Chapter 3

(a) find all zeros of the function, (b) write the polynomial as a product of linear factors, and (c) use your factorization to determine the \(x\) -intercepts of the graph of the function. Use a graphing utility to verify that the real zeros are the only \(x\) -intercepts. $$f(x)=x^{4}+25 x^{2}+144$$

Write the standard form of the quadratic function that has the indicated vertex and whose graph passes through the given point. Use a graphing utility to verify your result. Vertex: \(\left(\frac{1}{2}, 1\right)\) Point: \(\left(-2,-\frac{21}{5}\right)\)

(a) find the zeros algebraically, (b) use a graphing utility to graph the function, and (c) use the graph to approximate any zeros and compare them with those from part (a). \(f(x)=2 x^{4}-2 x^{2}-40\)

Find the zeros (if any) of the rational function. Use a graphing utility to verify your answer. $$f(x)=\frac{x^{2}+4 x-21}{x^{2}-4 x+3}$$

Use a graphing utility to graph the function. What do you observe about its asymptotes? $$f(x)=-\frac{x}{\sqrt{9+x^{2}}}$$

Use the Remainder Theorem and synthetic division to evaluate the function at each given value. Use a graphing utility to verify your results. \(g(x)=2 x^{6}+3 x^{4}-x^{2}+3\) (a) \(g(2)\) (b) \(g(1)\) (c) \(g(3)\) (d) \(g(-1)\)

(a) find all zeros of the function, (b) write the polynomial as a product of linear factors, and (c) use your factorization to determine the \(x\) -intercepts of the graph of the function. Use a graphing utility to verify that the real zeros are the only \(x\) -intercepts. $$f(x)=x^{4}-8 x^{3}+17 x^{2}-8 x+16$$

Write the standard form of the quadratic function that has the indicated vertex and whose graph passes through the given point. Use a graphing utility to verify your result. Vertex: \(\left(-\frac{1}{4},-1\right) ; \quad\) Point: \(\left(0,-\frac{17}{16}\right)\)

(a) find the zeros algebraically, (b) use a graphing utility to graph the function, and (c) use the graph to approximate any zeros and compare them with those from part (a). \(f(x)=5 x^{4}+15 x^{2}+10\)

The game commission introduces 100 deer into newly acquired state game lands. The population \(N\) of the herd is given by $$N=\frac{20(5+3 t)}{1+0.04 t}, \quad t \geq 0$$ where \(t\) is the time in years. (a) Use a graphing utility to graph the model. (b) Find the populations when \(t=5, t=10,\) and \(t=25.\) (c) What is the limiting size of the herd as time increases? Explain.

Use a graphing utility to graph the function. What do you observe about its asymptotes? $$g(x)=\frac{4|x-2|}{x+1}$$

Use the Remainder Theorem and synthetic division to evaluate the function at each given value. Use a graphing utility to verify your results. \(h(x)=x^{3}-5 x^{2}-7 x+4\) (a) \(h(3)\) (b) \(h(2)\) (c) \(h(-2)\) (d) \(h(-5)\)

Find a polynomial function with real coefficients that has the given zeros. (There are many correct answers.) $$5, i,-i$$

(a) find the zeros algebraically, (b) use a graphing utility to graph the function, and (c) use the graph to approximate any zeros and compare them with those from part (a). \(f(x)=x^{3}-4 x^{2}-25 x+100\)

The endpoints of the interval over which distinct vision is possible are called the near point and far point of the eye (see figure). With increasing age, these points normally change. The table shows the approximate near points \(y\) (in inches) for various ages \(x\) (in years). $$\begin{array}{|c|c|} \hline \text { Age, \(x\) } & \text { Near point, \(y\) } \\ \hline 16 & 3.0 \\ 32 & 4.7 \\ 44 & 9.8 \\ 50 & 19.7 \\ 60 & 39.7 \\ \hline \end{array}$$ (a) Find a rational model for the data. Take the reciprocals of the near points to generate the points \(\left(x, \frac{1}{y}\right).\) Use the regression feature of a graphing utility to find a linear model for the data. The resulting line has the form \(\frac{1}{y}=a x+b.\) Solve for \(y.\) (b) Use the table feature of the graphing utility to create a table showing the predicted near point based on the model for each of the ages in the original table. (c) Do you think the model can be used to predict the near point for a person who is 70 years old? Explain.

Use a graphing utility to graph the function. What do you observe about its asymptotes? $$f(x)=-\frac{8|3+x|}{x-2}$$

Use the Remainder Theorem and synthetic division to evaluate the function at each given value. Use a graphing utility to verify your results. \(f(x)=4 x^{4}-16 x^{3}+7 x^{2}+20\) (a) \(f(1)\) (b) \(f(-2)\) (c) \(f(5)\) (d) \(f(-10)\)

Find a polynomial function with real coefficients that has the given zeros. (There are many correct answers.) $$3,4 i,-4 i$$

(a) find the zeros algebraically, (b) use a graphing utility to graph the function, and (c) use the graph to approximate any zeros and compare them with those from part (a). \(y=4 x^{3}+4 x^{2}-7 x+2\)

Consider a physics laboratory experiment designed to determine an unknown mass. A flexible metal meter stick is clamped to a table with 50 centimeters overhanging the edge (see figure). Known masses \(M\) ranging from 200 grams to 2000 grams are attached to the end of the meter stick. For each mass, the meter stick is displaced vertically and then allowed to oscillate. The average time \(t\) (in seconds) of one oscillation for each mass is recorded in the table. $$\begin{array}{|c|c|} \hline \text { Mass, \(M\) } & \text { Time, \(t\) } \\ \hline 200 & 0.450 \\ 400 & 0.597 \\ 600 & 0.712 \\ 800 & 0.831 \\ 1000 & 0.906 \\ 1200 & 1.003 \\ 1400 & 1.088 \\ 1600 & 1.126 \\ 1800 & 1.218 \\ 2000 & 1.338 \\ \hline \end{array}$$ A model for the data is given by $$t=\frac{38 M+16,965}{10(M+5000)}$$ (a) Use the table feature of a graphing utility to create a table showing the estimated time based on the model for each of the masses shown in the table. What can you conclude? (b) Use the model to approximate the mass of an object when the average time for one oscillation is 1.056 seconds.

Use a graphing utility to graph the function. What do you observe about its asymptotes? $$f(x)=\frac{4(x-1)^{2}}{x^{2}-4 x+5}$$

Use synthetic division to show that \(x\) is a solution of the thirddegree polynomial equation, and use the result to factor the polynomial completely. List all the real solutions of the equation. Value of \(x\) \(x=4\) \(x=-6\) \(x=-\frac{3}{2}\) \(x=\frac{1}{3}\) \(x=\sqrt{3}\) \(x=2-\sqrt{5}\) Polynomial Equation $$x^{3}-13 x-12=0$$

(a) find the zeros algebraically, (b) use a graphing utility to graph the function, and (c) use the graph to approximate any zeros and compare them with those from part (a). \(y=4 x^{3}-20 x^{2}+25 x\)

The sales \(S\) (in thousands of units) of a tablet computer during the \(n\) th week after the tablet is released are given by $$S=\frac{150 n}{n^{2}+100}, \quad n \geq 0$$ (a) Use a graphing utility to graph the sales function. (b) Find the sales in week 5, week 10, and week 20. (c) According to this model, will sales ever drop to zero units? Explain.

Use a graphing utility to graph the function. What do you observe about its asymptotes? $$g(x)=\frac{3 x^{4}-5 x+3}{x^{4}+1}$$

Use synthetic division to show that \(x\) is a solution of the thirddegree polynomial equation, and use the result to factor the polynomial completely. List all the real solutions of the equation. Value of \(x\) \(x=4\) \(x=-6\) \(x=-\frac{3}{2}\) \(x=\frac{1}{3}\) \(x=\sqrt{3}\) \(x=2-\sqrt{5}\) Polynomial Equation $$x^{3}-31 x+30=0$$

(a) find the zeros algebraically, (b) use a graphing utility to graph the function, and (c) use the graph to approximate any zeros and compare them with those from part (a). \(y=x^{5}-5 x^{3}+4 x\)

The cost \(C\) (in dollars) of supplying recycling bins to \(p \%\) of the population of a rural township is given by $$C=\frac{25,000 p}{100-p}, \quad 0 \leq p<100$$ (a) Use a graphing utility to graph the cost function. (b) Find the costs of supplying bins to \(15 \%\) \(50 \%,\) and \(90 \%\) of the population. (c) According to the model, would it be possible to supply bins to \(100 \%\) of the population? Explain.

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, and slant asymptotes. $$f(x)=\frac{2 x^{2}+1}{x}$$

Use synthetic division to show that \(x\) is a solution of the thirddegree polynomial equation, and use the result to factor the polynomial completely. List all the real solutions of the equation. Value of \(x\) \(x=4\) \(x=-6\) \(x=-\frac{3}{2}\) \(x=\frac{1}{3}\) \(x=\sqrt{3}\) \(x=2-\sqrt{5}\) Polynomial Equation $$2 x^{3}-17 x^{2}+12 x+63=0$$

Use a graphing utility to graph the quadratic function and find the \(x\) -intercepts of the graph. Then find the \(x\) -intercepts algebraically to verify your answer. \(y=x^{2}-4 x\)

Find all the real zeros of the polynomial function. Determine the multiplicity of each zero. Use a graphing utility to verify your results. \(f(x)=x^{2}-25\)

Determine whether the statement is true or false. Justify your answer. A rational function can have infinitely many vertical asymptotes.

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, and slant asymptotes. $$g(x)=\frac{1-x^{2}}{x}$$

Use synthetic division to show that \(x\) is a solution of the thirddegree polynomial equation, and use the result to factor the polynomial completely. List all the real solutions of the equation. Value of \(x\) \(x=4\) \(x=-6\) \(x=-\frac{3}{2}\) \(x=\frac{1}{3}\) \(x=\sqrt{3}\) \(x=2-\sqrt{5}\) Polynomial Equation $$60 x^{3}-89 x^{2}+41 x-6=0$$

Use a graphing utility to graph the quadratic function and find the \(x\) -intercepts of the graph. Then find the \(x\) -intercepts algebraically to verify your answer. \(y=-2 x^{2}+10 x\)

Find all the real zeros of the polynomial function. Determine the multiplicity of each zero. Use a graphing utility to verify your results. \(f(x)=49-x^{2}\)

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, and slant asymptotes. $$h(x)=\frac{x^{2}}{x-1}$$

Use synthetic division to show that \(x\) is a solution of the thirddegree polynomial equation, and use the result to factor the polynomial completely. List all the real solutions of the equation. Value of \(x\) \(x=4\) \(x=-6\) \(x=-\frac{3}{2}\) \(x=\frac{1}{3}\) \(x=\sqrt{3}\) \(x=2-\sqrt{5}\) Polynomial Equation $$x^{3}+2 x^{2}-3 x-6=0$$

A polynomial function \(f\) with real coefficients has the given degree, zeros, and solution point. Write the function (a) in completely factored form and (b) in polynomial form. Degree 4 Zeros -1,2,\(i\) Solution Point \(f(1)=8\)

Use a graphing utility to graph the quadratic function and find the \(x\) -intercepts of the graph. Then find the \(x\) -intercepts algebraically to verify your answer. \(y=2 x^{2}-7 x-30 \quad\)

Find all the real zeros of the polynomial function. Determine the multiplicity of each zero. Use a graphing utility to verify your results. \(h(t)=t^{2}-6 t+9\)

Write a rational function \(f\) that has the specified characteristics. (There are many correct answers.) (a) Vertical asymptote: \(x=2\) Horizontal asymptote: \(y=0\) Zero: \(x=1\) (b) Vertical asymptote: \(x=-1\) Horizontal asymptote: \(y=0\) Zero: \(x=2\) (c) Vertical asymptotes: \(x=-2, x=1\) Horizontal asymptote: \(y=2\) Zeros: \(x=3, x=-3\) (d) Vertical asymptotes: \(x=-1, x=2\) Horizontal asymptote: \(y=-2\) Zeros: \(x=-2, x=3\) (c) Vertical asymptotes: \(x=0, x=\pm 3\) Horizontal asymptote: \(y=3\) Zeros: \(x=-1, x=1, x=2\)

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, and slant asymptotes. $$f(x)=\frac{x^{3}}{x^{2}-1}$$

Use synthetic division to show that \(x\) is a solution of the thirddegree polynomial equation, and use the result to factor the polynomial completely. List all the real solutions of the equation. Value of \(x\) \(x=4\) \(x=-6\) \(x=-\frac{3}{2}\) \(x=\frac{1}{3}\) \(x=\sqrt{3}\) \(x=2-\sqrt{5}\) Polynomial Equation $$x^{3}-x^{2}-13 x-3=0$$

A polynomial function \(f\) with real coefficients has the given degree, zeros, and solution point. Write the function (a) in completely factored form and (b) in polynomial form. Degree 4 Zeros $$1,-4, \sqrt{3} i$$ Solution Point $$f(0)=-6$$

Use a graphing utility to graph the quadratic function and find the \(x\) -intercepts of the graph. Then find the \(x\) -intercepts algebraically to verify your answer. \(y=-\frac{1}{2}\left(x^{2}-6 x-7\right) \quad\)

Find all the real zeros of the polynomial function. Determine the multiplicity of each zero. Use a graphing utility to verify your results. \(f(x)=x^{2}+10 x+25\)

A real zero of the numerator of a rational function \(f\) is \(x=c .\) Must \(x=c\) also be a zero of \(f ?\) Explain.

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, and slant asymptotes. $$g(x)=\frac{x^{3}}{2 x^{2}-8}$$