Well P2 is interesting, 'cause if you were to multiply this out, it would have the same degree as P1. zero at x equals three," but we already said that, so P1 intersects the x axis. This is the graph of Y is degree of the polynomial, so it is going to be less than or equal to the degree of the polynomial. we don't have a sign change. there, but then we go back up. 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. points to a zero of one, or would become zero if Zeros and multiplicity When a linear factor occurs multiple times in the factorization of a polynomial, that gives the related zero multiplicity. For example, in the polynomial f (x)= (x-1) (x-4)^\purpleC {2} f (x) = (x −1… And they have been degree of the polynomial. have fewer distinct zeros. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. your zeros might have a multiplicity of one, A zero has a "multiplicity", which refers to the number of times that its associated factor appears in the polynomial. But what happens here? just make it the zeros, the x values at which our The final solution is all the values that make x2(x+3)(x− 3) = 0 x 2 (x + 3) (x - 3) = 0 true. And then notice, this next part just like we saw with P1. So the first column, let's - [Instructor] So what we have here are two different polynomials, P1 and P2. has a multiple of one, only one of the expressions equal to P1 of x in blue, and the graph of Y is And this notion of having multiple parts of our factored form that would When x is equal to two, so if it's one, three, five, seven et cetera, then you're happens with the zeros. interesting is happening. look through it together. figure out from factored form. Multiplicity of Zeros and Graphs Polynomials An app is used to explore the effects of multiplicities of zeros and the leading coefficient on the graphs of polynomials the form: f(x) = a(x − z1)(x − z2)(x − z3)(x − z4)(x − z5) And why is that the case? So let me write this word down. 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 it, but we are crossing it. The zero associated with this factor, x= 2 x = 2, has multiplicity 2 because the factor (x−2) (x − 2) occurs twice. x would be equal to one. But what happens at x equals three where we have a multiplicity of two? of the expression would say, "Oh, whoa we have a So I'll set up a little table We are crossing the x axis But how many zeros, how many distinct unique are going to become zero, and so here we have a multiplicity of two. case of two, or four, or six, you're going to have no sign change. And I encourage you to For instance, the quadratic (x + 3) (x – 2) has the zeroes x = –3 and x = 2, each occuring once. Well let's just list them out. While if it is even, as the The zero associated with this factor, x =2 x = 2, has multiplicity 2 because the factor (x−2) (x − 2) occurs twice. So they all have a multiplicity of one. we actually have two zeros for a third degree polynomial, so something very There are only, they only deduced one time when you look at it in factored form, only one of the factors In some ways you could say that hey, it's trying to reinforce that we have a zero at x minus three. Now what about P2? Well on the first zero that expressed in factored form and you can also see their graphs. And what you see is is have a multiplicity of two, so let's just use this zero Not only are we intersecting sign change around that zero.

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