Multiplicity of zeros of polynomials | Polynomial graphs | Algebra 2 | Khan Academy

So what we have here are two different polynomials, p1 and p2, and they have been expressed in factored form. 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 of x in white.

What we're going to do in this video is continue our study of zeros, but we're going to look at a special case when something interesting happens with their zeros. So let's just first look at p1's zeros. I'll set up a little table here because it'll be useful.

The first column, let's just make it the zeros, the x values at which our polynomial is equal to zero. That's pretty easy to figure out from factored form. When x is equal to one, the whole thing is 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. 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 because 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 is 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. Then, if x is equal to three, this whole thing is going to be equal to zero, and we can see that it intersects the x-axis at x equals three.

Then notice this next part of the expression would say, oh, well, we have 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 it's trying to reinforce that we have a zero at x minus three.

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: multiplicity. Multiple! I'll write it out there, and I will write it over here: multiplicity.

For each of these zeros, we have a multiplicity of one; they're only deduced one time. When you look at it at 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 multiplicity 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.

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 can 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're crossing the x-axis there, we're 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 3 is 0, 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.

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—and let me write this down—if the multiplicity is odd, so if it's 1, 3, 5, 7, etc., then you're going to have a sign change. While if it is even, as in the case of two or four or six, you're going to have no sign change—no sign change.

One way to think about it is 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 a negative is a positive. When x is greater than three, both of them 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. What you see is that the 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.

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 going to be equal to the degree of the polynomial. But if you have a zero that has a higher than one multiplicity, then you're going to have fewer distinct zeros.

Another way to think about it is if you were to add all the multiplicities, that is going to be equal to the degree of your polynomial.