The population of a certain insect varies dramatically with the weather, with spring-like temperatures causing a population boom and extreme weather (summer heat and winter cold) adversely affecting the population. This phenomena can be modeled by the polynomial p(m) = -m4 + 26m3 – 217m2 + 588m, where p(m) represents the number of live insects (in hundreds of thousands) in month m (m = 1nu→ Jan).
(a). Use the remainder theorem to find the population of insects during the cool of spring (March) and the fair weather of fall (October).
(b). Use the rational zeroes theorem to find the times when the population of insects becomes dormant [p(m) = 0]. Use this information to graph p(m), then use the graph to estimate the maximum and minimum population of insects, and the month at which each occurs.
