Zeros of polynomials: plotting zeros | Polynomial graphs | Algebra 2 | Khan Academy

We're told we want to find the zeros of this polynomial, and they give us the polynomial right over here, and it's in factored form. They say plot all the zeros or the x-intercepts of the polynomial in the interactive graph. This is a screenshot from Khan Academy; if you're doing it on Khan Academy, you would click where the zeros are to plot the zeros. But I'm just going to draw it in, so pause this video and see if you can have a go at this before we work on this together.

All right, now let's work on this together. So the zeros are the x values that make our polynomial equal to zero. Another way to think about it is: for what x values are p of x equal to zero? Those would be the zeros. So essentially we have to say, "Hey, what x values would make 2x times (2x + 3) times (x - 2)"—because this is p of x—"what x values would make this equal to 0?"

Well, as we've talked about in previous videos, if you take the product of things and that equals zero, if any one of those things equals zero, at least one of those things equals zero would make the whole product equal zero. So, for example, if 2x is equal to zero, it would make the whole thing zero. So, 2x could be equal to 0, and if 2x is equal to 0, that means x is equal to 0. You could try that out; if x is equal to 0, this part right over here is going to be equal to 0. It doesn't matter what these other two things are; 0 times something times something is going to be equal to 0.

And then you could say, "Well, maybe 2x + 3 is equal to 0." So we could just write that: 2x + 3 is equal to 0. And if that were to be true, what would x—or what would x have to be—in order to make that true? Subtract 3 from both sides: 2x would have to be equal to negative 3, or x would be equal to negative 3 halves. So this is another x value that would make the whole thing zero because if x is equal to negative 3 halves, then 2x + 3 is equal to 0. You take a 0 times whatever this is and whatever that is; you're going to get 0.

And then, last but not least, x - 2 could be equal to 0. That would make the whole product equal to 0. So what x value makes x - 2 equal to 0? We'll add 2 to both sides and you would get x is equal to 2. If x equals 2, that equals 0. It doesn't matter what these other two things are; 0 times something times something is going to be equal to 0.

So just like that, we have the zeros of our polynomial. The reason why they have x-intercepts in parentheses here is that's where the graph of p of x—if you say y equals p of x—that's where it would intersect the x-axis and that's because that's where our polynomial is equal to zero.

So let's see, we have x equals 0, which is right over there. Once again, if you were doing this on Khan Academy, you would just click right over there and it would put a little dot there. We have x is equal to negative three halves, which is the same thing as negative one and a half, so that's right over there. And then we have x equals 2, which is right over there. So those are the x-intercepts or the zeros of that polynomial.

Now this is useful in life because you could use it to graph a function. I don't know exactly what this function looks like. Maybe it looks something like this; maybe it looks something like this. We would have to try out a few other values to get a sense of that, but we at least know where it's intersecting the x-axis. It's at the zeros.