ray and kelsey are working to graph a third-degree polynomial function that represents the first pattern in the coaster plan. ray says the third-degree polynomial has 4 intercepts. kelsey argues the function can have as many as 3 zeros only. is there a way for the both of them to be correct? explain your answer.

Thet contradict each other, that's why both of them are incorrect.
Suppose that a polynomial has four roots: s, t, u, and v. If the polynomial were evaluated at any of these values, it would have to be zero. Therefore, the polynomial can be written in this form. 

p(x)(x - s)(x - t)(x - u)(x - v), where p(x) is some non-zero polynomial 

This polynomial has a degree of at least 4. It therefore cannot be cubic. 

Now prove Kelsey correct. We have already proved that there can be no more than three roots. To prove that a cubic polynomial with three roots is possible, all we have to do is offer a single example of that. This one will do. 

(x - 1)(x - 2)(x - 3) 

This is a cubic polynomial with three roots, and four or more roots are not possible for a cubic polynomial. Kelsey is correct. 

Incidentally, if this is a roller coaster we are discussing, then a cubic polynomial is not such a good idea, either for a vertical curve or a horizontal curve. I hope this helps