Zeros of polynomials (with factoring): common factor | Polynomial graphs | Algebra 2 | Khan Academy

So we're given a p of x; it's a third degree polynomial, and they say plot all the zeros or the x-intercepts of the polynomial in the interactive graph. The reason why they say interactive graph, this is a screenshot from the exercise on Khan Academy, where you could click and place the zeros. But the key here is let's figure out what x values make p of x equal to zero; those are the zeros, and then we can plot them.

So pause this video and see if you can figure that out. 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 5x to the third plus 5x squared minus 30x is equal to zero. 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 the terms are divisible by 5x, so let's factor out a 5x. This is going to be 5x times... if we take a 5x out of 5x to the third, we're left with an x squared. If we take out a 5x out of 5x squared, we're left with an x, so plus x. And if we take out a 5x of negative 30x, we're left with a negative 6, is equal to 0.

Now we have 5x times this second degree expression, and to factor that, let's see what two numbers add up to 1. You could view this as a 1x here, and their product is equal to negative 6. Let's see, positive 3 and negative 2 would do the trick. So I can rewrite this as 5x times (x + 3)(x - 2), and if what I just did looks unfamiliar, I encourage you to review factoring quadratics on Khan Academy, and that is all going to be equal to zero.

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 5x equal zero. Because if 5x is zero, zero times anything else is going to be zero. So what makes 5x equal zero? Well, if we divide both sides by 5, you're going to get x is equal to 0. And it is the case if x equals 0, this becomes 0, and then it doesn't matter what these are; 0 times anything is 0.

The other possible x value that would make everything 0 is the x value that makes x + 3 equal to 0. Subtract 3 from both sides; you get x is equal to negative 3. And then the other x value is the x value that makes x - 2 equal to 0. Add 2 to both sides; that's going to be x equals 2.

So there you have it; we have identified the 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 0, we have one at x equals negative 3, we have one at x equals 2. The reason why we're done now with this exercise, if you're doing some kind of category 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 a graph has to intersect the x-axis at these points.

So the graph might look something like that; it might look something like that. 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.