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Find a polynomial function with the given real zeros whose graph contains the given point. Zeros: -4 (multiplicity 1), 0 (multiplicity 3), 2 (multiplicity 1); degree 5; contains the point (-2,64)

A rare species of insect was discovered in the Amazon Rain Forest. To protect the species, environmentalists declared the insect endangered and transplanted the insect into a protected area.The population \(P\) of the insect \(t\) months after being transplanted is $$ P(t)=\frac{50(1+0.5 t)}{2+0.01 t} $$ (a) How many insects were discovered? In other words, what was the population when \(t=0 ?\) (b) What will the population be after 5 years? (c) Determine the horizontal asymptote of \(P(t) .\) What is the largest population that the protected area can sustain?

Minimum cost A rectangular area adjacent to a river is to be fenced in; no fence is needed on the river side. The enclosed area is to be 1000 square feet. Fencing for the side parallel to the river is \(\$ 5\) per linear foot, and fencing for the other two sides is \(\$ 8\) per linear foot; the four corner posts are \(\$ 25\) apiece. Let \(x\) be the length of one of the sides perpendicular to the river. (a) Write a function \(C(x)\) that describes the cost of the project. (b) What is the domain of \(C ?\) (c) Use a graphing utility to graph \(C=C(x)\). (d) Find the dimensions of the cheapest enclosure.

Polynomial function: (a) List each real zero and its multiplicity. (b) Determine whether the graph crosses or touches the \(x\) -axis at each \(x\) -intercept. (c) Determine the maximum number of turning points on the graph. (d) Determine the end behavior; that is, find the power function that the graph of f resembles for large values of \(|x|\). $$ f(x)=7\left(x^{2}+4\right)^{2}(x-5)^{3} $$

Minimizing Surface Area United Parcel Service has contracted you to design an open box with a square base that has a volume of 5000 cubic inches. See the illustration. (a) Express the surface area \(S\) of the box as a function of \(x\). (b) Using a graphing utility, graph the function found in part (a). (c) What is the minimum amount of cardboard that can be used to construct the box? (d) What are the dimensions of the box that minimize the surface area? (e) Why might UPS be interested in designing a box that minimizes the surface area?

Polynomial function: (a) List each real zero and its multiplicity. (b) Determine whether the graph crosses or touches the \(x\) -axis at each \(x\) -intercept. (c) Determine the maximum number of turning points on the graph. (d) Determine the end behavior; that is, find the power function that the graph of f resembles for large values of \(|x|\). $$ f(x)=(x-5)^{3}(x+4)^{2} $$

Determine where the graph of \(f\) is below the graph of g by solving the inequality \(f(x) \leq g(x) .\) Graph \(f\) and g together. \(f(x)=x^{4}-1\) \(g(x)=x-1\)

Polynomial function: (a) List each real zero and its multiplicity. (b) Determine whether the graph crosses or touches the \(x\) -axis at each \(x\) -intercept. (c) Determine the maximum number of turning points on the graph. (d) Determine the end behavior; that is, find the power function that the graph of f resembles for large values of \(|x|\). $$ f(x)=(x+\sqrt{3})^{2}(x-2)^{4} $$

Find bounds on the real zeros of each polynomial function. $$ f(x)=x^{4}-3 x^{2}-4 $$

Polynomial function: (a) List each real zero and its multiplicity. (b) Determine whether the graph crosses or touches the \(x\) -axis at each \(x\) -intercept. (c) Determine the maximum number of turning points on the graph. (d) Determine the end behavior; that is, find the power function that the graph of f resembles for large values of \(|x|\). $$ f(x)=4 x\left(x^{2}-3\right) $$