PROFIT The demand equation for a certain product is where

The ________ ________ of ________ states that if is a polynomial of degree then has at least one zero in the complex number system.

The ________ ________ ________ states that if is a polynomial of degree then has precisely linear factors, where are complex numbers.

The test that gives a list of the possible rational zeros of a polynomial function is called the ________ ________ Test.

If is a complex zero of a polynomial with real coefficients, then so is its ________,

Every polynomial of degree with real coefficients can be written as the product of ________ and ________ factors with real coefficients, where the ________ factors have no real zeros.

A quadratic factor that cannot be factored further as a product of linear factors containing real numbers is said to be ________ over the ________.

The theorem that can be used to determine the possible numbers of positive real zeros and negative real zeros of a function is called ________ ________ of ________.

A real number is a(n) ________ bound for the real zeros of if no real zeros are less than and is a(n) ________ bound if no real zeros are greater than

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f x  x3 7x 6 f

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h t  t 3 8t 2 13t 6

p x  x3 9x 2 27x 27 h

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f x  3x3 19x 2 33x 9 C

f x  9x4 9x3 58x 2 4x 24 f

f x  2x4 15x3 23x 2 15x 25 f

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f x  4x3 15x 2 8x 3 f

f x  4x3 12x 2 x 15 f

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f x  4x4 17x 2 4

f x  32x3 52x 2 17x 3 f

f x  4x3 7x 2 11x 18 f

f x  x 2 12 4 3x 2 2

P t  t 4 7t f x  x 2 12 4

h x  x5 7x4 10x3 14x 2 24x P

g x  6x4 11x3 51x 2 99x 27 h

1, 5i

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f y  y f x  x 4 256 4

f z  z 2 2z 2

h(x)  x 3 3x 2 4x 2 f

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g x  3x3 2x 2 15x 10 f

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f x  x 3 2 3 1 f

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GEOMETRY An open box is to be made from a rectangular piece of material, 15 centimeters by 9 centimeters, by cutting equal squares from the corners and turning up the sides.

GEOMETRY A rectangular package to be sent by a delivery service (see figure) can have a maximum combined length and girth (perimeter of a cross section) of 120 inches.

ADVERTISING COST A company that produces MP3 players estimates that the profit (in dollars) for selling a particular model is given by where is the advertising expense (in tens of thousands of dollars). Using this model, find the smaller of two advertising amounts that will yield a profit of $2,500,000.

ADVERTISING COST A company that manufactures bicycles estimates that the profit (in dollars) for selling a particular model is given by where is the advertising expense (in tens of thousands of dollars). Using this model, find the smaller of two advertising amounts that will yield a profit of $800,000.

GEOMETRY A bulk food storage bin with dimensions 2 feet by 3 feet by 4 feet needs to be increased in size to hold five times as much food as the current bin. (Assume each dimension is increased by the same amount.) (a) Write a function that represents the volume of the new bin. (b) Find the dimensions of the new bin

GEOMETRY A manufacturer wants to enlarge an existing manufacturing facility such that the total floor area is 1.5 times that of the current facility. The floor area of the current facility is rectangular and measures 250 feet by 160 feet. The manufacturer wants to increase each dimension by the same amount. (a) Write a function that represents the new floor area (b) Find the dimensions of the new floor. (c) Another alternative is to increase the current floors length by an amount that is twice an increase in the floors width. The total floor area is 1.5 times that of the current facility. Repeat parts (a) and (b) using these criteria.

COST The ordering and transportation cost (in thousands of dollars) for the components used in manufacturing a product is given by where is the order size (in hundreds). In calculus, it can be shown that the cost is a minimum when Use a calculator to approximate the optimal order size to the nearest hundred units.

HEIGHT OF A BASEBALL A baseball is thrown upward from a height of 6 feet with an initial velocity of 48 feet per second, and its height (in feet) is where is the time (in seconds). You are told the ball reaches a height of 64 feet. Is this possible?

PROFIT The demand equation for a certain product is where is the unit price (in dollars) of the product and is the number of units produced and sold. The cost equation for the product is where is the total cost (in dollars) and is the number of units produced. The total profit obtained by producing and selling units is You are working in the marketing department of the company that produces this product, and you are asked to determine a price that will yield a profit of 9 million dollars. Is this possible? Explain.

ATHLETICS The attendance (in millions) at NCAA womens college basketball games for the years 2000 through 2007 is shown in the table. (Source: National Collegiate Athletic Association, Indianapolis, IN) (a) Use a graphing utility to create a scatter plot of the data. Let represent the year, with corresponding to 2000. (b) Use the regression feature of the graphing utility to find a quartic model for the data. (c) Graph the model and the scatter plot in the same viewing window. How well does the model fit the data? (d) According to the model in part (b), in what year(s) was the attendance at least 10 million? (e) According to the model, will the attendance continue to increase? Explain.

It is possible for a third-degree polynomial function with integer coefficients to have no real zeros.

If is a zero of the function given by then must also be a zero of

g x  f x g

g x  3f x

g x  f x 5 g

g x  f 2x

g x  3 f x

g x  f x

THINK ABOUT IT A third-degree polynomial function has real zeros and 3, and its leading coefficient is negative. Write an equation for Sketch the graph of How

CAPSTONE Use a graphing utility to graph the function given by for different values of Find values of such that the zeros of satisfy the specified characteristics. (Some parts do not have unique answers.) (a) Four real zeros (b) Two real zeros, each of multiplicity 2 (c) Two real zeros and two complex zeros (d) Four complex zeros (e) Will the answers to parts (a) through (d) change for the function where (f) Will the answers to parts (a) through (d) change for the function where g, g x)  f 2x?

THINK ABOUT IT Sketch the graph of a fifth-degree polynomial function whose leading coefficient is positive and that has a zero at of multiplicity 2.

WRITING Compile a list of all the various techniques for factoring a polynomial that have been covered so far in the text. Give an example illustrating each technique, and write a paragraph discussing when the use of each technique is appropriate.

THINK ABOUT IT Let be a quartic polynomial with leading coefficient and Write an equation for

THINK ABOUT IT Let be a cubic polynomial with leading coefficient and Write an equation for

y 1 123

y 2 12 3 1

Use the information in the table to answer each question. (a) What are the three real zeros of the polynomial function (b) What can be said about the behavior of the graph of at (c) What is the least possible degree of Explain. Can the degree of ever be odd? Explain. (d) Is the leading coefficient of positive or negative? Explain. (e) Write an equation for (There are many correct answers.) (f) Sketch a graph of the equation you wrote in part (e).

(a) Find a quadratic function (with integer coefficients) that has as zeros. Assume that is a positive integer. (b) Find a quadratic function (with integer coefficients) that has as zeros. Assume that is a positive integer

GRAPHICAL REASONING The graph of one of the following functions is shown below. Identify the function shown in the graph. Explain why each of the others is not the correct function. Use a graphing utility to verify your result. (a) (b) (c) (d) 2 4 40 30 20 10 x y k x  x 1) x 2 x 3.5