Chapter 10

Discuss the possibilities for the roots to each equation. Do not solve the equation. $$x^{3}+x-1=0$$

Find all asymptotes, \(x\) -intercepts, and \(y\) -intercepts for the graph of each rational function and sketch the graph of the function. $$f(x)=\frac{x+1}{x^{2}}$$

Discuss the possibilities for the roots to each equation. Do not solve the equation. $$-x^{4}+3 x^{3}+5 x+5=0$$

Find all asymptotes, \(x\) -intercepts, and \(y\) -intercepts for the graph of each rational function and sketch the graph of the function. $$f(x)=\frac{x-1}{x^{2}}$$

Discuss the possibilities for the roots to each equation. Do not solve the equation. $$x^{5}+x^{3}+3 x=0$$

Find all asymptotes, \(x\) -intercepts, and \(y\) -intercepts for the graph of each rational function and sketch the graph of the function. $$f(x)=\frac{2 x-1}{x^{3}-9 x}$$

Discuss the possibilities for the roots to each equation. Do not solve the equation. $$x^{3}-5 x^{2}+6 x=0$$

Find all asymptotes, \(x\) -intercepts, and \(y\) -intercepts for the graph of each rational function and sketch the graph of the function. $$f(x)=\frac{2 x^{2}+1}{x^{3}-x}$$

Establish the best integral bounds for the roots of each equation according to the theorem on bounds. $$x^{4}-5 x^{2}+7=0$$

Find all asymptotes, \(x\) -intercepts, and \(y\) -intercepts for the graph of each rational function and sketch the graph of the function. $$f(x)=\frac{x}{x^{2}-1}$$

Establish the best integral bounds for the roots of each equation according to the theorem on bounds. $$2 x^{3}-x^{2}-7 x+7=0$$

Find all asymptotes, \(x\) -intercepts, and \(y\) -intercepts for the graph of each rational function and sketch the graph of the function. $$f(x)=\frac{x}{x^{2}+x-2}$$

Find all asymptotes, \(x\) -intercepts, and \(y\) -intercepts for the graph of each rational function and sketch the graph of the function. $$f(x)=\frac{2}{x^{2}+1}$$

Establish the best integral bounds for the roots of each equation according to the theorem on bounds. $$x^{2}-7 x-16=0$$

Find all asymptotes, \(x\) -intercepts, and \(y\) -intercepts for the graph of each rational function and sketch the graph of the function. $$f(x)=\frac{x}{x^{2}+1}$$

Establish the best integral bounds for the roots of each equation according to the theorem on bounds. $$x^{2}+x-13=0$$

Find all asymptotes, \(x\) -intercepts, and \(y\) -intercepts for the graph of each rational function and sketch the graph of the function. $$f(x)=\frac{x^{2}}{x+1}$$

Find all asymptotes, \(x\) -intercepts, and \(y\) -intercepts for the graph of each rational function and sketch the graph of the function. $$f(x)=\frac{x^{2}}{x-1}$$

Solve each problem. Oscillating modulators. The number of oscillating modulators produced by a factory in \(t\) hours is given by the polynomial function \(n(t)=t^{2}+6 t\) for \(t \approx 1 .\) The cost in dollars of operating the factory for \(t\) hours is given by the function \(c(t)=36 t+500\) for \(t \geq 1 .\) The average cost per modulator is given by the rational function \(f(t)=\frac{36 t+500}{t^{2}+6 t}\) for \(t=1 .\) Graph the function \(f .\) What is the average cost per modulator at time \(t=20\) and time \(t=30 ?\) What can you conclude about the average cost per modulator after a long period of time?

We can find the zeros of a polynomial function by solving a polynomial equation. We can also work backward to find a polynomial function that has given zeros. a) Write a first-degree polynomial function whose zero is \(-2\) b) Write a second-degree polynomial function whose zeros are 5 and \(-5\) c) Write a third-degree polynomial function whose zeros are \(1,-3,\) and 4 d) Is there a polynomial function with any given number of zeros? What is its degree?

For each equation find the value of \(k\) given that 3 satisfies the equation. a) \(x^{4}-3 x^{3}+5 x^{2}-7 x+k=0\) b) \(x^{4}-x^{3}-2 x^{2}+k x-k=0\) c) \(5 x^{3}-k x^{2}-k x-3 k=0\)

Use the rational root theorem, Descartes' rule of signs, and the theorem on bounds as aids in finding all solutions to each equation. $$x^{3}+x+10=0$$

Sketch the graph of each polynomial function. First graph the function on a calculator and use the calculator graph as a guide. $$f(x)=x-20$$

Solve each problem. Average cost of an SUV. Mercedes-Benz spent \(\$ 700\) million to design its new SUV (Motor Trend, www.motortrend.com). If it costs \(\$ 25,000\) to manufacture each SUV, then the average cost per vehicle in dollars when \(x\) vehicles are manufactured is given by the rational function $$A(x)=\frac{25,000 x+700,000,000}{x}$$ a) What is the horizontal asymptote for the graph of this function? b) What is the average cost per vehicle when \(50,000\) vehicles are made? c) For what number of vehicles is the average cost \(\$ 30,000 ?\) d) Graph this function for \(x\) ranging from 0 to \(100,000\).

Use the rational root theorem, Descartes' rule of signs, and the theorem on bounds as aids in finding all solutions to each equation. $$x^{3}-7 x^{2}+17 x-15=0$$

Sketch the graph of each polynomial function. First graph the function on a calculator and use the calculator graph as a guide. $$f(x)=(x-20)^{2}$$

Solve each problem. Average cost of a pill. Assuming Pfizer spent a typical \(\$ 350\) million to develop its latest miracle drug and \(\$ 0.10\) each to make the pills, then the average cost per pill in dollars when \(x\) pills are made is given by the rational function $$A(x)=\frac{0.10 x+350,000,000}{x}$$ a) What is the horizontal asymptote for the graph of this function? b) What is the average cost per pill when 100 million pills are made? c) For what number of pills is the average cost per pill \(\$ 2 ?\) d) Graph this function for \(x\) ranging from 0 to 150 million.

With a graphing calculator an equation can be solved without the kind of hint that was given for Exercises 43–52. Solve each of the following equations by examining the graph of a corresponding function. Use synthetic division to check. a) \(x^{3}-4 x^{2}-7 x+10=0\) b) \(8 x^{3}-20 x^{2}-18 x+45=0\)

Use the rational root theorem, Descartes' rule of signs, and the theorem on bounds as aids in finding all solutions to each equation. $$2 x^{3}-5 x^{2}-6 x+4=0$$

Sketch the graph of each polynomial function. First graph the function on a calculator and use the calculator graph as a guide. $$f(x)=(x-20)^{2}(x+30)$$

Use the rational root theorem, Descartes' rule of signs, and the theorem on bounds as aids in finding all solutions to each equation. $$3 x^{3}-17 x^{2}+12 x+6=0$$

Sketch the graph of each polynomial function. First graph the function on a calculator and use the calculator graph as a guide. $$f(x)=(x-20)^{2}(x+30)^{2}$$

Use the rational root theorem, Descartes' rule of signs, and the theorem on bounds as aids in finding all solutions to each equation. $$4 x^{3}-6 x^{2}-2 x+1=0$$

Use the rational root theorem, Descartes' rule of signs, and the theorem on bounds as aids in finding all solutions to each equation. $$x^{3}+5 x^{2}-20 x-42=0$$

Sketch the graph of each polynomial function. First graph the function on a calculator and use the calculator graph as a guide. $$f(x)=(x-20)^{2}(x+30)^{2} x^{2}$$

Use the rational root theorem, Descartes' rule of signs, and the theorem on bounds as aids in finding all solutions to each equation. $$x^{4}-5 x^{3}+5 x^{2}+5 x-6=0$$

In each case, find a polynomial function whose graph behaves in the required manner. Answers may vary. The graph has only one \(x\) -intercept at \((3,0)\) and crosses the \(x\) -axis there.

Sketch the graph of each pair of functions in the same coordinate system. What do you observe in each case? $$f(x)=\sqrt{x}, g(x)=\sqrt{x}+1 / x$$

Use the rational root theorem, Descartes' rule of signs, and the theorem on bounds as aids in finding all solutions to each equation. $$x^{4}-2 x^{3}+5 x^{2}-8 x+4=0$$

Sketch the graph of each pair of functions in the same coordinate system. What do you observe in each case? $$f(x)=x^{3}, g(x)=x^{3}+1 / x^{2}$$

Use the rational root theorem, Descartes' rule of signs, and the theorem on bounds as aids in finding all solutions to each equation. $$x^{4}-7 x^{3}+17 x^{2}-17 x+6=0$$

In each case, find a polynomial function whose graph behaves in the required manner. Answers may vary. The graph has only two \(x\) -intercepts at \((-2,0)\) and \((1,0) .\) It crosses the \(x\) -axis at \((-2,0)\) but does not cross at \((1,0)\).

In each case find a rational finction whose graph has the required asymptotes. Answers may vary. The graph has the \(x\) -axis as a horizontal asymptote and the y-axis as a vertical asymptote.

Solve each problem The total profit in dollars on the sale of \(x\) Electronic Tummy Trimmers is given by the polynomial function \(P(x)=x^{3}-40 x^{2}+400 x .\) Find the profit when 10 are sold. How many must be sold to get a profit of 0 dollars?

Use the rational root theorem, Descartes' rule of signs, and the theorem on bounds as aids in finding all solutions to each equation. $$x^{4}+7 x^{3}+17 x^{2}+17 x+6=0$$

In each case, find a polynomial function whose graph behaves in the required manner. Answers may vary. The graph has only two \(x\) -intercepts at \((5,0)\) and \((-6,0)\) It does not cross the \(x\) -axis at either \(x\) -intercept.

In each case find a rational finction whose graph has the required asymptotes. Answers may vary. The graph has the \(x\) -axis as a horizontal asymptote and the line \(x=2\) as a vertical asymptote.

The velocity in feet per second (ft'sec) of a rocket \(t\) seconds (sec) after launching is given by the polynomial function \(v(t)=t^{3}-20 t^{2}+110 t .\) What is the velocity of the rocket 10 see after launching? For what value of \(t\) does the rocket have 0 velocity? (GRAPH CANT COPY)

Use the rational root theorem, Descartes' rule of signs, and the theorem on bounds as aids in finding all solutions to each equation. $$x^{6}-x^{5}+2 x^{4}-2 x^{3}-15 x^{2}+15 x=0$$

In each case find a rational finction whose graph has the required asymptotes. Answers may vary. The graph has the \(x\) -axis as a horizontal asymptote and lines \(x=3\) and \(x=-1\) as vertical asymptotes.