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Chapter 2: Problem 91
Perform the division by assuming that \(n\) is a positive integer. $$\frac{x^{3 n}+9 x^{2 n}+27 x^{n}+27}{x^{n}+3}$$
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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.
Find the rational zeros of the polynomial function. $$f(x)=x^{3}-\frac{3}{2} x^{2}-\frac{23}{2} x+6=\frac{1}{2}\left(2 x^{3}-3 x^{2}-23 x+12\right)$$
Determine whether the statement is true or false. Justify your answer. $$i^{44}+i^{150}-i^{74}-i^{109}+i^{61}=-1$$
The maximum safe load uniformly distributed over a one-foot section of a two- inch-wide wooden beam can be approximated by the model $$\text { Load }=168.5 d^{2}-472.1$$ where \(d\) is the depth of the beam. (a) Evaluate the model for \(d=4, d=6, d=8, d=10\) and \(d=12 .\) Use the results to create a bar graph. (b) Determine the minimum depth of the beam that will safely support a load of 2000 pounds.
Find a polynomial function with real coefficients that has the given zeros. (There are many correct answers.) $$2,5+i$$
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