[Music] good day everyone so in this video lesson we will discuss about graphs of polynomial function so our velocity is one of the property of graphs of polynomial function which is the multiplicity of roots or zeros so in multiplicity of roots or zeros it tells how many times a particular number of roots zero of a given polynomial so the characteristics of multiplicity kappa add number so it crosses the x-axis x-axis when the crop changes from positive to negative or vice versa even number no even number your multiplicity okay young tangent x-axis x-axis x axis so the graph is either positive on both side of the root or negative on both sides of the root okay so let's okay for example given the function y is equal to x plus 2 squared x plus 1 cube x minus 1 cube x minus 2 so and it's graph complete the table below okay first uh equate moon and not into zero so equate that is zero so x plus is equal to zero and that is negative two so your roots need or in solution at n or the value of x is negative two and then you x plus one that is negative one and then you x minus one that is positive one so hop again equate nothing and z zero that is x is equal to one and then x minus two kappa g equates a zero that is x is equal to two so your multiplicity ebits have been elam bases uh exponentially so it's a negative two that is two it's a negative one that is uh okay so you say say nothing by the class since um root negative two the multiplicity is two and that is multiplicity is three so that is add number so add number your multiplicity class so it is a negative one right so you multiplicity not n kappa event that is tangent to x axis uh your multiplicity nothing is add number that is uh crosses to crosses to x-axis okay next is okay so this is four okay paritank four okay so the next one so you multiplicity then x minus one and that is four so that is even number so up an even number eb sabinion graph nathan is so italian okay so that is tangent x-axis since even number your multiplicity that is tangent to x axis next so two the multiplicity of two is one okay itaeon and that one is add number and kapag add number okay it crosses to x axis okay so the crosses are x axis nothing so gangnam chakras uh this is my behavior in a graph x axis another example we have y is equal to x plus three times x minus two squared times x plus one cubed okay so so first equate that into the zero x plus three is equal to zero that is negative three and then x minus two equal to zeros x is equal to posit uh that is okay okay you it's a number in line so x plus one equal to zero that is negative one and then x minus two equal to zero that is positive two okay so uh in a nut and in arrangement and so you will set us you there's a number line so it's a negative three and multiplicity is one okay so that is add number so a pack add number so that is okay closest to x axis and there's a negative one that is three and it's still add number okay on the ban and it is a negative one so that is still closest to x axis and then the last one young two okay so your multiplicity number two is to then and that is even number so upon even number so and grab nothing tangent to x axis thank you for watching this video i hope you learned something don't forget to like subscribe and hit the bell button put updated ko for more video tutorial this is your guide in learning your math lesson your walmart channel