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22 February, 04:16

Anonymous a year ago

1. 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

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  1. 22 February, 04:21
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    Of course there is no way for them to be both correct, since they contradict each other. Here is how to prove Ray 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.
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