Chapter 3

Graph the function using a graphing utility, and find its zeros. $$p(x)=x^{3}+(3+\sqrt{2}) x^{2}+4 x+6.7$$

For each polynomial function, (a) find a function of the form \(y=c x^{2}\) that has the same end behavior. (b) find the \(x\) - and \(y\) -intercept(s) of the graph. (c) find the interval(s) on which the value of the function is positive. (d) find the interval(s) on which the value of the function is negative. (e) use the information in parts ( \(a\) ) \(-\) (d) to sketch a graph of the function. $$f(x)=-\left(x^{2}-1\right)(x-2)(x+3)$$

Use the given information to (a) sketch a possible graph of the polynomial function; (b) indicate on your graph roughly where the local maxima and minima, if any, might occur; (c) find a possible expression for the polynomial; and \((d)\) use a graphing utility to check your anstroers to parts \((a)-(c)\) The polynomial \(p(x)\) has real zeros at \(x=-1\) and \(x=3\) and the graph crosses the \(x\) -axis at both of these zeros. As \(x \rightarrow \pm \pm \infty, p(x) \rightarrow \infty\)

Sketch a graph of the rational function and find all intercepts and slant asymptotes. Indicate all asymptotes onthe graph. $$h(x)=\frac{x^{2}+2 x+1}{x+3}$$

You will use polynomials to study real-world problems. Geometry A rectangle has length \(x^{2}-x+6\) units and width \(x+1\) units. Find \(x\) such that the area of the rectangle is 24 square units.

For each polynomial function, (a) find a function of the form \(y=c x^{2}\) that has the same end behavior. (b) find the \(x\) - and \(y\) -intercept(s) of the graph. (c) find the interval(s) on which the value of the function is positive. (d) find the interval(s) on which the value of the function is negative. (e) use the information in parts ( \(a\) ) \(-\) (d) to sketch a graph of the function. $$f(x)=x\left(x^{2}-4\right)(x+1)$$

Use the given information to (a) sketch a possible graph of the polynomial function; (b) indicate on your graph roughly where the local maxima and minima, if any, might occur; (c) find a possible expression for the polynomial; and \((d)\) use a graphing utility to check your anstroers to parts \((a)-(c)\) The only points at which the graph of the polynomial \(f(s)\) crosses the s-axis are (-1,0) and \((2,0),\) and the only point at which it just touches the s-axis is \((0,0) .\) The function is positive on the intervals \((-\infty,-1)\) and \((2, \infty)\)

Sketch a graph of the rational function and find all intercepts and slant asymptotes. Indicate all asymptotes onthe graph. $$f(x)=\frac{3 x^{2}+5 x-2}{x+1}$$

You will use polynomials to study real-world problems. Geometry The length of a rectangular box is 10 inches more than the height, and its width is 5 inches more than the height. Find the dimensions of the box if the volume is 168 cubic inches.

For each polynomial function, (a) find a function of the form \(y=c x^{2}\) that has the same end behavior. (b) find the \(x\) - and \(y\) -intercept(s) of the graph. (c) find the interval(s) on which the value of the function is positive. (d) find the interval(s) on which the value of the function is negative. (e) use the information in parts ( \(a\) ) \(-\) (d) to sketch a graph of the function. $$g(x)=2 x^{2}(x+3)$$

Use the given information to (a) sketch a possible graph of the polynomial function; (b) indicate on your graph roughly where the local maxima and minima, if any, might occur; (c) find a possible expression for the polynomial; and \((d)\) use a graphing utility to check your anstroers to parts \((a)-(c)\) The real zeros of the polynomial \(h(x)\) are \(x=3\) and \(x=0.5,\) each of multiplicity \(1,\) and \(x=\sqrt{2},\) of multiplicity 2. As \(|x|\) gets large, \(h(x) \rightarrow+\infty\)

Sketch a graph of the rational function and find all intercepts and slant asymptotes. Indicate all asymptotes onthe graph. $$g(x)=\frac{2 x^{2}+11 x+5}{x-3}$$

You will use polynomials to study real-world problems. Manufacturing An open rectangular box is constructed by cutting a square of length \(x\) from each corner of a 12-inch by 15-inch rectangular piece of cardboard and then folding up the sides. For this box, \(x\) must be greater than or equal to 1 inch. (a) What is the length of the square that must be cut from each corner if the volume of the box is to be 112 cubic inches? (b) What is the length of the square that must be cut from each corner if the volume of the box is to be 150 cubic inches?

For each polynomial function, (a) find a function of the form \(y=c x^{2}\) that has the same end behavior. (b) find the \(x\) - and \(y\) -intercept(s) of the graph. (c) find the interval(s) on which the value of the function is positive. (d) find the interval(s) on which the value of the function is negative. (e) use the information in parts ( \(a\) ) \(-\) (d) to sketch a graph of the function. $$f(x)=-3 x^{2}(x-1)$$

Use the given information to (a) sketch a possible graph of the polynomial function; (b) indicate on your graph roughly where the local maxima and minima, if any, might occur; (c) find a possible expression for the polynomial; and \((d)\) use a graphing utility to check your anstroers to parts \((a)-(c)\) The polynomial \(q(x)\) has exactly one real zero and no local maxima or minima.

Sketch a graph of the rational function and find all intercepts and slant asymptotes. Indicate all asymptotes onthe graph. $$h(x)=\frac{x^{3}+1}{x^{2}+3 x}$$

You will use polynomials to study real-world problems. Manufacturing The height of a right circular cylinder is 5 inches more than its radius. Find the dimensions of the cylinder if its volume is 1000 cubic inches.

For each polynomial function, (a) find a function of the form \(y=c x^{2}\) that has the same end behavior. (b) find the \(x\) - and \(y\) -intercept(s) of the graph. (c) find the interval(s) on which the value of the function is positive. (d) find the interval(s) on which the value of the function is negative. (e) use the information in parts ( \(a\) ) \(-\) (d) to sketch a graph of the function. $$f(x)=(2 x+1)(x-3)\left(x^{2}+1\right)$$

Sketch a graph of the rational function and find all intercepts and slant asymptotes. Indicate all asymptotes onthe graph. $$h(x)=\frac{x^{3}-1}{x^{2}-2 x}$$

For each polynomial function, (a) find a function of the form \(y=c x^{2}\) that has the same end behavior. (b) find the \(x\) - and \(y\) -intercept(s) of the graph. (c) find the interval(s) on which the value of the function is positive. (d) find the interval(s) on which the value of the function is negative. (e) use the information in parts ( \(a\) ) \(-\) (d) to sketch a graph of the function. $$g(x)=-(x-2)(3 x-1)\left(x^{2}+1\right)$$

Sketch a graph of the rational function involving common factors and find all intercepts and asymptotes. Indicate all asymptotes on the graph. $$f(x)=\frac{3 x+9}{x^{2}-9}$$

Find at least two different cubic polynomials whose only real zero is \(-1 .\) Graph your answers to check them.

Sketch a graph of the rational function involving common factors and find all intercepts and asymptotes. Indicate all asymptotes on the graph. $$f(x)=\frac{2 x-4}{x^{2}-4}$$

Let \(p(x)=x^{5}+x^{3}-2 x\). (a) Show that \(p\) is symmetric with respect to the origin. (b) Find a zero of \(p\) by inspection of the polynomial expression. (c) Use a graphing utility to find the other zeros. (d) How do you know that you have found all the zeros of \(p ?\)

Sketch a graph of the rational function involving common factors and find all intercepts and asymptotes. Indicate all asymptotes on the graph. $$f(x)=\frac{x^{2}+x-2}{x^{2}+2 x-3}$$

Sketch a graph of the rational function involving common factors and find all intercepts and asymptotes. Indicate all asymptotes on the graph. $$f(x)=\frac{2 x^{2}-5 x+2}{x^{2}-5 x+6}$$

Sketch a graph of the rational function involving common factors and find all intercepts and asymptotes. Indicate all asymptotes on the graph. $$f(x)=\frac{x^{2}+3 x-10}{x-2}$$

Consider the function \(f(x)=0.001 x^{3}+2 x^{2} .\) Answer the following questions. (a) Graph the function in a standard window of a graphing utility. Explain why this window setting does not give a complete graph of the function. (b) Using the \(x\) -intercepts and the end behavior of the function, sketch an approximate graph of the function by hand. (c) Find a graphing window that shows a correct graph for this function.

Sketch a graph of the rational function involving common factors and find all intercepts and asymptotes. Indicate all asymptotes on the graph. $$f(x)=\frac{x^{2}+2 x+1}{x+1}$$

You will use polynomial functions to study real-world problems. The following model gives the supply of wine from France, based on data for the years \(1994-2001:\) \(w(x)=0.0437 x^{4}-0.661 x^{3}+3.00 x^{2}-4.83 x+62.6\) where \(w(x)\) is in kilograms per capita and \(x\) is the number of years since \(1994 .\) (Source: Food and Agriculture Organization of the United Nations) (a) According to this model, what was the per capita wine supply in \(1994 ?\) How close is this value to the actual value of 62.5 kilograms per capita? (b) Use this model to compute the wine supply from France for the years 1996 and \(2000 .\) (c) The actual wine supplies for the years 1996 and 2000 were 60.1 and 54.6 kilograms per capita respectively. How do your calculated values compare with the actual values? (d) Use end behavior to determine if this model will be accurate for long-term predictions.

The concentration \(C(t)\) of a drug in a patient's bloodstream \(t\) hours after administration is given by $$ C(t)=\frac{10 t}{1+t^{2}} $$ where \(C(t)\) is in milligrams per liter. (a) What is the drug concentration in the patient's bloodstream 8 hours after administration? (b) Find the horizontal asymptote of \(C(t)\) and explain its significance.

You will use polynomial functions to study real-world problems. The numbers of burglaries (in thousands) in the United States can be modeled by the following cubic function, where \(x\) is the number of years since \(1985 .\) $$b(x)=0.6733 x^{3}-22.18 x^{2}+113.9 x+3073$$ (a) What is the \(y\) -intercept of the graph of \(b(x)\), and what does it signify? (b) Find \(b(6)\) and interpret it. (c) Use this model to predict the number of burglaries that occurred in the year 2004 (d) Why would this cubic model be inaccurate for predicting the number of burglaries in the year \(2040 ?\)

The annual cost, in millions of dollars, of removing arsenic from drinking water in the United States can be modeled by the function $$ C(x)=\frac{1900}{x} $$ where \(x\) is the concentration of arsenic remaining in the water, in micrograms per liter. A microgram is \(10^{-6}\) gram. (Source: Environmental Protection Agency) (a) Evaluate \(C(10)\) and explain its significance. (b) Evaluate \(C(5)\) and explain its significance. (c) What happens to the cost function as \(x\) gets closer to zero?

You will use polynomial functions to study real-world problems. The number of species on the U.S. endangered species list during the years \(1998-2005\) can be modeled by the function $$f(t)=0.308 t^{3}-5.20 t^{2}+32.2 t+921$$ where \(t\) is the number of years since \(1998 .\) (Source: U.S. Fish and Wildlife Service) (a) Find and interpret \(f(0)\). (b) How many species were on the list in \(2004 ?\) (c) \(\quad\) Use a graphing utility to graph this function for \(0 \leq t \leq 7 .\) Judging by the trend seen in the graph, is this model reliable for long- term predictions? Why or why not?

A truck rental company charges a daily rate of \(\$ 15\) plus \(\$ 0.25\) per mile driven. What is the average cost per mile of driving \(x\) miles per day? Use this expression to find the average cost per mile of driving 50 miles per day.

You will use polynomial functions to study real-world problems. An open box is to be made by cutting four squares of equal size from a 10 -inch by 15 -inch rectangular piece of cardboard (one at each corner) and then folding up the sides. (a) Let \(x\) be the length of a side of the square cut from each corner. Find an expression for the volume of the box in terms of \(x\). Leave the expression in factored form. (b) What is a realistic range of values for \(x ?\) Explain.

To print booklets, it costs \(\$ 300\) plus an additional \(\$ 0.50\) per booklet. What is the average cost per booklet of printing \(x\) booklets? Use this expression to find the average cost per booklet of printing 1000 booklets.

You will use polynomial functions to study real-world problems. Construction A cylindrical container is to be constructed so that the sum of its height and its diameter is 10 feet. (a) Write an equation relating the height of the cylinder, \(h\), to its radius, \(r\). Solve the equation for \(h\) in terms of \(r\) (b) The volume of a cylinder is given by \(V=\pi r^{2} h .\) Use your answer from part (a) to express the volume of the cylindrical container in terms of \(r\) alone. Leave your expression in factored form so that it will be easier to analyze. (c) What are the values of \(r\) for which this problem makes sense? Explain.

A wireless phone company has a pricing scheme that includes 250 minutes worth of phone usage in the basic monthly fee of \(\$ 30 .\) For each minute over and above the first 250 minutes of usage, the user is charged an additional \(\$ 0.60\) per minute. (a) Let \(x\) be the number of minutes of phone usage per month. What is the expression for the average cost per minute if the value of \(x\) is in the interval (0,250)\(?\) (b) What is the expression for the average cost per minute if the value of \(x\) is above \(250 ?\) (c) If phone usage in a certain month is 600 minutes, what is the average cost per minute?

Body-mass index (BMII) is a measure of body fat based on height and weight that applies to both adultmales and adult females. It is calculated using the following formula: $$ \mathrm{BMI}=\frac{703 w}{h^{2}} $$ where \(w\) is the person's weight in pounds and \(h\) is the person's height in inches. A BMI in the range 18.5 24.9 is considered normal. (Source: National Institutes of Health) (a) Calculate the BMI for a person who is 5 feet 5 inches tall and weighs 140 pounds. Is this person's BMI within the normal range? (b) Calculate the weight of a person who is 6 feet tall and has a BMI of 24 (c) Calculate the height of a person who weighs 170 pounds and has a BMI of 24.3.

How much pure gold should be added to a 2-ounce alloy that is presently \(25 \%\) gold to make it \(60 \%\) gold?

Show that all polynomial functions have a \(y\) -intercept. Can the same be said of \(x\) -intercepts?

Can the graph of a function with range \([4, \infty)\) cross the \(x\) -axis?

A gift box company wishes to make a small open box by cutting four equal squares from a 3 inch by 5 -inch card, one from each corner. (a) Let \(x\) denote the length of the square cut from each corner. Write an expression for the volume of the box in terms of \(x\). Call this function \(V(x) .\) What is the realistic domain of this function? (b) Write an expression for the surface area of the box in terms of \(x .\) Call this function \(S(x)\) (c) Write an expression in terms of \(x\) for the ratio of the volume of the box to its surface area. Call this function \(r(x)\) (d) Fill in the following table giving the values of \(r(x)\) for the given values of \(x\) (TABLE CANNOT COPY) (e) What do you observe about the ratio of the volume to the surface area as \(x\) increases? From your table, approximate the value of \(x\) that would give the maximum ratio of volume to surface area. (f) \(=\) Use a graphing utility to find the value of \(x\) that would give the maximum ratio of volume to surface area.

Explain why all polynomial functions of odd degree must have range \((-\infty, \infty)\).

Sketch a possible graph of a rational function \(r(x)\) of the following description: the graph of \(r\) has a horizontal asymptote \(y=-2\) and a vertical asymptote \(x-1,\) with \(y\) -intercept at (0,0).

Explain why all polynomial functions of odd degree must have at least one real zero.

Sketch a possible graph of a rational function \(r(x)\) of the following description: the graph of \(r\) has a horizontal asymptote \(y=-2\) and a vertical asymptote \(x=1,\) with \(y\) -intercept at (0,0) and \(x\) -intercept at (2,0).

Give a possible expression for a rational function \(r(x)\) of the following description: the graph of \(r\) has a horizontal asymptote \(y=2\) and a vertical asymptote \(x=1,\) with \(y\) intercept at \((0,0) .\) It may be helpful to sketch the graph of \(r\) first. You may check your answer with a graphing utility.

Give a possible expression for a rational function \(r(x)\) of the following description: the graph of \(r\) has a horizontal asymptote \(y=0\) and a vertical asymptote \(x=0,\) with no \(x\) - or \(y\) -intercepts. It may be helpful to sketch the graph of \(r\) first. You may check your answer with a graphing utility.