーンアカデミー (Describing numerical relationships with polynomial identities | Algebra 2 | Khan Academy)

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    林宜悉 に公開 2021 年 01 月 14 日
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