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• - [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.

• 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

# 多項式零的多重性｜多項式圖形｜代數2｜可汗學院 (Multiplicity of zeros of polynomials | Polynomial graphs | Algebra 2 | Khan Academy)

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林宜悉 發佈於 2021 年 01 月 14 日