 In this video, I want to talk about solving polynomial inequalities. How does one solve an inequality involves a polynomial of some kind? Now, whether you're solving an inequality involving polynomials or rational functions or any function type whatsoever, remember that the graph can be a very powerful tool because when you look at things like, I need this polynomial to be greater than zero, I really want you to think of it the following way. This right here is y equals some polynomial, it's a function. And so with that perspective, our inequality really is solved inequality, y is greater than zero. Well, if you're looking for points for which y is greater than zero, you're looking for those things which are above, excuse me, above the x-axis. We want to look for those points on the graph that are above the x-axis. On the other hand, if you had like y is less than zero, you're looking for things which are below the x-axis. So we can actually think of the graph as a very powerful tool to help us solve these inequalities. So this is going to be our plan of attack. When you have a polynomial inequality, you're going to first solve the equation. So take x plus three times x minus one squared equals zero. We want to solve the equation, that's already factored, so that's great for us. We can see that the x-intercepts very quickly are going to be negative three and one. Next, once we have it solved, we have these markers. Now these markers are not the solutions to the inequality, but they do help us with the graph substantially. So when we start marking things up here, we're going to have our x-axis. I'm not even going to even bother with the y-axis. You can have it if it helps you, but we don't really need it so much. What we're going to do is we're going to plot our points on our x-axis. So we'll have our point, we'll call this one one, we have another point negative three. And again, if you feel so naked without your y-axis here, put it on the graph here, but we're not going to need it. So basically, we're trying to graph right now the function. We want to graph f of x equals x plus three times x minus one squared. So we have our x-intercepts. We can also think about the in behavior. This function as x goes to infinity is approximately an x cubed as x approaches plus or minus infinity. The in behavior, this thing is going to be approximately an x cubed, which means it's going to point up on the right and it's going to point down on the left. That's what we get from the in behavior. We also know about its intercepts, right, that negative three had an odd multiplicity. So that means it's going to cross the x-axis, but at one, we have an even multiplicity. So we know we're going to touch the x-axis. So our picture has to look something like the following. We come up and cross, we come back down and touch, and then we go off towards infinity. Okay, so we get the graph of our function, or the graph of our function f of x. Now remember, from what we're trying to say earlier, we're trying to graph, we're trying to solve the inequality f of x is greater than zero. The greater than zero in this situation means that we're looking for things which are above the x-axis. Which x coordinates, excuse me, which, which y coordinates are above the x-axis. So what we see here is that we're going to be above the x-axis when we're to the right of negative three, coming along to the x-intercept one there. And then after one, we're above the x-axis forever afterwards. And so what that tells us about our solution is that our solution is going to be negative three to one. I'm writing this, of course, in interval notation. All the points between negative three to one is above the x-axis union one to infinity. That portion is also above the x-axis. Now notice here that we're looking for things above the x-axis. The x-intercepts are not included in the solution here because they're not above the x-axis. They're on the x-axis. Their y-coordinate is zero. Zero is not greater than or equal to zero. Now, of course, if we had modified this problem, if we look for the points which are greater than or equal to zero, then now it tell us that negative three is actually part of the solution. And one would also be part of the solution. In fact, the solution would change to be negative three to infinity. We never actually include infinity there. So if we want to greater than or equal to zero, we get negative three to infinity. But sticking with the original problem, if we want to solve the polynomials greater than zero, we get negative three to one and one to infinity. Let's take another look at a different example this time. Let's take x to the fourth is greater than or equal to x. So as this is a polynomial inequality, my recommendations to first solve the polynomial equation, which whether you do that or not, I want you to set one side equal to zero. So I kind of jumped a step there, right? You have x to the fourth is greater than x. If you subtract x from both sides, that's where this thing came from. And again, if you want to do that with the inequality, that's a great idea too. So the x to the fourth minus x is greater than or equal to zero. It's critical here because we want some comparison to zero with our inequality whenever possible. Because this one right here tells you that we want to be above or in this case on the x-axis. Because if we're equal to zero, we're on the x-axis where x intercepts. So continuing to solve this thing, if you look at x to the fourth minus x, I would factor out the common divisor of x between them. So you get x times x cubed minus one equals zero. And then x cubed minus one, that's a difference of cubes. And so I'm going to use that difference of cubes factorization. You're going to get x minus one times x squared plus x plus one equals zero. Now we can find markers from this pretty quickly. So the first factor x, if x equals zero, that means x equals zero. If x minus one equals zero, that means x would equal one. Now if you try to solve this one right here, x squared plus x plus one, you're going to get a little stuck. And this happens when you do your difference of cubes factorization. The quadratic that factors out is irreducible. It doesn't have any real solutions there. And you can see that if you try to solve this, like, say, with the quadratic formula, you get x equals negative b plus or minus the square root of b squared minus four ac all over 2a. This gets you negative one plus or minus the square root of negative three over two. That's an imaginary number. This isn't real. And so when it comes to solving inequalities, we can't get an imaginary number or something to be greater than zero. We have to be working with real numbers in this situation. If I was solving a rational equation, I would accept a complex solution, a non-real solution. But for inequalities, our numbers do have to be real. Basically what we're seeing here is that because the discriminant of this quadratic was negative, we see that the picture, if we just graph this function y equals this quadratic, then we would see that the graph looks something like this. It's this concave up parabola. It's entirely above the x-axis. Notice here that it's always positive. What that affects here is the following. Whatever this number turns out to be, let's say that x times x minus one turns out to be positive. If you times that by a positive, it's positive positive. That's a double positive. It's going to be positive. On the other hand, if x times x minus one would say a negative number and you multiply that by a positive, well, then the net effect is going to be negative. So when you multiply something by a positive, it's always the sign never changes. So the sign, whether we're above or below the x-axis depends entirely on this portion of the function. Essentially, this quadratic portion really, it doesn't have any bearing. It doesn't have any effect. This is why we can get away with ignoring. We can ignore the non-real roots because they're not going to have any effect on the inequality in play here. So we're then going to start graphing this function and we have these two markers. We have the two markers. We're going to have x equals zero. So I'm going to put a point right here and we're going to have x equals one. Feel free to space them out as much as you want. So now our function here, our original function, so f of x equals x to the fourth minus x. Be aware that as x goes to infinity, this thing will be approximately the same thing as its leading term as x approaches plus or minus infinity. And so this function will essentially look like up on the right and up on the left and points up in that situation. What about our multiplicities? Well, we have an odd multiplicity. We have an odd multiplicity. That indicates that we're going to cross the x-axis in both situations. So if we put that information together, our graph is going to look something like the following. We're going to cross at zero. We have to turn around and come back and cross at one. Again, this graph does have some more nuance to it than what you see on the screen right now. But in terms of solving inequality, this is sufficient information for us. If we want to figure out where is this graph above the x-axis, we're going to take x equals zero and go to the left. All of those points are above the x-axis there. And we're also above the x-axis if we start at one and go to the right. So we're going to clip off the wings of our bird right here. And so then our solution set would look like negative infinity up to zero inclusive, because we do allow for x-intercepts on this one. Union, you're going to take then the interval from one to infinity, like so. And this then gives us the solution to our inequalities, our inequality here. And so this shows us how one can solve a polynomial inequality. If I have a polynomial inequality, I would compare one side to zero, probably the right hand side. Then we have a polynomial greater than or equal to less than or equal to less than greater than zero, something like that. Then graph the polynomial on the left hand side, focusing on those x-intercepts, right? And then once you graph it, then you can highlight very quickly who's above, who's below, depending on which direction inequality is going. And then you can find your solution from there. Basically, to solve a polynomial inequality, you should graph it. And you don't need a perfect computer animated graph. These type of simple things using x-intercepts, their multiplicities and in behavior, is sufficient to give us whether we're above or below the x-axis.