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Magazine Problem 58
Sketch the graph of each polynomial function. First graph the function on a calculator and use the calculator graph as a guide. $$f(x)=(x-20)^{2}(x+30)^{2}$$
Problem 58
Use the rational root theorem, Descartes' rule of signs, and the theorem on bounds as aids in finding all solutions to each equation. $$3 x^{3}-17 x^{2}+12 x+6=0$$
Problem 59
Use the rational root theorem, Descartes' rule of signs, and the theorem on bounds as aids in finding all solutions to each equation. $$4 x^{3}-6 x^{2}-2 x+1=0$$
Problem 60
Use the rational root theorem, Descartes' rule of signs, and the theorem on bounds as aids in finding all solutions to each equation. $$x^{3}+5 x^{2}-20 x-42=0$$
Problem 60
Sketch the graph of each polynomial function. First graph the function on a calculator and use the calculator graph as a guide. $$f(x)=(x-20)^{2}(x+30)^{2} x^{2}$$
Problem 61
Sketch the graph of each pair of functions in the same coordinate system. What do you observe in each case? $$f(x)=\sqrt{x}, g(x)=\sqrt{x}+1 / x$$
Problem 61
Use the rational root theorem, Descartes' rule of signs, and the theorem on bounds as aids in finding all solutions to each equation. $$x^{4}-5 x^{3}+5 x^{2}+5 x-6=0$$
Problem 61
In each case, find a polynomial function whose graph behaves in the required manner. Answers may vary. The graph has only one \(x\) -intercept at \((3,0)\) and crosses the \(x\) -axis there.
Problem 62
Use the rational root theorem, Descartes' rule of signs, and the theorem on bounds as aids in finding all solutions to each equation. $$x^{4}-2 x^{3}+5 x^{2}-8 x+4=0$$
Problem 62
Sketch the graph of each pair of functions in the same coordinate system. What do you observe in each case? $$f(x)=x^{3}, g(x)=x^{3}+1 / x^{2}$$
