Hey, everyone. Whenever we worked with quadratic functions and their parabolas, we were able to find our x-intercepts by setting f(x) equal to 0 and most often getting either 1 or 2 x-intercepts by solving that quadratic equation. Now for polynomial functions, we could have 3 x-intercepts or even more than that. So how are we going to find all of them? Well, we're actually going to do the exact same thing and set f(x) equal to 0, but the problem still remains, how do I solve that for some complicated polynomial? And it's actually something that we've done a million times before and that is simply by factoring. So really, we're just going to factor, set each factor equal to 0, and then we're going to look at something called the multiplicity of each zero. So I'm going to walk you through all of this. Let's go ahead and get started.

So I mentioned the multiplicity of the zero and this is really just the number of times that a factor occurs. So if I have a factor, x-12 that I use in order to get the zero x is equal to 1, my zero x equals 1 is going to have a multiplicity of 2 because that factor occurs twice. This is really just x minus 1 times x minus 1, and that's it. So let's go ahead and look at a more complicated polynomial here. So I have f(x)=2x×x-32×x+43. Let's go ahead and solve for each zero. This is already factored for me, so I can go ahead and just take each individual factor and set them equal to 0. So first, I have 2x is equal to 0, then I have this x minus 3 squared, which I know that squared isn't going to do anything, so I can just set x minus 3 equal to 0. And then I have x + 4 cubed, so I can set x + 4 equal to 0. So let's go ahead and do 2x is equal to 0 first. So if I divide both sides by 2, I'm simply left with x is equal to 0. Now over here with x minus 3, I can go ahead and add 3 to both sides, leaving me with x is equal to 3. Lastly, I have x + 4 equals 0. Subtracting 4 from both sides, I am left with x is equal to negative 4, and those are going to be my 3 zeros or my 3 x-intercepts.

Let's go ahead and identify the multiplicity of each of these zeros as well. So looking at this first factor or my first zero of x equals 0 that I got from 2x. Now, this factor only happens once, so it has a multiplicity of 1. Now looking at my second factor, for my second zero of x equals 3, which I got from the factor x minus 3 squared. Now this factor happens twice, so it has a multiplicity of 2. Now looking at my last zero, x is equal to negative 4, I got that from the factor x + 4 cubed, so this has a multiplicity of 3.

Now what's the point of multiplicity? Why do we even need to find it? Well, multiplicity is actually going to tell us the behavior of our graph at each zero. So if our multiplicity is even, that tells us that our graph is going to touch our x-axis and simply bounce right back off. So at this point here, it just touches our x-axis, bounces back off in the same direction it came from. Now if our multiplicity is instead odd, our graph is going to fully cross our x-axis, going from one side to the other like we see at these two points. So let's go ahead and identify that in our example. So if I had this multiplicity of 1, one is an odd number, so that means that my graph is going to fully cross the x-axis at that zero. Now here, I have a multiplicity of 2, which is an even number, so it's simply going to touch the x-axis and bounce right back off. Lastly, here, I have a multiplicity of 3, which is again an odd number, so my graph is going to fully cross my x-axis again at that point. So that's all you need to know about zeros and their multiplicity. Let's get some practice.