Final answer:
To build the polynomial, we use the form y = ax(x - b) with roots at x = 500 and x = 1000. The resulting equation is y = a(x - 500)(x - 1000), where 'a' is a positive constant that would be determined based on additional information, like a given y-intercept or particular coordinates on the graph.
Step-by-step explanation:
To build a polynomial with the given characteristics, we can start by noting the given intercepts and the behavior of the graph. The polynomial must have one root at x = 500 (this is an x-intercept), must rise to a maximum, cross the x-axis at another point, and then rise again through x = 1000.
The general form of the polynomial given is y = ax(x – b). To satisfy the condition of crossing the x-axis again, we must have another root; let's consider this root to be x = 1000. The equation thus becomes y = a(x-500)(x-1000).
The complete polynomial would then look like y = a(x – 500)(x – 1000), where a is a positive constant because it needs to open downwards (indicating a maximum) and then rise. You can determine the value of a and fine-tune the equation further if any other specific conditions are given. For example, if we want a coefficient such as the y-intercept is 50 or any particular value at a certain x-value, we could solve for a accordingly.