 ## 字幕列表 影片播放

• - All right, now let's work through this together.

• And we can see that all of the choices

• are expressed as a polynomial in factored form.

• And factored form is useful when we're thinking

• about the roots of a polynomial,

• the x-values that make that polynomial equal to zero.

• The roots are also evident when we look at this graph here.

• We have a root at x equals -4,

• a root at x equals -1.5, or -3/2,

• and a root at x is equal to 1.

• So really what we have to do is say,

• "Which of these factors are consistent

• with the roots that we see?"

• So let's go root by root.

• So here on the left we have a root at x equals -4.

• In order for this polynomial to be zero

• when x is equal to -4, that means that x + 4

• must be a factor, or some multiple,

• or some constant times x + 4,

• must be a factor of our polynomial.

• Now we can see in the choices that

• we have a bunch of x + 4s,

• but they have different exponents on them.

• The first one has a 2 as an exponent, it's being squared,

• while the others have a 1 as the exponent.

• Now what we've talked about in other videos,

• when we talked about multiplicity,

• we said, "Hey, if we see a sign change

• around a root, like we're seeing right over here

• around x equals -4,"

• that means that we are going to see

• an odd exponent on the corresponding factor.

• But if we didn't see a sign change,

• as we see in this other root over here,

• that means we would see an even exponent.

• Now, we clearly see the sign change,

• so we would expect an odd exponent,

• and, of course, 1 is an odd number and 2 isn't.

• So if you just have a straight up x + 4,

• you would have a sign change around x equals -4.

• So I can rule out this first choice,

• these other three choices are still looking good

• based on just the x + 4 factor.

• Now lets move on to the next factor right over here,

• so, or the next root.

• The next root is at x is equal to -3/2,

• and so one way to think about it is you could have

• a factor that looks like x + 3/2,

• or this times some constant.

• Now, when we look at the choices, or the remaining choices,

• we don't see x + 3/2, but we do see something

• that involves a 2 and a 3, and so one way to think about it

• is just, "Hey, if I just multiply this by the constant 2,

• that would get us 2x + 3," well I do see that

• right over here, and then the next question is

• what should be the exponent?

• Well, once again we have a sign change

• around x equals -3/2, so we would expect

• an odd exponent there.

• And you can see out of the choices,

• only two of them have an exponent of 1,

• which is an odd number, while the other one

• has an even exponent there, so we can rule

• that one out as well.

• And then we go to this last root.

• Ah, we'll do this in an orange color.

• We have a root at x equals 1,

• so we would expect x - 1, or this

• multiplied by some constant, to be one of the factors.

• And what's interesting here is we don't see a sign change

• around x equals 1, so we would expect an even exponent.

• And so, out of the remaining choices,

• we see an x - 1 in both of them,

• but only choice C has the even exponent

• that we would expect, so choice C is looking good.

• If we were to look at choice D,

• where this is to the first power,

• we would expect a sign change

• around x is equal to 1, so this would be

• a situation where the curve would keep going down,

• something like that, so we like choice C.

• Let's do another example.

• So, once again, we are asked,

• "What could be the equation of p?"

• and we're given a graph, so again

• pause this video and try to work through that.

• All right, we're going to do the same idea.

• Let's go to this first root right over here.

• We have a root at x equals -3,

• so we would expect some multiple

• of x + 3 to be one of the factors.

• We also have a sign change around x equals -3,

• so we would expect an odd multiplicity

• and we would expect an odd exponent

• on the x + 3 factor.

• When we look at all the choices,

• C and D have an even exponent,

• so if we had x + 3 to the fourth,

• then you wouldn't have a sign change here,

• you would just touch the x-axis

• and then go back to where it was coming from.

• So we can rule out these choices.

• And now, let's look at the second root.

• Right over here, at x is equal to 2,

• so we would expect x - 2 to be one of the factors,

• or a multiple of this, and because we don't have

• a sign change around x equals 2,

• the graph just touches the x-axis

• and goes back to where it was coming from,

• we would expect an even exponent here.

• And so when we look at the choices around the x -2 factor,

• we see only one of them has an even exponent,

• so I am liking Choice B,

• and we are done.

- All right, now let's work through this together.

# 多項式的零點（倍率）｜多項式圖形｜代數2｜可汗學院 (Zeros of polynomials (multiplicity) | Polynomial graphs | Algebra 2 | Khan Academy)

• 5 0
林宜悉 發佈於 2021 年 01 月 14 日