Final answer:
The polynomial function representing a negative odd function with zeros at -2 and 1, with a double root at 1, is f(x) = -k(x + 2)(x - 1)^2, where k > 0.
Step-by-step explanation:
To find the polynomial function representing a negative odd function with zeros at -2 and 1, and a double root at 1, we need to understand the properties of odd functions and how they relate to their zeros. An odd function satisfies the condition y(x) = −y(−x), meaning it is symmetric about the origin. If a function has a zero at x = a, then it must also have a zero at x = -a. A double root at 1 indicates that the zero at x = 1 has a multiplicity of two.
The simplest odd function with a zero at -2 is (x + 2), and since the function must also have a double root at 1, the corresponding factors are (x - 1)². However, to ensure the function is negative, we multiply by a negative constant. Hence, the polynomial function representing the conditions is f(x) = -k(x + 2)(x - 1)², where k > 0 to adhere to the negative requirement of the function.