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  • - 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.

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A2 初級

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)

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    林宜悉 發佈於 2021 年 01 月 14 日
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