Zeros of polynomials
- So we're given a p of x, it's a third degree polynomial, and they say, plot all the zeroes or the x-intercepts of the polynomial in the interactive graph. And the reason why they say interactive graph, this is a screen shot from the exercise on Kahn Academy, where you could click and place the zeroes. But the key here is, lets figure out what x values make p of x equal to zero, those are the zeroes. And then we can plot them. So pause this video, and see if you can figure that out. So the key here is to try to factor this expression right over here, this third degree expression, because really we're trying to solve the X's for which five x to third plus five x squared minus 30 x is equal to zero. And the way we do that is by factoring this left-hand expression. So the first thing I always look for is a common factor across all of the terms. It looks like all of the terms are divisible by five x. So let's factor out a five x. So this is going to be five x times, if we take a five x out of five x to the third, we're left with an x squared. If we take out a five x out of five x squared, we're left with an x, so plus x. And if we take out a five x of negative 30 x, we're left with a negative six is equal to zero. And now, we have five x times this second degree, the second degree expression and to factor that, let's see, what two numbers add up to one? You could use as a one x here. And their product is equal to negative six. And let's see, positive three and negative two would do the trick. So I can rewrite this as five x times, so x plus three, x plus three, times x minus two, and if what I did looks unfamiliar, I encourage you to review factoring quadratics on Kahn Academy, and that is all going to be equal to zero. And so if I try to figure out what x values are going to make this whole expression zero, it could be the x values or the x value that makes five x equal zero. Because if five x zero, zero times anything else is going to be zero. So what makes five x equal zero? Well if we divide five, if you divide both sides by five, you're going to get x is equal to zero. And it is the case. If x equals zero, this becomes zero, and then doesn't matter what these are, zero times anything is zero. The other possible x value that would make everything zero is the x value that makes x plus three equal to zero. Subtract three from both sides you get x is equal to negative three. And then the other x value is the x value that makes x minus two equal to zero. Add two to both sides, that's gonna be x equals two. So there you have it. We have identified three x values that make our polynomial equal to zero and those are going to be the zeros and the x intercepts. So we have one at x equals zero. We have one at x equals negative three. We have one at x equals, at x equals two. And the reason why it's, we're done now with this exercise, if you're doing this on Kahn Academy or just clicked in these three places, but the reason why folks find this to be useful is it helps us start to think about what the graph could be. Because the graph has to intercept the x axis at these points. So the graph might look something like that, it might look something like that. And to figure out what it actually does look like we'd probably want to try out a few more x values in between these x intercepts to get the general sense of the graph.