Placeholder Image

字幕表 動画を再生する

  • - [Instructor] So what we have here

  • are two different polynomials, P1 and P2.

  • And they have been expressed in factored form

  • and you can also see their graphs.

  • This is the graph of Y is equal to P1 of x in blue,

  • and the graph of Y is equal to P2 x in white.

  • What we're going to do in this video

  • is continue our study of zeros,

  • but we're gonna look at a special case

  • when something interesting happens with the zeros.

  • So let's just first look at P1's zeros.

  • So I'll set up a little table here, because it'll be useful.

  • So the first column, let's just make it the zeros,

  • the x values at which our polynomial is equal to zero

  • and that's pretty easy to figure out from factored form.

  • When x is equal to one,

  • the whole thing's going to be equal to zero

  • because zero times anything is zero.

  • When x is equal to two, by the same argument,

  • and when x is equal to three.

  • And we can see it here on the graph,

  • when x equals one,

  • the graph of y is equal to P1 intersects the x axis.

  • It does it again at the next zero, x equals two.

  • And at the next zero, x equals three.

  • We can also see the property

  • that between consecutive zeros our function,

  • our polynomial maintains the same sign.

  • So between these first two,

  • or actually before this first zero it's negative,

  • then between these first two it's positive,

  • then the next two it's negative,

  • and then after that it is positive.

  • Now what about P2?

  • Well P2 is interesting,

  • 'cause if you were to multiply this out,

  • it would have the same degree as P1.

  • In either case, you would have an x to the third term,

  • you would have a third degree polynomial.

  • But how many zeros,

  • how many distinct unique zeros does P2 have?

  • Pause this video and think about that.

  • Well let's just list them out.

  • So our zeros,

  • well once again if x equals one,

  • this whole expression's going to be equal to zero,

  • so we have zero at x equals one,

  • and we can see that our white graph also intersects

  • the x axis at x equals one.

  • And then if x is equal to three,

  • this whole thing's going to be equal to zero,

  • and we can see that it intersects

  • the x axis at x equals three.

  • And then notice, this next part of the expression would say,

  • "Oh, whoa we have a zero at x equals three,"

  • but we already said that, so we actually have two zeros

  • for a third degree polynomial,

  • so something very interesting is happening.

  • In some ways you could say that hey,

  • it's trying to reinforce

  • that we have a zero at x minus three.

  • And this notion of having multiple parts

  • of our factored form that would all point to the same zero,

  • that is the idea of multiplicity.

  • So let me write this word down.

  • So multiplicity.

  • Multiplicity,

  • I'll write it out there.

  • And I will write it over here, multiplicity.

  • And so for each of these zeros,

  • we have a multiplicity of one.

  • There are only, they only deduced one time

  • when you look at it in factored form,

  • only one of the factors points to each of those zeros.

  • So they all have a multiplicity of one.

  • For P2, the first zero has a multiple of one,

  • only one of the expressions points to a zero of one,

  • or would become zero if x would be equal to one.

  • But notice, out of our factors,

  • when we have it in factored form,

  • out of our factored expressions,

  • or our expression factors I should say,

  • two of them become zero when x is equal to three.

  • This one and this one are going to become zero,

  • and so here we have a multiplicity of two.

  • And I encourage you to pause this video again

  • and look at the behavior of graphs,

  • and see if you can see a difference

  • between the behavior of the graph

  • when we have a multiplicity of one

  • versus when we have a multiplicity of two.

  • All right, now let's look through it together.

  • We could look at P1 where all of the zeros

  • have a multiplicity of one,

  • and you can see every time we have a zero

  • we are crossing the x axis.

  • Not only are we intersecting it, but we are crossing it.

  • We are crossing the x axis there, we are crossing it again,

  • and we're crossing it again,

  • so at all of these we have a sign change around that zero.

  • But what happens here?

  • Well on the first zero that has a multiplicity of one,

  • that only makes one of the factors equal zero,

  • we have a sign change, just like we saw with P1.

  • But what happens at x equals three

  • where we have a multiplicity of two?

  • Well there, we intersect the x axis still,

  • P of three is zero, but notice we don't have a sign change.

  • We were positive before, and we are positive after.

  • We touch the x axis right there, but then we go back up.

  • And the general idea, and I encourage you to test this out,

  • and think about why this is true,

  • is that if you have an odd multiplicity,

  • now let me write this down.

  • If the multiplicity is odd, so if it's one, three, five,

  • seven et cetera, then you're going to have a sign change.

  • Sign change.

  • While if it is even, as the case of two, or four, or six,

  • you're going to have no sign change.

  • No sign,

  • no sign change.

  • One way to think about it,

  • in an example where you have a multiplicity of two,

  • so let's just use this zero here, where x is equal to three,

  • when x is less than three,

  • both of these are going to be negative,

  • and a negative times and negative is a positive.

  • And when x is greater than three,

  • both of 'em are going to be positive,

  • and so in either case you have a positive.

  • So notice, you saw no sign change.

  • Another thing to appreciate is thinking about the number

  • of zeros relative to the degree of the polynomial.

  • And what you see is is that the number of zeros,

  • number of zeros is

  • at most equal to the degree of the polynomial,

  • so it is going to be less than or equal to

  • the degree of the polynomial.

  • And why is that the case?

  • Well you might not, all your zeros might have

  • a multiplicity of one, in which case the number

  • of zeros is equal, is going to be equal

  • to the degree of the polynomial.

  • But if you have a zero

  • that has a higher than one multiplicity,

  • well then you're going to have fewer distinct zeros.

  • Another way to think about it is,

  • if you were to add all the multiplicities,

  • then that is going to be equal

  • to the degree of your polynomial.

- [Instructor] So what we have here

字幕と単語

動画の操作 ここで「動画」の調整と「字幕」の表示を設定することができます

A2 初級

多項式のゼロの多重度|多項式グラフ|代数学2|カーンアカデミー (Multiplicity of zeros of polynomials | Polynomial graphs | Algebra 2 | Khan Academy)

  • 7 1
    林宜悉 に公開 2021 年 01 月 14 日
動画の中の単語