## 字幕表 動画を再生する

• - [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.

• 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.

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