Please help!!!
Ray and Kelsey have summer internships at an engineering firm. As part of their internship, they get to assist in the planning of a brand new roller coaster. For this assignment, you help Ray and Kelsey as they tackle the math behind some simple curves in the coaster's track.
Part A
The first part of Ray and Kelsey's roller coaster is a curved pattern that can be represented by a polynomial function.
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.
Yes. A polynomial can have, at most, a number of zeroes equal to it's degree. A third degree polynomial can therefore have three zeroes, however, Ray is arguing that the polynomial has four INTERCEPTS. While the terms "zeroes" and "intercepts" are often used synonymously, technically a zero is an x-intercept. A function can also have a y-intercept, so in total, a cubic can have four intercepts, though only three of them will be zeroes.
No, there is no way for both Ray and Kelsey to be correct. A polynomial function of degree n can have at most n real zeros or intercepts.
In this case, Ray says the third-degree polynomial has 4 intercepts, which means it can have at most 4 real zeros. On the other hand, Kelsey argues that the function can have as many as 3 zeros only. Since 3 is less than 4, they cannot both be correct.
A third-degree polynomial function can have either 3 or fewer real zeros, or it can have complex zeros as well. However, it cannot have more than 3 real zeros. So, in this scenario, Kelsey is correct that the function can have at most 3 zeros.
To determine if Ray and Kelsey can both be correct about the number of intercepts and zeros of a third-degree polynomial function, we first need to understand what these terms mean in the context of polynomial functions.
Intercepts: The intercepts of a polynomial function are the points where the graph of the function intersects the x-axis. These points correspond to the values of x for which the function evaluates to zero.
Zeros: The zeros of a polynomial function are the solutions to the equation f(x) = 0. In other words, they are the values of x that make the function equal to zero.
Now, let's consider the degree of a polynomial function. The degree of a polynomial refers to the highest exponent of its variable term(s). For example, a third-degree polynomial function has a highest exponent of 3.
Ray claims that the third-degree polynomial has 4 intercepts. This statement implies that the graph of the function intersects the x-axis at four distinct points. However, since a third-degree polynomial has a maximum of three possible roots or zeros, Ray's claim is not accurate. The number of zeros of a polynomial function is limited by its degree.
On the other hand, Kelsey argues that a third-degree polynomial can have as many as 3 zeros only. This statement is correct because the Fundamental Theorem of Algebra states that a polynomial function of degree n can have at most n distinct zeros.
To summarize, Ray is incorrect in claiming that a third-degree polynomial has 4 intercepts, as the maximum number of zeros for a third-degree polynomial is 3. Kelsey is correct in stating that a third-degree polynomial can have as many as 3 zeros. Therefore, Ray and Kelsey cannot both be correct in this context.