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Chapter 2: Problem 105
Determine whether the statement is true or false. Justify your answer. A fifth-degree polynomial function can have five turning points in its graph.
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Find all real zeros of the function. $$f(z)=12 z^{3}-4 z^{2}-27 z+9$$
The mean salaries \(S\) (in thousands of dollars) of public school classroom teachers in the United States from 2000 through 2011 are shown in the table. $$\begin{array}{|c|c|}\hline \text { Year } & \text { Salary, \(S\) } \\\\\hline 2000 & 42.2 \\\2001 & 43.7 \\\2002 & 43.8 \\\2003 & 45.0 \\\2004 & 45.6 \\\2005 & 45.9 \\\2006 & 48.2 \\\2007 & 49.3 \\\2008 & 51.3 \\\2009 & 52.9 \\\2010 & 54.4 \\\2011 & 54.2 \\\\\hline\end{array}$$ A model that approximates these data is given by $$S=\frac{42.16-0.236 t}{1-0.026 t}, \quad 0 \leq t \leq 11$$ where \(t\) represents the year, with \(t=0\) corresponding to 2000. (a) Use a graphing utility to create a scatter plot of the data. Then graph the model in the same viewing window. (b) How well does the model fit the data? Explain. (c) Use the model to predict when the salary for classroom teachers will exceed \(\$ 60,000\). (d) Is the model valid for long-term predictions of classroom teacher salaries? Explain.
Match the cubic function with the numbers of rational and irrational zeros. (a) Rational zeros: \(0 ;\) irrational zeros: 1 (b) Rational zeros: \(3 ;\) irrational zeros: 0 (c) Rational zeros: \(1 ;\) irrational zeros: 2 (d) Rational zeros: \(1 ;\) irrational zeros: 0 $$f(x)=x^{3}-1$$
Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. $$f(x)=2 x^{4}+5 x^{3}+4 x^{2}+5 x+2$$
Use the position equation $$s=-16 t^{2}+v_{0} t+s_{0}$$ where \(s\) represents the height of an object (in feet), \(v_{0}\) represents the initial velocity of the object (in feet per second), \(s_{0}\) represents the initial height of the object (in feet), and \(t\) represents the time (in seconds). A projectile is fired straight upward from ground level \(\left(s_{0}=0\right)\) with an initial velocity of 160 feet per second. (a) At what instant will it be back at ground level? (b) When will the height exceed 384 feet?
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