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CCSS.Math: ,

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 alright 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 0 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 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 one 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 and because it's in factored form each of the parts of the product will probably make our polynomial 0 for one of these zeros so let's see if in order for our polynomial to be equal to 0 when X is equal to negative 4 we probably want to have a term that has an X plus 4 in it or we want to have it I should say a product that has an X plus 4 in it because X plus 4 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 3 halves is equal to 0 so I would look for something like X minus 3 halves in our product I don't see an X minus 3 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 our 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 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 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 three halves two X minus three is equal to zero which makes the entire product equal to 0 and when exit - and when an X is equal to 3 it makes X minus 3 equals to 0 0 times something times something is going to be equal to 0