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### Course: Integrated math 3 > Unit 4

Lesson 2: Positive and negative intervals of polynomials- Positive and negative intervals of polynomials
- Positive & negative intervals of polynomials
- Multiplicity of zeros of polynomials
- Zeros of polynomials (multiplicity)
- Zeros of polynomials (multiplicity)
- Zeros of polynomials & their graphs
- Positive & negative intervals of polynomials

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# Multiplicity of zeros of polynomials

The polynomial p(x)=(x-1)(x-3)² is a 3rd degree polynomial, but it has only 2 distinct zeros. This is because the zero x=3, which is related to the factor (x-3)², repeats twice. This is called multiplicity. It means that x=3 is a zero of multiplicity 2, and x=1 is a zero of multiplicity 1. Multiplicity is a fascinating concept, and it is directly related to graphical behavior of the polynomial around the zero.

## Want to join the conversation?

- Quick question out of curiosity, how would you be able to find out how the graph of a polynomial comes from the third quadrant (bottom left) instead of the second (top left) as the x axis approaches negative infinity?(8 votes)
- Remember that the graph of a function can 'bend' n-1 times, where n is the degree of the polynomial. After plotting the 'bends', plug in large negative values for x. If the y values are trending towards negative infinity as well, the function will come from the third quadrant. If the y values are increasing, it will come from the second quadrant. Or, if you know the end behavior on the positive end, you could determine whether it is an even or an odd function.(11 votes)

- I am still confused why an even multiplicity will result in no sign change, while an odd multiplicity will. Could someone explain?(5 votes)
- You can break a polynomial into "linear factors." For example, we can break x^3 - 4x into (x + 2)(x)(x - 2).

Imagine you are driving along the number line from left to right. As you drive onto the screen from the left, all three factors will be negative numbers.

For example, if x = -100, the polynomial will equal (-100 + 2)(-100)(-100 - 2) = (negative)(negative)(negative). When you multiply three negative numbers together, you get a negative result, so the entire polynomial will come out negative.

Now imagine you cross x = -2. The first linear factor, (x + 2), goes from negative to zero to positive. The very instant you cross x = 2, the polynomial becomes (positive)(negative)(negative) = positive.

Every time you "drive across" a zero, exactly one of the linear factors changes sign from negative to positive, and that flips the sign of the polynomial.

But when you have two identical roots, then TWO of the factors change sign from negative to positive at the same instant. So in that case, the sign of the polynomial DOESN'T change.

For example, say we have x^3 + 2x^2 = (x + 2)(x)(x). When we drive in from the left, the three factors start out as (negative)(negative)(negative), so the polynomial is negative. When we drive across x = -2, the factors become (positive)(negative)(negative), so the polynomial becomes positive.

But when we drive across x = 0, BOTH of the remaining factors flip at the same instant, so the factors become (positive)(positive)(positive), and the polynomial stays positive.(9 votes)

- what is multiplicity(2 votes)
- Multiplicity is just the degree (exponent) of each factor, or the amount of times a factor appears.

If the multiplicity is even, the graph will “bounce” off the x-axis where that factor is zero.

And if the multiplicity is odd, the graph will cross through the x-axis.(4 votes)

- If p(x)= (x-1) (7x-21) (x-3)

here zeros are 2 since 2 parts of the factored expression point to the same zero. But, (7x-21) & (x-3) are different expressions as the latter is a scaled-down version of the first one. Will it make any difference to the idea of multiplicity?(3 votes)- No. p(x) has two distinct zeroes, 1 (of multiplicity 1) and 3 (of multiplicity 2).

This is why, when writing polynomials in factored form, we usually factor out any constants so that the coefficients of all of the x's are 1. So p would be written as

7(x-1)(x-3)².(0 votes)

- Did he spell zeroes wrong i dont know if its an american thing or not
**bold**(2 votes)- it depends on which way your looking at it.If you are from a different ethical background it may seem misspelled but if your just a person who hates English you can notice the difference(1 vote)

- What if the multiplicity is 3, then how would the graph of that polynomial look like?(2 votes)
- I got most of the video, but the end just completely flew me off track. What does Sal mean by the number of zeros is equal to degree of a polynomial? Please help.(1 vote)
- The degree of a polynomial is the highest exponent that appears in it. The degree of x³-5x²+1 is 3.

A zero of a polynomial is a value that you can plug in for x to make the whole expression equal 0. -1 is a zero of the polynomial x⁵+1, since (-1)⁵+1=0.

Most polynomials have multiple different zeroes. 1 and 2 are both zeroes of x²-3x+2. Sal is saying that any polynomial has exactly as many zeroes as its degree, assuming you count the multiplicity of zeroes and allow complex zeroes.(2 votes)

- Is the reason that a multiplicity of 2 curves upwards because any number that is exponentiated to the power of an even number cannot be negative? EX: x^2 can never be a neg #.(1 vote)
- I don't get how he knows the multiplicity. In a equation like (x+5)^2(2x-3)^5 what would -5 be a multiplicity of? it doesn't make sense(1 vote)
- f(x)=(x+1)^3(x-2)^2(x-4) The multiplicity of a zero(0 votes)
- 𝑓(𝑥) = (𝑥 + 1)³(𝑥 − 2)²(𝑥 − 4)

has the zeros 𝑥 = −1, 𝑥 = 2, and 𝑥 = 4

with the multiplicities 3, 2, and 1 respectively.(3 votes)

## Video transcript

- [Instructor] So what we have here are two different polynomials, P1 and P2. And they have been
expressed in factored form and you can also see their graphs. This is the graph of Y is
equal to P1 of x in blue, and the graph of Y is
equal to P2 x in white. What we're going to do in this video is continue our study of zeros, but we're gonna look at a special case when something interesting
happens with the zeros. So let's just first look at P1's zeros. So I'll set up a little table
here, because it'll be useful. So the first column, let's
just make it the zeros, the x values at which our
polynomial is equal to zero and that's pretty easy to
figure out from factored form. When x is equal to one, the whole thing's going
to be equal to zero because zero times anything is zero. When x is equal to two,
by the same argument, and when x is equal to three. And we can see it here on the graph, when x equals one, the graph of y is equal to
P1 intersects the x axis. It does it again at the
next zero, x equals two. And at the next zero, x equals three. We can also see the property that between consecutive
zeros our function, our polynomial maintains the same sign. So between these first two, or actually before this
first zero it's negative, then between these
first two it's positive, then the next two it's negative, and then after that it is positive. Now what about P2? Well P2 is interesting, 'cause if you were to multiply this out, it would have the same degree as P1. In either case, you would
have an x to the third term, you would have a third degree polynomial. But how many zeros, how many distinct unique
zeros does P2 have? Pause this video and think about that. Well let's just list them out. So our zeros, well once again if x equals one, this whole expression's
going to be equal to zero, so we have zero at x equals one, and we can see that our
white graph also intersects the x axis at x equals one. And then if x is equal to three, this whole thing's going
to be equal to zero, and we can see that it intersects the x axis at x equals three. And then notice, this next part
of the expression would say, "Oh, whoa we have a
zero at x equals three," but we already said that, so
we actually have two zeros for a third degree polynomial, so something very
interesting is happening. In some ways you could say that hey, it's trying to reinforce that we have a zero at x minus three. And this notion of having multiple parts of our factored form that would
all point to the same zero, that is the idea of multiplicity. So let me write this word down. So multiplicity. Multiplicity, I'll write it out there. And I will write it
over here, multiplicity. And so for each of these zeros, we have a multiplicity of one. There are only, they only deduced one time when you look at it in factored form, only one of the factors
points to each of those zeros. So they all have a multiplicity of one. For P2, the first zero
has a multiple of one, only one of the expressions
points to a zero of one, or would become zero if
x would be equal to one. But notice, out of our factors, when we have it in factored form, out of our factored expressions, or our expression factors I should say, two of them become zero
when x is equal to three. This one and this one
are going to become zero, and so here we have a multiplicity of two. And I encourage you to
pause this video again and look at the behavior of graphs, and see if you can see a difference between the behavior of the graph when we have a multiplicity of one versus when we have a multiplicity of two. All right, now let's
look through it together. We could look at P1 where all of the zeros have a multiplicity of one, and you can see every time we have a zero we are crossing the x axis. Not only are we intersecting
it, but we are crossing it. We are crossing the x axis
there, we are crossing it again, and we're crossing it again, so at all of these we have a
sign change around that zero. But what happens here? Well on the first zero that
has a multiplicity of one, that only makes one of
the factors equal zero, we have a sign change,
just like we saw with P1. But what happens at x equals three where we have a multiplicity of two? Well there, we intersect the x axis still, P of three is zero, but notice
we don't have a sign change. We were positive before,
and we are positive after. We touch the x axis right
there, but then we go back up. And the general idea, and I
encourage you to test this out, and think about why this is true, is that if you have an odd multiplicity, now let me write this down. If the multiplicity is odd,
so if it's one, three, five, seven et cetera, then you're
going to have a sign change. Sign change. While if it is even, as the
case of two, or four, or six, you're going to have no sign change. No sign, no sign change. One way to think about it, in an example where you
have a multiplicity of two, so let's just use this zero
here, where x is equal to three, when x is less than three, both of these are going to be negative, and a negative times and
negative is a positive. And when x is greater than three, both of 'em are going to be positive, and so in either case you have a positive. So notice, you saw no sign change. Another thing to appreciate
is thinking about the number of zeros relative to the
degree of the polynomial. And what you see is is
that the number of zeros, number of zeros is at most equal to the
degree of the polynomial, so it is going to be less than or equal to the degree of the polynomial. And why is that the case? Well you might not, all
your zeros might have a multiplicity of one,
in which case the number of zeros is equal, is going to be equal to the degree of the polynomial. But if you have a zero that has a higher than one multiplicity, well then you're going to
have fewer distinct zeros. Another way to think about it is, if you were to add all the multiplicities, then that is going to be equal to the degree of your polynomial.