Zeros of polynomials
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- [Instructor] We are asked, what could be the equation of p? And we have 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, and 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. And 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 four, we have a zero because our polynomial is zero there. So we know p of negative four is equal to zero. We also know that p of, looks like 1 1/2, or I could say 3/2. p of 3/2 is equal to zero, and we also know that p of three is equal to zero. So let's look for an expression where that is true. And because it's in factored form, each of the parts of the product will probably make our polynomial zero for one of these zeroes. So let's see if, 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 a, I should say, a product that has an x plus four in it. Because x plus four is equal to zero when x is equal to negative four. Well we have an x plus four there, and we have an x plus four there. So I'm liking choices B and D so far. Now for this second root, we have p of 3/2 is equal to zero so I would look for something like x minus 3/2 in our product. I don't see an x minus 3/2 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 to get rid of this fraction here, if I multiply by two this would be the same thing as, let me scroll down a little bit, same thing as two x minus three. And you could test that out, two x minus three is equal to zero when x is equal to 3/2. And let's see, we have a two x minus three right over there. So choice D is looking awfully good, but let's just verify it with this last one. For p of three to be equal to zero, we could have an expression like x minus three in the product because this is equal to zero when x is equal to three, 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 3/2, two x minus three is equal to zero which makes the entire product equal to zero. And when x minus, and when an 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.