Zeros of polynomials: matching equation to graph | Polynomial graphs | Algebra 2 | Khan Academy

We are asked what could be the equation of p, and we have the graph of our polynomial p right over here. You could view this as the graph of y is equal to p of x. So pause this video and see if you can figure that out.

All right, now let's work on this together. You can see that all the choices have p of x in factored form, where it's very easy to identify the zeros or the x values that would make our polynomial equal to zero. We could also look at this graph and we can see what the zeros are. This is where we're going to intersect the x-axis, also known as the x-intercepts.

So you can see when x is equal to negative 4, we have a 0 because our polynomial is 0 there. So we know p of negative 4 is equal to 0. We also know that p of, it looks like 1 and a half, or I could say 3 halves, p of 3 halves is equal to 0. And we also know that p of 3 is equal to 0.

So let's look for an expression where that is true. Because it's in factored form, each of the parts of the product will probably make our polynomial zero for one of these zeros.

So let's see if, in order for our polynomial to be equal to zero when x is equal to negative four, we probably want to have a term that has an x plus four in it. Or we want to have, I should say, a product that has an x plus four in it because x plus four is equal to 0 when x is equal to negative 4. Well, we have an x plus 4 there, and we have an x plus 4 there. So I'm liking choices B and D so far.

Now for this second root, we have p of three halves is equal to zero. So I would look for something like x minus three halves in our product. I don't see an x minus three halves here, but as we've mentioned in other videos, you can also multiply these times constants.

So if I were to multiply, let's see, if I to get rid of this fraction here, if I multiply by 2, this would be the same thing as, let me scroll down a little bit, the same thing as 2x minus 3. And you could test that out; 2x minus 3 is equal to 0 when x is equal to 3 halves. And let's see, we have a 2x minus 3 right over there. So choice D is looking awfully good.

But let's just verify it with this last one. For p of 3 to be equal to 0, we could have an expression like x minus 3 in the product because this is equal to 0 when x is equal to 3. And we indeed have that right over there.

So choice D is looking very good. When x is equal to negative four, this part of our product is equal to zero, which makes the whole thing equal to zero. When x is equal to three halves, 2x minus three is equal to zero, which makes the entire product equal to zero. And when x is equal to three, it makes x minus three equal to zero. Zero times something times something is going to be equal to zero.