Finding zeros of polynomials
- [Voiceover] So, we have a fifth-degree polynomial here, p of x, and we're asked to do several things. First, find the real roots. And let's sort of remind ourselves what roots are. So root is the same thing as a zero, and they're the x-values that make the polynomial equal to zero. So the real roots are the x-values where p of x is equal to zero. So, the x-values that satisfy this are going to be the roots, or the zeros, and we want the real ones. As you'll learn in the future, there's also going to be imaginary roots, or zeros, or there might be. Then we want to think about how many times, how many times we intercept the x-axis. Well as we'll see, however many real roots we have that's how many times we are going to intercept... However many unique real roots we have, that's however many times we're going to intercept the x-axis. How do I know that? Well, let's just think about an arbitrary polynomial here. So those are my axes. This is the x-axis, that's my y-axis. And let me just graph an arbitrary polynomial here. So, let's say it looks like that. Well, what's going on right over here. At this x-value, we see, based on the graph of the function, that p of x is going to be equal to zero. So that's going to be a root. This is also going to be a root, because at this x-value, the function is equal to zero. At this x-value the function's equal to zero. At this x-value the function is equal zero. If we're on the x-axis then the y-value is zero. So the function is going to be equal to zero. This is a graph of y is equal, y is equal to p of x. Not necessarily this p of x, but I'm just drawing some arbitrary p of x. So there's some x-value that makes the function equal to zero. Well, that's going to be a point at which we are intercepting the x-axis. So we want to know how many times we are intercepting the x-axis. As we'll see, it's gonna be the same number of real roots, or the same number of real zeros we have. And then they want us to figure out the smallest of those x-intercepts, and we'll figure it out for this particular polynomial. So, let me give myself a little bit more space. So, let's get to it. So we really want to solve p of x is equal to zero. So we really want to set, that right over there, equal to zero, and solve this. So we want to solve this equation. The x-values that make this equal to zero, if I input them into the function I'm gonna get the function equaling zero. All right. So the first thing that might jump out at you is that all of these terms are divisible by x. So I like to factor that out from the get-go. So, we can rewrite this as x times x to the fourth power plus nine x-squared minus two x-squared minus 18 is equal to zero. Now there's something else that might have jumped out at you. It actually just jumped out of me as I was writing this down is that we have two third-degree terms. After we've factored out an x, we have two second-degree terms. Now, it might be tempting to just add these two together, and actually that it would be a completely legitimate way of trying to factor this so that we can solve this equation. But instead of doing it that way, we might take this as a clue that maybe we can factor by grouping. Remember, factor by grouping, you split up that middle degree term and see if you can reverse the distributive property twice. So, let's see if we can do that. Can we group together these first two terms and factor something interesting out? And group together these second two terms and factor something interesting out? And then maybe we can factor something out after that. What am I talking about? Well, this is going to be the same thing as x times... Well, this one, actually let me write a big parenthesis, here, this one right over here is the same thing as... I can factor out an x-squared. So, it's gonna be x-squared plus... Sorry. It's gonna be x-squared, if I factor out an x-squared, I'm gonna get an x-squared plus nine. And then over here, if I factor out a, let's see, negative two. I don't want to... If I factor out a, yep, negative two, I'm gonna get, so minus two times... I'm gonna get an x-squared plus nine, again. Now this is interesting, because this is telling us maybe we can factor out an x-squared plus nine. So, let me factor out an x-squared plus nine from each of these terms, and I'm going to get, I am going to get x... I'll leave these big green parentheses here for now, If we factor out an x-squared plus nine, it's going to be x-squared plus nine times x-squared, x-squared minus two. X-squared minus two, and I gave myself a little bit too much space. So, let me delete that. So let me delete that right over there and then close the parentheses. Then close the parentheses. Actually, I can even get rid of those green parentheses now, if I want to, optimally, make this a little bit simpler. So far we've been able to factor it as x times x-squared plus nine times x-squared minus two. And the whole point that I'm factoring this is if I can find the product of a bunch of expressions equaling zero, then I can say, "Well, the product of those expressions "are going to be zero if one or more of those expressions "are equal to zero", and I can solve for x. Well, let's see. This one's completely factored. This one is completely factored if we're thinking about real roots. This one, you can view it as a difference of squares if you view two as a square root of two-squared. So, we can rewrite this as, and of course all of this is equal to zero. Let me just write equals. So we could write this as equal to x times times x-squared plus nine times... Let's see, I can factor this business into x plus the square root of two times x minus the square root of two. I'm just recognizing this as a difference of squares. And, once again, we just want to solve this whole, all of this business, equaling zero. All of this equaling zero. So how can this equal to zero? Well any one of these expressions, if I take the product, and if any one of them equals zero then I'm gonna get zero. So, x could be equal to zero. X could be equal to zero, and that actually gives us a root. When x is equal to zero, this polynomial is equal to zero, and that's pretty easy to verify. Let's see, can x-squared plus nine equal zero? X-squared plus nine equal zero. Well, if you subtract nine from both sides, you get x-squared is equal to negative nine. And that's why I said, there's no real solution to this. So, no real, let me write that, no real solution. There are some imaginary solutions, but no real solutions. Now, can x plus the square root of two equal zero? X plus the square root of two equal zero. Sure, if we subtract square root of two from both sides, you get x is equal to the negative square root of two. And can x minus the square root of two equal zero? Sure, you add square root of two to both sides, you get x is equal to the square root of two. So, there we have it. We have figured out our zeros. X could be equal to zero. P of zero is zero. P of negative square root of two is zero, and p of square root of two is equal to zero. So, those are our zeros. Their zeros are at zero, negative squares of two, and positive squares of two. And so those are going to be the three times that we intercept the x-axis. And what is the smallest of those intercepts? Well, the smallest number here is negative square root, negative square root of two. And you could tackle it the other way. You could take this part right over... Which part? Yeah, this part right over here and you could add those two middle terms, and then factor in a non-grouping way, and I encourage you to do that. But just to see that this makes sense that zeros really are the x-intercepts. I went to Wolfram|Alpha and I graphed this polynomial and this is what I got. So, this is what I got, right over here. If you see a fifth-degree polynomial, say, it'll have as many as five real zeros. But, if it has some imaginary zeros, it won't have five real zeros. Instead, this one has three. And that's because the imaginary zeros, which we'll talk more about in the future, they come in these conjugate pairs. So, if you don't have five real roots, the next possibility is that you're going to have three real roots. And, if you don't have three real roots, the next possibility is you're gonna have one real root. So, that's an interesting thing to think about. And so, here you see, your three real roots. You see your three real roots which correspond to the x-values at which the function is equal to zero, which is where we have our x-intercepts.