Zeros of polynomials introduction | Polynomial graphs | Algebra 2 | Khan Academy

Let's say that we have a polynomial ( p ) of ( x ) and we can factor it. We can put it in the form ( (x - 1)(x + 2)(x - 3)(x + 4) ). What we are concerned with are the zeros of this polynomial. 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.

Another way to think about it is: for what ( x ) values is ( p(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 - 1)(x + 2)(x - 3)(x + 4) ) going to be equal to zero? I encourage you to pause this video, think about that a little bit before we work through it together.

Well, the key realization here is that 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. 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 equal to zero."

So the zeros of this polynomial are going to be the ( x ) values that could make ( x - 1 = 0 ). So if ( x - 1 = 0 ), we know what ( x ) value would make that happen: if ( x = 1 ), if you add 1 to both sides here, ( x = 1 ). So ( x = 1 ) is a zero of this polynomial. Another way to say that is ( p(1) ) when ( x = 1 ), that whole polynomial is going to be equal to zero.

How do I know that? Well, if I put a 1 in right over here, this expression ( (x - 1) ) is going to be equal to 0. So you have 0 times a bunch of other stuff, which is going to be equal to 0. 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 + 2 = 0 ? Well, ( x = -2 ), ( x = -2 ) would make ( (x + 2) = 0 ). So, ( x = -2 ) is another zero of this polynomial.

We could keep going; what would make ( x - 3 = 0 ? Well, if ( x = 3 ), that would make ( (x - 3) = 0 ) and that would then make the entire expression equal to zero. And then last but not least, what would make ( x + 4 = 0 ? Well, if ( x = -4 ).

And just like that, we have found four zeros for this polynomial. When ( x = 1 ), the polynomial is equal to zero. When ( x = -2 ), the polynomial is equal to zero. When ( x = 3 ), the polynomial is equal to zero. And when ( x = -4 ), the polynomial is equal to zero.

One of the interesting things about the zeros of a polynomial is that 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. So, let's see at ( x = 1 ), so let me just do it this way. So we have 1, 2, 3, and 4, and then you have -1, -2, -3, and then last but not least -4. We know that this polynomial ( p(x) ) is going to be equal to zero at ( x = 1 ) so it's going to intersect the x-axis right there. It's going to be equal to zero at ( x = -2 ); so right over there at ( x = 3 ); right over there, and ( x = -4 ).

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 -4?" 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 = -4).

Actually, let me color-code it: ( x = -4 ) and that is that zero right over there. ( x = -2 ), that's this zero right there. ( x = 1 ) right over there, and then ( x = 3 ) right over there. In future videos, we will study this in even more depth.