Find the zeros of the given polynomial function and give the multiplicity of each. State whether the graph crosses or touches the x-axis at each zero. f(x)=2x4−12x3+18x2f\left(x\right)=2x^4-12x^3+18x^2f(x)=2x4−12x3+18x2

Touch at x = 0 x=0 x = 0 , Touch at x = 3 x=3 x = 3

Find the zeros of the given polynomial function and give the multiplicity of each. State whether the graph crosses or touches the x-axis at each zero. f ( x ) = 2 x 4 − 12 x 3 + 18 x 2 f\left(x\right)=2x^4-12x^3+18x^2 f ( x ) = 2 x 4 − 12 x 3 + 18 x 2

In this problem, we wanna find the zeros of the polynomial function and then find the multiplicity of each zero which will then tell us whether a graph is going to touch or fully cross the x axis at each of those points. So let's go ahead and get started. Now the function that I have here is f of x is equal to 2x to the power of 4 minus 12x cubed plus 18x squared. Now remember, finding zeros, we're just taking our function and setting it equal to 0 and solving that equation. Now here, I'm going to go ahead and factor. So I'm going to first look for the greatest common factor between all of these terms. Now you won't always have one, but here, looking at these terms, I do have a greatest common factor, gcf. It looks like all of these terms have in common the term 2 x squared, which I can go ahead and pull out of each of these terms. Now factoring out that 2x squared out of everything I have here, I'm left with x squared minus 6x plus 9, and all of that is still equal to 0. Now I have this 2x squared. I don't really need to worry about that anymore, but I can further factor this x squared minus 6x+9 because it is actually a perfect square trinomial. So if I have my 2x squared here, but then this x squared minus 6x+9 will fully factor into x minus 3 squared, and that's still all equal to 0. So this is now fully factored. So I can go ahead and take each individual factor and set them equal to 0 to actually get the zeros of my polynomial function. So starting with this first x squared term, I can go ahead and set that equal to 0. And if I take the square root of 0, I know that that's not going to do anything. I'm simply left with x is equal to 0. There's my first zero. Now looking at my second factor here, I have this x minus 3 squared, and I can go ahead and just take x minus 3 and set that equal to 0 because I know that that squared isn't going to do anything. I would just be taking the square root of 0. So this x minus 3 equals 0, I can go ahead and add 3 to both sides, leaving me with x is equal to 3 as my second zero. Okay. Now that I found my zeros, let's go ahead and identify the multiplicity of each of them. So looking at my first factor from my first zero of x equals 0, which came from this factor, x squared. Now x squared, since it is squared, that means it occurs twice, so it has a multiplicity of 2. Whenever our multiplicity is even, that tells us that we are simply going to touch or bounce off of our x axis going right back in the direction we came from. So here with my multiple c of 2, we're simply going to touch the x axis. Now looking at my second factor, x equals 3, it came from the factor x minus 3, which is squared. Now because, again, this factor occurs twice, it also has a multiplicity of 2, which we just said is an even number, which then tells us that it is also simply going to touch the x axis at this point. So that's all for this one. I'll see you in the next one.

Find the zeros of the given polynomial function and give the mult...

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Find the zeros of the given polynomial function and give the multiplicity of each. State whether the graph crosses or touches the x-axis at each zero. $f\left(x\right)=2x^4-12x^3+18x^2$