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Chapter 2: Problem 28
Perform the operation and write the result in standard form. $$(8+\sqrt{-18})-(4+3 \sqrt{2} i)$$
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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.
A rectangular playing field with a perimeter of 100 meters is to have an area of at least 500 square meters. Within what bounds must the length of the rectangle lie?
Write the polynomial as the product of near factors and list all the zeros of the function. $$f(x)=x^{4}+29 x^{2}+100$$
For each function, identify the degree of the function and whether the degree of the function is even or odd. Identify the leading coefficient and whether the leading coefficient is positive or negative. Use a graphing utility to graph each function. Describe the relationship between the degree of the function and the sign of the leading coefficient of the function and the right-hand and left-hand behavior of the graph of the function. (a) \(f(x)=x^{3}-2 x^{2}-x+1\) (b) \(f(x)=2 x^{5}+2 x^{2}-5 x+1\) (c) \(f(x)=-2 x^{5}-x^{2}+5 x+3\) (d) \(f(x)=-x^{3}+5 x-2\) (e) \(f(x)=2 x^{2}+3 x-4\) (f) \(f(x)=x^{4}-3 x^{2}+2 x-1\) (g) \(f(x)=x^{2}+3 x+2\)
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. $$f(x)=2 x^{4}+5 x^{3}+4 x^{2}+5 x+2$$
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