Chapter 3

Use the zero or root feature of a graphing utility to approximate (accurate to the nearest thousandth) the zeros of the function, (b) determine one of the exact zeros and use synthetic division to verify your result, and (c) factor the polynomial completely. $$g(x)=6 x^{4}-11 x^{3}-51 x^{2}+99 x-27$$

Use a graphing utility to graph the function and find its domain and range. $$f(x)=\sqrt{121-x^{2}}$$

Use a graphing utility to graph the function. Identify any symmetry with respect to the \(x\) -axis, \(y\) -axis, or origin. Determine the number of \(x\) -intercepts of the graph. \(g(x)=\frac{1}{8}(x+1)^{2}(x-3)^{3}\)

The table shows the numbers \(S\) of cellular phone subscriptions per 100 people in the United States from 1995 through 2012 . The data can be approximated by the model \(S=-0.0223 t^{3}+0.825 t^{2}-3.58 t+12.6\) \(5 \leq t \leq 22\) where \(t\) represents the year, with \(t=5\) corresponding to 1995 (a) Use a graphing utility to plot the data and graph the model in the same viewing window. (b) How well does the model fit the data? (c) Use the Remainder Theorem to evaluate the model for the year \(2020 .\) Is the value reasonable? Explain.

Use a graphing utility to graph the function and find its domain and range. $$f(x)=-|x+9|$$

Use a graphing utility to graph the function. Identify any symmetry with respect to the \(x\) -axis, \(y\) -axis, or origin. Determine the number of \(x\) -intercepts of the graph. \(f(x)=x^{3}-4 x\)

The numbers of employees \(E\) (in thousands) in education and health services in the United States from 1960 through 2013 are approximated by \(E=-0.088 t^{3}+10.77 t^{2}+14.6 t+3197\) \(0 \leq t \leq 53,\) where \(t\) is the year, with \(t=0\) corresponding to \(1960. (a) Use a graphing utility to graph the model over the domain. (b) Estimate the number of employees in education and health services in \)1960 .\( Use the Remainder Theorem to estimate the number in \)2010 .$ (c) Is this a good model for making predictions in future years? Explain.

Use a graphing utility to graph the function and find its domain and range. $$f(x)=-x^{2}+9$$

Use a graphing utility to graph the function. Identify any symmetry with respect to the \(x\) -axis, \(y\) -axis, or origin. Determine the number of \(x\) -intercepts of the graph. \(f(x)=x^{4}-2 x^{2}\)

A rectangular package sent by a delivery service can have a maximum combined length and girth (perimeter of a cross section) of 120 inches (see figure). (a) Show that the volume of the package is given by the function \(V(x)=4 x^{2}(30-x)\) (b) Use a graphing utility to graph the function and approximate the dimensions of the package that yield a maximum volume. (c) Find values of \(x\) such that \(V=13,500 .\) Which of these values is a physical impossibility in the construction of the package? Explain.

Use a graphing utility to graph the function. Identify any symmetry with respect to the \(x\) -axis, \(y\) -axis, or origin. Determine the number of \(x\) -intercepts of the graph. \(g(x)=\frac{1}{5}(x+1)^{2}(x-3)(2 x-9)\)

The number of parts per million of nitric oxide emissions \(y\) from a car engine is approximated by \(y=-5.05 x^{3}+3857 x-38,411.25\) \(13 \leq x \leq 18,\) where \(x\) is the air-fuel ratio. (a) Use a graphing utility to graph the model. (b) There are two air-fuel ratios that produce 2400 parts per million of nitric oxide. One is \(x=15\) Use the graph to approximate the other. (c) Find the second air-fuel ratio from part (b) algebraically. (Hint: Use the known value of \(x=15\) and synthetic division.)

Use a graphing utility to graph the function. Identify any symmetry with respect to the \(x\) -axis, \(y\) -axis, or origin. Determine the number of \(x\) -intercepts of the graph. \(h(x)=\frac{1}{5}(x+2)^{2}(3 x-5)^{2}\)

Determine whether the statement is true or false. Justify your answer. If \((7 x+4)\) is a factor of some polynomial function \(f\) then \(\frac{4}{7}\) is a zero of \(f\).

An open box is to be made from a square piece of material 36 centimeters on a side by cutting equal squares with sides of length \(x\) from the corners and turning up the sides (see figure). (a) Verify that the volume of the box is given by the function \(V(x)=x(36-2 x)^{2}\) (b) Determine the domain of the function \(V\). (c) Use the table feature of a graphing utility to create a table that shows various box heights \(x\) and the corresponding volumes \(V\). Use the table to estimate a range of dimensions within which the maximum volume is produced. (d) Use the graphing utility to graph \(V\) and use the range of dimensions from part (c) to find the \(x\) -value for which \(V(x)\) is maximum.

Determine whether the statement is true or false. Justify your answer. The value \(x=\frac{1}{7}\) is a zero of the polynomial function \(f(x)=3 x^{5}-2 x^{4}+x^{3}-16 x^{2}+3 x-8\).

The growth of a red oak tree is approximated by the function $$G=-0.003 t^{3}+0.137 t^{2}+0.458 t-0.839$$ where \(G\) is the height of the tree (in feet) and \(t(2 \leq t \leq 32)\) is its age (in years). Use a graphing utility to graph the function and estimate the age of the tree when it is growing most rapidly. This point is called the point of diminishing returns because the increase in growth will be less with each additional year. (Hint: Use a viewing window in which \(0 \leq x \leq 35\) and \(0 \leq y \leq 60 .\)

The U.S. production of crude oil \(y_{1}\) (in quadrillions of British thermal units) and of solar and photovoltaic energy \(y_{2}\) (in trillions of British thermal units) are shown in the table for the years 2004 through 2013 where \(t\) represents the year, with \(t=4\) corresponding to \(2004 .\) These data can be approximated by the models $$y_{1}=0.00281 t^{4}-0.0850 t^{3}+1.027 t^{2}-5.71 t+22.7$$ and $$y_{2}=0.618 t^{3}-10.80 t^{2}+66.2 t-71$$ (a) Use a graphing utility to plot the data and graph the model for \(y_{1}\) in the same viewing window. How closely does the model represent the data? (b) Extend the viewing window of the graphing utility to show the right-hand behavior of the model \(y_{1} .\) Would you use the model to estimate the production of crude oil in \(2015 ?\) in \(2020 ?\) Explain. (c) Repeat parts (a) and (b) for \(y_{2}\).

Find the value of \(k\) such that \(x-3\) is a factor of \(x^{3}-k x^{2}+2 k x-12\).

Determine whether the statement is true or false. Justify your answer. It is possible for a sixth-degree polynomial to have only one zero.

Complete each polynomial division. Write a brief description of the pattern that you obtain, and use your result to find a formula for the polynomial division \(\left(x^{n}-1\right) /(x-1) .\) Create a numerical example to test your formula. $$\text { (a) } \frac{x^{2}-1}{x-1}=$$ $$\text { (b) } \frac{x^{3}-1}{x-1}=$$ $$\text { (c) } \frac{x^{4}-1}{x-1}=$$

Determine whether the statement is true or false. Justify your answer. It is possible for a fifth-degree polynomial to have no real zeros.

A graph of \(y=f(x)\) is shown, where \(f(x)=2 x^{5}-3 x^{4}+x^{3}-8 x^{2}+5 x+3\) and \(f(-x)=-2 x^{5}-3 x^{4}-x^{3}-8 x^{2}-5 x+3\). (a) How many negative real zeros does \(f\) have? Explain. (b) How many positive real zeros are possible for \(f ?\) Explain. What does this tell you about the eventual right-hand behavior of the graph? (c) Is \(x=-\frac{1}{3}\) a possible rational zero of \(f ?\) Explain. (d) Explain how to check whether \(\left(x-\frac{3}{2}\right)\) is a factor of \(f\) and whether \(x=\frac{3}{2}\) is an upper bound for the real zeros of \(f\).

Determine whether the statement is true or false. Justify your answer. It is possible for a polynomial with an even degree to have a range of \((-\infty, \infty)\)

Use any convenient method to solve the quadratic equation. $$4 x^{2}-17=0$$

Determine whether the statement is true or false. Justify your answer. The graph of the function \(f(x)=x^{6}-x^{7}\) rises to the left and falls to the right.

Use any convenient method to solve the quadratic equation. $$25 x^{2}-1=0$$

Determine whether the statement is true or false. Justify your answer. The graph of the function \(f(x)=2 x(x-1)^{2}(x+3)^{3}\) crosses the \(x\) -axis at \(x=1\).

Use any convenient method to solve the quadratic equation. $$3 x^{2}-11 x-20=0$$

Use any convenient method to solve the quadratic equation. $$6 x^{2}+4 x-3=0$$

Use a graphing utility to graph $$y_{1}=x+2 \text { and } y_{2}=(x+2)(x-1)$$ Predict the shape of the graph of $$y_{3}=(x+2)(x-1)(x-3)$$ Use the graphing utility to verify your answer.

Let \(f(x)=14 x-3\) and \(g(x)=8 x^{2} .\) Find the indicated value. \((f+g)(-4)\)

Let \(f(x)=14 x-3\) and \(g(x)=8 x^{2} .\) Find the indicated value. \((g-f)(3)\)

Let \(f(x)=14 x-3\) and \(g(x)=8 x^{2} .\) Find the indicated value. \((f g)\left(-\frac{4}{7}\right)\)

Let \(f(x)=14 x-3\) and \(g(x)=8 x^{2} .\) Find the indicated value. \((f \circ g)(-1)\)

Let \(f(x)=14 x-3\) and \(g(x)=8 x^{2} .\) Find the indicated value. \((g \circ f)(0)\)

Solve the inequality and sketch the solution on the real number line. Use a graphing utility to verify your solution graphically. \(3(x-5)<4 x-7\)

Solve the inequality and sketch the solution on the real number line. Use a graphing utility to verify your solution graphically. \(2 x^{2}-x \geq 1\)

Solve the inequality and sketch the solution on the real number line. Use a graphing utility to verify your solution graphically. \(\frac{5 x-2}{x-7} \leq 4\)

Solve the inequality and sketch the solution on the real number line. Use a graphing utility to verify your solution graphically. \(|x+8|-1 \geq 15\)