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Chapter 3: Problem 42
Solve the equation \(3 x^{3}+7 x^{2}-22 x-8=0\) given that \(-\frac{1}{3}\) is a root.
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Use a graphing utility to obtain a complete graph for each polynomial function in Exercises \(58-61 .\) Then determine the number of real zeros and the number of nonreal complex zeros for each function. $$ f(x)=x^{6}-64 $$
In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=-x^{4}+4 x^{2}$$
In Exercises \(1-10\), determine which functions are polynomial functions. For those that are, identify the degree. $$f(x)=\frac{x^{2}+7}{x^{3}}$$
In Exercises \(78-79,\) use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. Then use the \(200 \mathrm{M} \text { OUT }]\) feature to show that \(f\) and g have identical end behavior. $$f(x)=x^{3}-6 x+1, \quad g(x)=x^{3}$$
Write the equation of a rational function \(f(x)=\frac{p(x)}{q(x)}\) having the indicated properties, in which the degrees of \(p\) and \(q\) are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. \(f\) has vertical asymptotes given by \(x=-2\) and \(x=2, a\) horizontal asymptote \(y=2, y\) -intercept at \(\frac{9}{2}, x\) -intercepts at \(-3\) and \(3,\) and \(y\) -axis symmetry.
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