Get ready for Precalculus
- 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)
The zeros of a polynomial p(x) are all the x-values that make the polynomial equal to zero. They are interesting to us for many reasons, one of which is that they tell us about the x-intercepts of the polynomial's graph. We will also see that they are directly related to the factors of the polynomial.
- [Instructor] Let's say that we have a polynomial, p of x, and we can factor it, and we can put it in the form x minus one times x plus two times x minus three times x plus four. And what we are concerned with are the zeros of this polynomial, and you might say, "What is a zero of a polynomial?" Well, those are the x-values that are going to make the polynomial equal to zero. So another way to think about it is, for what x-values is p of x going to be equal to zero, or another way you can think about it is, for what x-values is this expression going to be equal to zero. So for what x-values is x minus one times x plus two times x minus three times x plus four, going to be equal to zero. I encourage you to pause this video, and think about that a little bit before we work through it together. Well the key realization here is if you have the product of a bunch of expressions, if any one of them is equal to zero, it doesn't matter what the others are, because zero times anything else is going to be equal to zero. So the fancy term for that is the zero product property. But all it says is, hey, if you can find an x-value that makes any one of these expressions equal to zero, well that's going to make the entire expression going to be, it's going to make the entire expression equal to zero. So, the zeros of this polynomial are gonna be the x-values that could make x minus one equal to zero. So x minus one equals zero. Well we know what x-value would make that happen, if x is equal to one, if you add one to both sides here, x equals one, so x equals one is a zero of this polynomial. Another way to say that is p of one when x equals one, that whole polynomial is going to be equal to zero. How do I know that? Well if I put a one in, right over here, this expression right over here, x minus one, that is going to be equal to zero. So you're gonna have zero times a bunch of other stuff which is going to be equal to zero. And so by the same idea, we can figure out what the other zeros are. What would make this part equal to zero? What x-value would make x plus two equal to zero? Well, x equals negative two, x equals negative two, would make x plus two equal zero. So x equals negative two is another zero of this polynomial. And we could keep going. What would make x minus three equal to zero? Well if x is equal to three, that would make x minus three equal to zero, and that would then make the entire expression equal to zero. And then last but not least, what would make x plus four equal to zero? Well if x is equal to negative four. And just like that we have found four zeros for this polynomial, when x equals one the polynomial's equal to zero, when x equals negative two the polynomial's equal to zero, when x equals three the polynomial's equal to zero, and when x equals negative four the polynomial's equal to zero. And one of the interesting things about the zeros of a polynomial you could actually use that to start to sketch out what the graph might look like. So, for example, we know, that this polynomial is going to take on the value zero at these zeros. So let me just draw a rough sketch right over here. So this is my x-axis, that's my y-axis, that's my y-axis. And so, let's see, at x equals one, so let me just do it this way, so we have one, two, three, and four. Then you have negative one, negative two, negative three, and then last but not least, negative four. We know that this polynomial, p of x, is going to be equal to zero at x equals one. So it's going to intersect the x-axis right there. It's going to be equal to zero at x equals negative two, so right over there. At x equals three, right over there. And x equals negative four. Now we don't know exactly what the graph looks like, just based on this. We could try out some values on either side to figure out, hey, is it above the x-axis, or below the x-axis, for x-values less than negative four. And we can try things out like that but we know it intersects the x-axis at these points. So it might look something like this, this is a very rough sketch. It might look something like this, we don't know without doing a little bit more work. But ahead of time, I took a look at what this looks like, I went on to Desmos, and I graphed it, and you can see, it looks exactly as what we would expect. The graph of this polynomial intersects the x-axis at x equals negative four, actually let me color code it, x equals negative four, and that is that zero right over there, x equals negative two, that's this zero right there, x equals one, right over there, and then x equals three, right over there. In future videos we will study this in even more depth.