- Zeros of polynomials introduction
- Zeros of polynomials: plotting zeros
- Zeros of polynomials: matching equation to zeros
- Zeros of polynomials: matching equation to graph
- Zeros of polynomials (factored form)
- Zeros of polynomials (with factoring): grouping
- Zeros of polynomials (with factoring): common factor
- Zeros of polynomials (with factoring)
When we are given a polynomial in factored form, we can quickly find the polynomial's zeros. Then, we can represent them as the x-intercepts of the polynomial's graph.
- [Instructor] 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. And they say plot all the zeros, or the x-intercepts, of the polynomial in the interactive graph. And so 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 gonna draw it in. So pause this video and see if you could 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. So 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 two x times two x plus three times x minus two, 'cause this is p of x, what x values would make this equal to zero? 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 equal zero, at least one of those things equal zero, make the whole product equal zero. So for example, if two x is equal to zero, it would make the whole thing zero, so two x could be equal to zero, and if two x is equal to zero, that means x is equal to zero, and you could try that out. If x is equal to zero, this part right over here is going to be equal zero. Doesn't matter what these other two things are. Zero times something times something is going to be equal to zero. And then you could say, well, well maybe two x plus three is equal to zero, so we could just write that. Two x plus three is equal to zero, and if that were true, what would x, or what would x have to be in order to make that true? Subtract three from both sides, two x would have to be equal to negative three, or x would be equal to negative 3/2. So this is another x value that would make the whole thing zero, 'cause if x is equal to negative 3/2, then two x plus three is equal to zero, you take a zero times whatever this is and whatever that is, you're gonna get zero. And then last but not least, x minus two could be equal to zero. That would make the whole product equal to zero. So what x value makes x minus two equal zero? We'll add two to both sides, and you would get x is equal to two. If x equals two, that equals zero, doesn't matter what these other two things are. Zero times something times something is going to be equal to zero. So just like that, we have the zeros of our polynomial, and 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 equal zero which is right over there. Once again, if you're 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 3/2, which is the same thing as negative 1/2, so that's right over there. And then, we have x equals two, 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.