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  • - A polynomial P has zeros when X is equal to negative four,

  • X is equal to three,

  • and X is equal to one-eighth.

  • What could be the equation of P?

  • So pause this video and think about it on your own

  • before we work through it together.

  • All right.

  • So the fact that we have zeros at these values,

  • that means that P of X,

  • when X is equal to one of these values is equal to zero.

  • So P of negative four is equal to zero,

  • P of three is equal to zero,

  • and P of one-eighth is equal to zero.

  • And before I even look at these choices,

  • I could think about constructing a polynomial for

  • which that is true.

  • That's going to be true if I can express this polynomial

  • as the product of expressions where each of these

  • would make each of those expressions equal to zero.

  • So what's an expression that would be zero

  • when X is equal to negative four?

  • Well the expression X plus four,

  • this is equal to zero when X is equal to negative four,

  • so I like that.

  • What would be an expression that would be equal to zero

  • when X is equal to three?

  • Well what about the expression X minus three?

  • If X is equal to three,

  • then this is going to equal to be equal zero.

  • Zero times anything is going to be equal to zero.

  • So P of three would be zero in this case.

  • And then what is an expression that would be equal to zero

  • when X is equal to one-eighth?

  • Well that would be X minus one-eighth.

  • Now tho--

  • these aren't the only expressions.

  • You could multiply them by constants and still

  • the principles that I just talked about would be true.

  • But our polynomial would look something like this.

  • You could try it out.

  • If X is equal to negative fou--

  • (chuckles)

  • if X is equal to negative four,

  • well then this first expression is zero.

  • Zero times something times something is zero.

  • Same thing for X equals three.

  • If this right over--

  • If X equals three,

  • then X minus three is equal to zero,

  • and then zero times something times something is zero.

  • And then if X is equal to one-eighth,

  • this expression's going to be equal to zero.

  • Zero times something times something

  • is going to be equal to zero.

  • So which of these choices look like that?

  • So let's see.

  • X plus

  • four,

  • I actually see that in choices

  • B,

  • and I see that in choices D.

  • Choice C has X minus four there.

  • So that would have a zero at X equals four.

  • If X equals four, this first--

  • this first expression--

  • this first part of the expression

  • would be equal to zero.

  • But we care about that happening when

  • X is equal to negative four.

  • So I would actually rule out C,

  • and then for the same reason I would rule out A.

  • So we're between B and D,

  • and now let's see.

  • Which of these have an X minus three in them.

  • Well I see an X minus three here.

  • I see an X minus three there.

  • So I like the--

  • I still like B and D.

  • I'll put another check mark right over there.

  • And then last but not least,

  • which of these would be equal to zero when

  • X is equal to one-eighth?

  • Well, let's see.

  • If I do one-eighth times one-eighth here

  • I'm gonna get one-sixty-fourth for

  • this part of the expression.

  • And so that's not going to be equal to zero.

  • And these other two things aren't going to be

  • equal to zero when X is equal to one-eighth,

  • so this one is not looking so good,

  • but let's verify this one.

  • This has--

  • If X is equal to one-eighth,

  • we have eight times one-eighth which is one,

  • minus one.

  • That is going to be equal to zero,

  • so this one checks out.

  • And you might be thinking hey!

  • This last polynomial looks a little bit different

  • than this polynomial that I wrote up here

  • when I just tried to come up with a polynomial for which

  • this would be true.

  • And as I mentioned,

  • you could take this and multiply it by constants

  • and it would still be true. So if you just take this,

  • and if we were to multiply it by eight,

  • you would get P of X down here,

  • because if we were to multiply this times eight,

  • which wouldn't change the zeros,

  • well then if you distribute this eight,

  • this last expression would become eight X minus one.

  • Which is exactly what we have down here.

- A polynomial P has zeros when X is equal to negative four,

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A2 初級

多項式のゼロ:方程式をゼロに合わせる|多項式グラフ|代数学2|カーンアカデミー (Zeros of polynomials: matching equation to zeros | Polynomial graphs | Algebra 2 | Khan Academy)

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    林宜悉 に公開 2021 年 01 月 14 日
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