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the polynomial P of X has four distinct zeros four distinct zeros which of the following graphs could represent y is equal to P of X well four distinct zeros means that the graph is going to intersect the x-axis exactly four times so this first one there's intersects the x-axis one two three times so we can rule that one out this one intersects the x-axis one two three four five times it's going to rule that one out this one intersects the x-axis one two three four times it has four distinct zeros so this looks like this one right over here it looks like our choice and this one right over here it actually only has two two distinct zeros or at least that's what it looks like so I would rule that one out as well so this is definitely going to be the one that has four distinct zeros zeros are the are the x values at which the function is equal or the polynomial in this case which is a function of X is going to be equal to zero and you can see that's happening exactly four times