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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?

• 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,

• 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

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