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  • - [Instructor] Let's say that we have the polynomial p of x.

  • And when expressed in factored form, it is x plus two

  • times two x minus three

  • times x minus four.

  • What we're going to do in this video is use our knowledge

  • of the roots of this polynomial to think about intervals

  • where this polynomial would be positive or negative.

  • And the key realization is, is that the sign

  • of a polynomial stays the same between consecutive zeros.

  • Let me just draw an arbitrary graph of a polynomial here

  • to make you appreciate why that is true.

  • So x-axis, y-axis.

  • And if I were to draw some arbitrary polynomial like that,

  • you can see that between consecutive zeros

  • the sign is the same.

  • Between this zero and this zero, the polynomial is positive.

  • Between this zero and this zero, the polynomial is negative.

  • And that's almost intuitively true.

  • Because if the sign did not stay the same,

  • that means you would have to cross the x-axis.

  • So you would have a zero,

  • but we're saying between consecutive zeros.

  • So between this zero and this zero, it is positive again.

  • Then after that zero, it stays negative.

  • Once again, the only way it wouldn't stay negative is

  • if there were another zero.

  • So now let's go back to this example here.

  • And let me delete this because this is not the graph

  • of p of x, which I have just written down.

  • Let's first think about its zeros.

  • So the zeros are the x-values

  • that would either make x plus two equal zero,

  • two x minus three equal zero, or x minus four equal zero.

  • So first, we can think about,

  • well, what x-values would make x plus two,

  • x plus two equal to zero?

  • Well, that, of course, would be x equals negative two.

  • What x-values would make two x minus three equal zero?

  • Two x minus three equal to zero,

  • add three to both sides, you get two x equals three.

  • Divide both sides by two, you get x equals 3/2.

  • And then last but not least, what x-values would make

  • x minus four equal to zero? (distant siren wailing)

  • Add four to both sides, you get x is equal to four.

  • And so if we were to plot this,

  • it would look something like this.

  • So this x equals negative two,

  • (distant siren wailing) x equals negative one.

  • This is zero.

  • This is one, two,

  • three, and four.

  • And let me draw the y-axis here.

  • So the y-axis would look something like this, x and y.

  • We have a zero at x equals negative two,

  • so our graph will intersect the x-axis there.

  • We have a zero at x is equal to 3/2, which is 1 1/2,

  • which is right over there.

  • And we have a zero at x equals hour,

  • which is right over there.

  • And so we have several candidate intervals.

  • And actually, let me write this down in a table.

  • So the intervals

  • over which and this is really,

  • be between consecutive zeros,

  • intervals to consider.

  • So I'll draw a little table here.

  • So you have x is less than negative two.

  • That's one interval.

  • X is less than, actually, let me color-code this.

  • So if I were to say the interval

  • for x is less than negative two,

  • so that's this yellow that I draw at the extreme left there.

  • We could have an interval where

  • x is between negative two and 3/2,

  • so negative two is less than x, is less than positive 3/2.

  • That would be this interval right over here.

  • You have the interval, I'm trying to use all my colors,

  • between 3/2 and four, this interval here.

  • So that would be 3/2 is less than x, is less than four.

  • And then last but not least,

  • you have the interval where x is greater than four,

  • that interval right over there.

  • So x is greater,

  • greater than four.

  • Now, there's a couple of ways of thinking about whether,

  • over that interval, our function is positive or negative.

  • One method is to just evaluate our p,

  • our function at a point in the interval.

  • And if it's positive,

  • well, that's means that that whole interval is positive.

  • It's negative,

  • that means that that whole interval is negative.

  • And once again, it's intuitive because,

  • if for whatever reason, it were to switch,

  • we would have another zero.

  • I know I keep saying that.

  • But another way to think about it is,

  • over that interval, what is the behavior of x plus two,

  • two x minus three, and x minus four?

  • Think about whether they're positive or negative,

  • and use our knowledge of multiplying positives

  • and negatives together to figure out whether we're dealing

  • with a positive or negative.

  • So let's do it, we could do it both ways.

  • So let's think of this as our sample x,

  • sample x-value.

  • And then let's see what we can intuit about

  • or deduce about whether, over that interval,

  • we are positive or negative.

  • So for x is less than negative two,

  • maybe an easy one or an obvious one to use,

  • it could be any value where x is less than negative two,

  • but let's try x is equal to negative three.

  • So you could try to evaluate p of negative three.

  • You could just evaluate that.

  • Actually, let's just do that.

  • So that's going to be equal to negative one times,

  • two times negative three is negative six,

  • minus three is negative nine,

  • negative nine times negative three minus four,

  • that is negative seven.

  • So if you were to multiply all of this out,

  • this would give you negative 63, which is clearly negative.

  • So over this interval right over here,

  • our polynomial is going to be negative.

  • So then we can move on to the next one.

  • An interesting thing is we didn't even have

  • to figure out the 63 part.

  • We can just see that there's a negative times a negative

  • times a negative, which is going to be a negative.

  • And so let's just do that going forward.

  • Let's just think about whether each

  • of these are going to be positive or negative

  • and what would happen when you multiply

  • those positive and negatives together.

  • Now, in this second interval between negative two and 3/2,

  • what is going to happen?

  • Well, we could do a sample point.

  • Let's say x is equal to zero.

  • That might be pretty straightforward.

  • Well, when x is equal to zero,

  • we're going to be dealing with a positive

  • times a negative times a negative,

  • a positive times a negative times a negative.

  • And the reason why I did that is in my head.

  • I said, okay, that's going to be a positive two

  • times a negative three times a negative four.

  • So a positive times a negative times a negative,

  • so I could write it this way.

  • It's going to be a positive times a negative

  • times a negative.

  • Well, a negative times a negative is a positive,

  • and a positive times a positive is a positive.

  • So we are positive over that interval.

  • And if you were to evaluate p of zero,

  • you will get a positive value.

  • Now what about this next interval?

  • What about this next interval here between 3/2 and four?

  • We could try x is equal to two.

  • At when x is equal to two,

  • we are going to get a positive

  • times a positive

  • times a, two minus four is negative, times a negative.

  • So this is going to be negative over that interval.

  • And then last but not least, when x is greater than four,

  • we could try x is equal to let's say five.

  • We are going to have a positive,

  • positive times a positive

  • times a positive.

  • So we are going to have a positive.

  • And as I mentioned,

  • you could also do it without the sample points.

  • You could say, okay, when x is greater than four,

  • you could say, okay, for any x greater than four,

  • if you add two to it, that for sure is going to be positive.

  • For any x greater than four,

  • if you multiply it by two and subtract three,

  • well, that's still going to be positive

  • 'cause two times something greater than four

  • is definitely greater than three.

  • And for any x greater than four,

  • if you subtract four from it,

  • you're still going to have a positive value.

  • So that's another way to think about it,

  • even if you don't use a sample point.

  • But there you have it, we figured out the intervals

  • over which the function is negative or positive.

  • And we don't know exactly what the function looks like,

  • but generally speaking it's negative

  • over this first interval.

  • So it might look something like this.

  • It's positive over that next interval.

  • And then it's negative over that third interval.

  • And then it's positive over that last interval.

  • So we'd have a general shape like this.

  • We don't know, without trying out more points,

  • exactly how high or low it would actually go.

- [Instructor] Let's say that we have the polynomial p of x.

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A2 初級

多項式の正負の区間|多項式グラフ|代数2|カーンアカデミー (Positive and negative intervals of polynomials | 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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