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Current time:0:00Total duration:6:36

Video transcript

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 in this video is continue our study of zeroes but we're gonna look at a special case when something interesting happens with there with the zeroes so let's just first look at P ones zeroes so I set up a little table here because it'll be useful so the first column let's just make it the zeroes the x-values at which at which our polynomial is equal to 0 and that's pretty easy to figure out from factored form when X is equal to 1 the whole thing is going to be equal to 0 because 0 times anything is 0 when X is equal to 2 by the same argument and when X is equal to 3 and we could see it here on the graph when x equals 1 the graph of y is equal to p1 intersects the x axis it does it again at the next 0 x equals 2 and at the next 0 x equals 3 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 0 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 P 2 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 p2f pause this video and think about that well let's just list them out so our zeros well once again if x equals 1 this whole expression is going to be equal to 0 so we have a 0 at x equals 1 and we can see that our white graph also intersects the x axis at x equals 1 and then if X is equal to 3 this whole thing is going to be equal to 0 and we can see that it intersects the x axis at x equals 3 and then notice this next this next part of the expression would say oh well we have a zero at x equals three oh 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 [Music] multi-city I'll write it out there and I will write it over here mul T plus a T and so for each of these zeros we have a multiplicity of one there only there 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 P to 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 1 but notice out of our factors when we have an 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 2 all right now let's look through it together we can look at P 1 where all of the zeros have a multiplicity of 1 and you can see every time we have a 0 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 year that has a multiplicity of 1 that only makes one of the factors equal zero we have a sign change just like we saw with p1 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 with 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 and let me write this down if the multiplicity is odd so if it's one three five seven etc 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 a negative is a positive and 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 now 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 that that is going to be equal to the degree of your polynomial