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1 Quadratic Functions and Models 135 In Exercises 29β36, use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and -intercepts. Then check your results algebraically by writing the quadratic function in standard form. x 50. Vertex: 51. Vertex: 52. Vertex: 4 ; 5 2, 3 2, 0; 5 6, 6... |
find the Graphical Reasoning the x x -intercepts algebraically to confirm your results. 53. y x 2 16 y β8 β4 x 8 55. y x 2 4x 5 y 54. y x 2 6x 56. y 2x 2 5x 3 x β6 β4 8 β4 β4 β8 y 2 β2 β4 x 2 58. 57. In Exercises 57β64, use a graphing utility to graph the quadratic function. Find the -intercepts of the graph and x com... |
table showing possible values of and the corresponding areas of the corral. Use the table to estimate the dimensions that will produce the maximum enclosed area. x (c) Use a graphing utility to graph the area function. Use the graph to approximate the dimensions that will produce the maximum enclosed area. (d) Write t... |
is the number of units sold. What sales level will x where yield a maximum profit? 333202_0201.qxd 12/7/05 9:10 AM Page 137 82. Maximum Profit The profit (in hundreds of dollars) that a company makes depends on the amount (in hundreds of dollars) the company spends on advertising according to the model P x P 230 20x 0... |
Statistics) y Year 1994 1995 1996 1997 1998 1999 2000 2001 2002 Number of hairdressers and cosmetologists, y 753 750 737 748 763 784 820 854 908 (a) Use a graphing utility to create a scatter plot of the corre- represent the year, with x 4 x data. Let sponding to 1994. (b) Use the regression feature of a graphing util... |
Synthesis In Exercises 89 and 90, determine whether True or False? the statement is true or false. Justify your answer. f x 12x2 1 89. The function given by has no x -intercepts. 90. The graphs of f x 4x2 10x 7 and gx 12x2 30x 1 have the same axis of symmetry. 91. Write the quadratic function f x ax2 bx c in standard ... |
12/7/05 9:11 AM Page 139 Section 2.2 Polynomial Functions of Higher Degree 139 2.2 Polynomial Functions of Higher Degree What you should learn β’ Use transformations to sketch graphs of polynomial functions. β’ Use the Leading Coefficient Test to determine the end behavior of graphs of polynomial functions. β’ Find and u... |
Page 140 140 Chapter 2 Polynomial and Rational Functions if For power functions given by f x xn, n is even, then the graph of the function is symmetric with respect to the y n -axis, and if is odd, then the graph of the function is symmetric with respect to the origin. (β1, 1) β1 f x x n, The polynomial functions that... |
function. Describe the relationship between the degree and the sign of the leading coefficient of the function and the right-hand and left-hand behavior of the graph of the function. a. b. c. d. e. f. g. f x x3 2x2 x 1 f x 2x5 2x2 5x 1 f x 2x5 x2 5x 3 f x x3 5x 2 f x 2x2 3x 4 f x x 4 3x2 2x 1 f x x2 3x 2 as f x β β in... |
the graph. 333202_0202.qxd 12/7/05 9:11 AM Page 142 142 Chapter 2 Polynomial and Rational Functions Example 2 Applying the Leading Coefficient Test A polynomial function is written in standard form if its terms are written in descending order of exponents from left to right. Before applying the Leading Coefficient Tes... |
Section 2.5.) f 2. The graph of has, at most, turning points. (Turning points, also called relative minima or relative maxima, are points at which the graph changes from increasing to decreasing or vice versa.) n 1 Finding the zeros of polynomial functions is one of the most important problems in algebra. There is a s... |
. This is consistent with the fact that a fourth-degree polynomial can have at most three turning points. x 1. x 1, x 0, and 2 y = β 2x 4 + 2x 2 β3 3 Now try Exercise 27. β2 FIGURE 2.19 k In Example 3, note that because x 0. x The graph touches the -axis at is even, the factor x 0, zero 2x2 yields the repeated as shown... |
. Apply the Leading Coefficient Test. Because the leading coefficient is positive and the degree is even, you know that the graph eventually rises to the left and to the right (see Figure 2.20). 2. Find the Zeros of the Polynomial. By factoring you can see that the zeros of are 0, 0 f x x 33x 4, of odd multiplicity). S... |
x2x 32 x 0 you can see that the zeros of x multiplicity). So, the -intercepts occur at to your graph, as shown in Figure 2.22. are f (odd multiplicity) and 2, 0. 3 0, 0 x 3 (even 2 Add these points and 3. Plot a Few Additional Points. Use the zeros of the polynomial to find the test intervals. In each test interval, ch... |
f b and (See Figure 2.24.) and f a f b, f c d. b, f b such that b d c y f b( ) FIGURE 2.24 a c b x Intermediate Value Theorem a Let and be real numbers such that such that between b f a f b, and f b. f a then, in the interval a < b. f If a, b, is a polynomial function f takes on every value The Intermediate Value Theo... |
any desired accuracy. as shown in Figure 2.26. For a and and apply the Intermediate Value Theorem again. By continuing this f 0.8 0.8 f x ( ) = 3 x 2β x + 1 y 2 (0, 1) (1, 1) x 2 1 f has a zero between 0.8β β and 0.7. β1 β1 β β ( 1, 1) FIGURE 2.26 Now try Exercise 85. Te c h n o l o g y You can use the table feature o... |
) ________ is a factor of the polynomial f x. (c) a, 0 is an ________ of the graph f. 5. If a real zero of a polynomial function is of even multiplicity, then the graph of ________ the -axis at x f x a, and if it is of odd multiplicity then the graph of ________ the -axis at x f x a. 6. A polynomial function is written... |
(b) (d) f x x 3 2 f x x 23 2 (b) (d 15 (bd) 2 (f) fx 1 x 14 2 x4 2 β4 β2 2 4 x β4 333202_0202.qxd 12/7/05 9:12 AM Page 149 Section 2.2 Polynomial Functions of Higher Degree 149 12. y x 6 (a) 8x 6 f x 1 f x x 6 4 fx 1 4 x6 2 (c) (e) (b) (d) (f) f x x 26 4 f x 1 4x 6 1 fx 2x6 1 In Exercises 13β22, describe the right-han... |
and (d) compare the results of part (c) with any -intercepts of the graph. y 0 x x 43. 44. 45. 46. y 4x3 20x 2 25x y 4x 3 4x 2 8x 8 y x 5 5x 3 4x 4x 3x 2 9 y 1 In Exercises 47β56, find a polynomial function that has the given zeros. (There are many correct answers.) 47. 49. 51. 53. 55. 0, 10 2, 6 0, 2, 3 4, 3, 3, 0 1 ... |
2 2t 15 4 gx x 2 10x 16 f x x 3 3x 2 f x 3x3 15x2 18x f x 4x 3 4x2 15x f x 5x2 x3 f x x 2x 4 gt 1 t 22t 22 79. 4 x 12x 33 80. gx 1 10 77. 75. 74. 68. gx x 4 4x2 72. f x 1 x 3 76. 78. f x 48x2 3x4 hx 1 3x 3x 42 333202_0202.qxd 12/7/05 9:12 AM Page 150 150 Chapter 2 Polynomial and Rational Functions In Exercises 81β84, ... |
91. Construction A roofing contractor is fabricating gutters from 12-inch aluminum sheeting. The contractor plans to use an aluminum siding folding press to create the gutter by creasing equal lengths for the sidewalls (see figure). x x 12 β 2x x x 36 2β x x (a) Let x (a) Verify that the volume of the box is given by ... |
(a) Write a function that represents the total volume V of the tank in terms of r. (b) Find the domain of the function. (c) Use a graphing utility to graph the function. (d) The total volume of the tank is to be 120 cubic feet. Use the graph from part (c) to estimate the radius and length of the cylindrical portion of... |
. Revenue The total revenue R (in millions of dollars) for a company is related to its advertising expense by the function R 1 100,000 x3 600x 2, 0 β€ x β€ 400 x is the amount spent on advertising (in tens of thouwhere sands of dollars). Use the graph of this function, shown in the figure, to estimate the point on the gr... |
so, is this behavior determined by Explain. a, h, or g (c) Use a graphing utility to graph the function given by Hx x 5 3x 3 2x 1. Use the graph and the result of part (b) to determine whether form can be written Hx ax h5 k. Explain. the in H Skills Review In Exercises 105β108, factor the expression completely. 105. 1... |
x x 2 qx. x 2, x 2 qx as shown in Figure 2.27. Because x 2 This means that there fx. is a factor of such that To find qx, you can use long division, as illustrated in Example 1. Example 1 Long Division of Polynomials 6x3 19x 2 16x 4 Divide mial completely. Solution by x 2, and use the result to factor the polyno- 6x2. ... |
1. which illustrates the following theorem, called the Division Algorithm. and f x dx The Division Algorithm If less than or equal to the degree of and rx f x dxqx rx such that are polynomials such that dx 0, and the degree of there exist unique polynomials dx is qx f x, Dividend Quotient Divisor Remainder where remai... |
can write the result as 2x4 4x3 5x 2 3x 2 x 2 2x 3 2x 2 1 x 1 x 2 2x 3. Now try Exercise 15. 333202_0203.qxd 12/7/05 9:23 AM Page 156 156 Chapter 2 Polynomial and Rational Functions Synthetic Division There is a nice shortcut for long division of polynomials when dividing by This shortcut is called synthetic division.... |
can be used to f x as illustrated in evaluate a polynomial function. That is, to evaluate a polynomial function x k, when Example 5. The remainder will be x k. divide f k, f x by Example 5 Using the Remainder Theorem Use the Remainder Theorem to evaluate the following function at x 2. f x 3x3 8x2 5x 7 Solution Using s... |
32x 3x 1. Note that this factorization implies that has four real zeros: x 2, x 3, x 3 2, and f x 1. FIGURE 2.28 This is confirmed by the graph of which is shown in Figure 2.28. f, Now try Exercise 57. Uses of the Remainder in Synthetic Division f x The remainder obtained in the synthetic division of provides the foll... |
Exercises 1 and 2, use long division 1. y1 x2, x 2 y2 x 2 4 2. y1 x4 3x 2 1, x2 5 y2 x 2 x 2 8 39 x2 5 Graphical Analysis In Exercises 3 and 4, (a) use a graphing utility to graph the two equations in the same viewing window, (b) use the graphs to verify that the expressions are equivalent, and (c) use long division t... |
x2 x3 x 1 3x3 4x2 5 x 3 2 In Exercises 37β 44, write the function in the form f x x kqx r for the given value of and demonstrate that f k r. k, Function fx x3 x2 14x 11 fx x3 5x2 11x 8 37. 38. Value of k k 4 k 2 333202_0203.qxd 12/7/05 9:23 AM Page 160 160 Chapter 2 Polynomial and Rational Functions Function fx 15x 4 1... |
24 48. f x 0.4x4 1.6x3 0.7x2 2 (a) f 1 (b) f 2 (c) f 5 (d) f 10 In Exercises 49β56, use synthetic division to show that is a solution of the third-degree polynomial equation, and use the result to factor the polynomial completely. List all real solutions of the equation. x In Exercises 69β72, simplify the rational exp... |
203.qxd 12/7/05 9:23 AM Page 161 Model It (co n t i n u e d ) (a) Use a graphing utility to create a scatter plot of the data. (b) Use the regression feature of the graphing utility to find a cubic model for the data. Graph the model in the same viewing window as the scatter plot. (c) Use the model to create a table of... |
Exercises 79 and 80, perform the 79. x3n 9x2n 27xn 27 xn 3 80. x3n 3x2n 5xn 6 x n 2 81. Writing Briefly explain what it means for a divisor to divide evenly into a dividend. 82. Writing Briefly explain how to check polynomial divi- sion, and justify your reasoning. Give an example. Exploration In Exercises 83 and 84, ... |
is x To overcome this deficienno real number cy, mathematicians created an expanded system of numbers using the imaginary i, unit defined as i 1 that can be squared to produce Imaginary unit 1. i 2 1. where By adding real numbers to real multiples of this imaginary unit, the set of complex numbers is obtained. Each co... |
bi a bi 0 0i 0. Example 1 Adding and Subtracting Complex Numbers a. 4 7i 1 6i 4 7i 1 6i (4 1) (7i 6i) 5 i b. (1 2i) 4 2i 1 2i 4 2i 1 4 2i 2i 3 0 3 Remove parentheses. Group like terms. Write in standard form. Remove parentheses. Group like terms. Simplify. Write in standard form. c. 3i 2 3i 2 5i 3i 2 3i 2 5i 2 2 3i 3i... |
6i 4i 31 8 3 6i 4i 11 2i (3 2i)(3 2i) 33 2i 2i3 2i c. 9 6i 6i 4i 2 9 6i 6i 41 9 4 13 d. 3 2i2 3 2i3 2i 33 2i 2i3 2i 9 6i 6i 4i 2 9 6i 6i 41 9 12i 4 5 12i Now try Exercise 27. Distributive Property Simplify. Distributive Property Distributive Property i2 1 Group like terms. Write in standard form. Distributive Property... |
by complex conjugate of denominator. 8 4i 12i 6i 2 16 4i 2 8 6 16i 16 4 2 16i 20 4 5 1 10 i Now try Exercise 49. Expand. i2 1 Simplify. Write in standard form. 333202_0204.qxd 12/7/05 9:30 AM Page 166 166 Chapter 2 Polynomial and Rational Functions Complex Solutions of Quadratic Equations When using the Quadratic Form... |
number (i) (ii) (iii) a bi, a bi, a bi, a 0, a 0, b 0 b 0 b 0 In Exercises 2β5, fill in the blanks. 2. The imaginary unit i is defined as i ________, where i2 ________. 3. If a is a positive number, the ________ ________ root of the negative number a is defined as 4. The numbers a bi and a bi are called ________ _____... |
6 3i 1 5 i 20 8 38. 40. 42. 44. 7 12i 3 2 i 15 1 8 In Exercises 45β54, write the quotient in standard form. 45. 47. 49. 51. 53. 5 i 2 4 5i 3 i 3 i 6 5i i 3i 4 5i 2 46. 48. 50. 52. 54. 14 2i 5 1 i 6 7i 1 2i 8 16i 2i 5i 2 3i2 In Exercises 55β58, perform the operation and write the result in standard form. 55. 57. 3 1 i ... |
a parallel circuit with two pathways satisfies the equation z 1 z 1 z1 1 z 2 z1 is the impedance of pathway 2. is the impedance (in ohms) of pathway 1 and where z2 (a) The impedance of each pathway in a parallel circuit is found by adding the impedances of all compoz2. nents in the pathway. Use the table to find and z... |
eros of Polynomial Functions Section 2.5 Zeros of Polynomial Functions 169 What you should learn β’ Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions. β’ Find rational zeros of polyno- mial functions. β’ Find conjugate pairs of com- plex zeros. β’ Find zeros of polynomials by f... |
Example 1 Zeros of Polynomial Functions a. The first-degree polynomial b. Counting multiplicity, the second-degree polynomial function has exactly one zero: f x x 2 x 2. f x x 2 6x 9 x 3x 3 x 3 x 3. has exactly two zeros: and (This is called a repeated zero.) c. The third-degree polynomial function f x x3 4x xx 2 4 xx... |
(x) = x3 + x + 1 y Solution Because the leading coefficient is 1, the possible rational zeros are the factors of the constant term. By testing these possible zeros, you can see that neither works. Β±1, f 1 13 1 1 3 f 1 13 1 1 1 2 3 x 1 So, you can conclude that the given polynomial has no rational zeros. Note from the g... |
either by hand or with a graphing utility, can give a good estimate of the locations of the zeros; (3) the Intermediate Value Theorem along with a table generated by a graphing utility can give approximations of zeros; and (4) synthetic division can be used to test the possible rational zeros. Finding the first zero i... |
0. Testing these by synthetic division shows that x 1 2, x 2. and Using the Quadratic Formula for the second factor, you find that the two additional solutions are irrational numbers. x 5 265 20 1.0639 and x 5 265 20 0.5639 Now try Exercise 23. Example 5 Solving a Polynomial Equation 1 x Find all the real solutions of... |
see Proofs in Mathematics on page 214. f x ci Factors of a Polynomial with real coefficients can be written as Every polynomial of degree the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros. n > 0 333202_0205.qxd 12/7/05 9:36 AM Page 174 174 Chapter 2 Poly... |
1 3i, and x 2. x 3, f are x 1 3i, Now try Exercise 47. f to obtain y = x4 β 3x3 + 6x2 + 2x β 60 80 β80 β4 FIGURE 2.33 5 2 You can see that and 3 appear to be zeros of the graph of the function. Use the zero or root feature or the zoom and trace features of the and graphing utility to confirm that x 3 are zeros of the ... |
4 as x 2 4 x 4x 4 x 2ix 2i you obtain f x x 1x 1x 2x 2ix 2i (1, 0) 2 4 x which gives the following five zeros of f. x 1, x 1, x 2, x 2i, and x 2i From the graph of only ones that appear as -intercepts. Note that shown in Figure 2.34, you can see that the real zeros are the x 1 is a repeated zero. x f Now try Exercise ... |
k x 3 3x 2 k should be has two variations x3 3x 2 x 1x 1x 2 you can see that the two positive real zeros are x 1 of multiplicity 2. Example 9 Using Descartesβs Rule of Signs Describe the possible real zeros of f x 3x3 5x2 6x 4. Solution The original polynomial has three variations in sign. to to f x 3x3 5x2 6x 4 to Th... |
x 2 3x 2. Solution The possible real zeros are as follows. Factors of 2 Factors of 6 Β±1, Β±2 Β±1, Β±2, Β±3, Β±6 f x Β±12 The original polynomial has three variations in sign. The polynomial f x 6x3 4x2 3x 2 6x3 4x2 3x 2 has no variations in sign. As a result of these two findings, you can apply Descartesβs Rule of Signs to c... |
βs square base. What should the dimensions of your candle mold be? V 1 x2 B 3 Bh, where and the height is is the x 2. So, the volume of the Substituting 25 for the volume yields the following. h is the area of the base and 3 x2x 2. Solution The volume of a pyramid is height. The area of the base is V 1 pyramid is 25 1 ... |
used to determine the possible numbers of positive real zeros and negative real zeros of a function is called ________ ________ of ________. 7. A real number b bound if no real zeros are greater than is a(n) ________ bound for the real zeros of b. f if no real zeros are less than b, and is a(n) ________ PREREQUISITE S... |
. 23. 24. z4 z3 2z 4 0 x 4 13x 2 12x 0 2y4 7y 3 26y 2 23y 6 0 x5 x4 3x3 5x 2 2x 0 f, In Exercises 25β28, (a) list the possible rational zeros of (b) sketch the graph of so that some of the possible zeros in part (a) can be disregarded, and then (c) determine all real zeros of f. f 25. 26. 27. 28. f x x3 x 2 4x 4 f x 3x... |
(b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. 43. 44. f x x 4 6x 2 27 f x x 4 2x 3 3x 2 12x 18 (Hint: One factor is x 2 6. ) 45. 46. f x x 4 4x 3 5x 2 2x 6 x 2 2x 2. (Hint: One factor is ) f x x 4 3x 3 x 2 12x 20 (Hint: One factor is x 2 4.... |
is an extended list of possible rational zeros, use a graphing utility to graph the function in order to discard any rational zeros that are obviously not zeros of the function. 73. 74. 75. 76. f x x3 24x 2 214x 740 f s 2s3 5s2 12s 5 f x 16x 3 20x 2 4x 15 f x 9x 3 15x 2 11x 5 f x 2x4 5x 3 4x 2 5x 2 77. 78. gx x 5 8x 4... |
2 x 1 6 z2 1 2z 1 4x4 25x 2 36 2x33x 2 23x 12 4x3 x 2 4x 1 6z311z2 3z 2 3 4 In Exercises 99β102, match the cubic function with the numbers of rational and irrational zeros. (a) Rational zeros: 0; irrational zeros: 1 (b) Rational zeros: 3; irrational zeros: 0 (c) Rational zeros: 1; irrational zeros: 2 (d) Rational zero... |
2,500,000. 106. Advertising Cost A company that manufactures bicy(in dollars) for selling a P cles estimates that the profit particular model is given by P 45x 3 2500x 2 275,000, 0 β€ x β€ 50 x is the advertising expense (in tens of thousands where of dollars). Using this model, find the smaller of two advertising amount... |
of 64 feet. Is this possible? p 111. Profit The demand equation for a certain product is where is the unit price (in dollars) is the number of units produced and is is the total cost (in dollars) is the number of units produced. The total profit p 140 0.0001x, x of the product and sold. The cost equation C 80x 150,000... |
12/7/05 9:36 AM Page 183 117. 119. gx f x 5 gx 3 f x 118. 120. gx f 2x gx f x 121. Exploration Use a graphing utility to graph the funcf x x 4 4x 2 k for different values of tion given by k. satisfy the specified characteristics. (Some parts do not have unique answers.) such that the zeros of Find values of k f (a) Fo... |
zeros. Assume that (with integer coefficients) b is a positive that has integer. 128. 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) (... |
on their graphical behavior near the -values excluded from the domain. x x Example 1 Finding the Domain of a Rational Function Find the domain of x -values. f x 1 x and discuss the behavior of f near any excluded Solution Because the denominator is zero when except f x to the left and right of x 0, x 0. is all real nu... |
ymptote of the graph of f if a, y b as x 2. The line f x as x b x or. Eventually (as ), the distance between the horizontal asymptote and the points on the graph must approach zero. Figure 2.38 shows the horizontal and vertical asymptotes of the graphs of three rational functions. x x or f(x) = 2x + 1 x + 1 Vertical as... |
Solution a. For this rational function, the degree of the numerator is equal to the degree of the denominator. The leading coefficient of the numerator is 2 and the leadas a ing coefficient of the denominator is 1, so the graph has the line horizontal asymptote. To find any vertical asymptotes, set the denominator equ... |
f 0. 3. Find the zeros of the numerator (if any) by solving the equation Then plot the corresponding -intercepts. Nx 0. x 4. Find the zeros of the denominator (if any) by solving the equation Dx 0. Then sketch the corresponding vertical asymptotes. 5. Find and sketch the horizontal asymptote (if any) by using the rule... |
in Example 3 is a vertical stretch and a right shift of the graph of because f x 1x g gx 3 x 2 3 1 x 2 3f x 2. Horizontal asymptote: y = 0 y 4 2 β2 β4 g(x Vertical asymptote: x = 2 FIGURE 2.40 y 3 2 1 β1 Horizontal asymptote2 f x( ) = 2 x β x 1 β1 β4 β3 β2 Vertical asymptote: = 0 x Example 3 Sketching the Graph of a R... |
oring the denominator, you have f x x x 1x 2. because f 0 0 y-intercept: x-intercept: Vertical asymptotes: Horizontal asymptote: Additional points: 0, 0, 0, 0 x 1, y 0, x 2, zeros of denominator because degree of Nx < degree of Dx Test interval, 1 1, 0 0, 2 2, Representative x-value Value of f Sign 3 0.5 1 3 f 3 0.3 f ... |
2, 1 2, 1.67 The graph is shown in Figure 2.43. Notice that there is a hole in the graph at x 3 because the function is not defined when x 3. FIGURE 2.43 HOLE AT x 3 Now try Exercise 41. 333202_0206.qxd 12/7/05 9:56 AM Page 190 190 Chapter 2 Polynomial and Rational Functions f x Vertical asymptote: x β = 1 β 8 β6 β4 2... |
ptote3 β2 11 3 4 5 x β2 β3 Vertical asymptote: x = 1 f(x) = x2 β x β 2 x β 1 FIGURE 2.45 Now try Exercise 61. 333202_0206.qxd 12/7/05 9:56 AM Page 191 Section 2.6 Rational Functions 191 Applications There are many examples of asymptotic behavior in real life. For instance, Example 8 shows how a vertical asymptote can b... |
? 11 2 y 11 in. 2 1 in. x 1 in. 11 in. 2 FIGURE 2.47 Graphical Solution Let be the area to be minimized. From Figure 2.47, you can write A Numerical Solution A Let be the area to be minimized. From Figure 2.47, you can write A x 3y 2. A x 3 y 2. The printed area inside the margins is modeled by 48 xy y 48x. To find the... |
the exact value of value is x 62 8.485. 333202_0206.qxd 12/7/05 9:56 AM Page 193 2.6 Exercises Section 2.6 Rational Functions 193 VOCABULARY CHECK: Fill in the blanks. 1. Functions of the form f x NxDx, where Nx and Dx are polynomials and Dx is not the zero polynomial, are called ________ ________. 2. If 3. If x β a a... |
x 6 β8 β6 β4 β2 (c) y (d) y 4 2 β2 β4 y 4 2 β2 x 4 6 β4 β2 4 2 β2 β4 x x 13. 15 14. 16 In Exercises 17β20, find the zeros (if any) of the rational function. 17. 19. gx 18. hx 2 5 x 2 2 20. gx x3 8 x 2 1 333202_0206.qxd 12/7/05 9:56 AM Page 194 194 194 Chapter 2 Chapter 2 Polynomial and Rational Functions Polynomial an... |
the following. In Exercises (a) Determine the domains of and f g. (b) Simplify graph of f f. and find any vertical asymptotes of the (c) Compare the functions by completing the table. (d) Use a graphing utility to graph and f g in the same viewing window. (e) Explain why the graphing utility may not show the differenc... |
the graph x to determine any -intercepts of the graph of the rational and solve the resulting equation function and (b) set to confirm your result in part (a). y 0 69. y x 1 x 3 y 70. y 2x x 3 y 6 4 2 β2 β4 x 4 6 8 6 4 2 β2 β4 x 2 4 6 8 71. y 1 x x 724 β2 β4 y 8 4 x 4 β8 β4 β4 x 4 8 73. Pollution The cost (in millions... |
the physical constraints of the problem. (c) Sketch a graph of the concentration function. (d) As the tank is filled, what happens to the rate at which the concentration of brine is increasing? What percent does the concentration of brine appear to approach? 333202_0206.qxd 12/7/05 9:56 AM Page 196 196 196 Chapter 2 C... |
ptotes. 82. The graph of a rational function can never cross one of its asymptotes. Think About It In Exercises 83 and 84, write a rational function that has the specified characteristics. (There are many correct answers.) f 83. Vertical asymptote: None 84. Vertical asymptote: Horizontal asymptote: y 2 x 2, Horizontal ... |
nomial Inequalities x2 2x 3 < 0, x To solve a polynomial inequality such as you can use the fact that a polynomial can change signs only at its zeros (the -values that make the polynomial equal to zero). Between two consecutive zeros, a polynomial must be entirely positive or entirely negative. This means that when the... |
207.qxd 12/7/05 9:40 AM Page 198 198 Chapter 2 Polynomial and Rational Functions Example 1 Solving a Polynomial Inequality Solve x2 x 6 < 0. Solution By factoring the polynomial as x2 x 6 x 2x 3 you can see that the critical numbers are polynomialβs test intervals are x 2 and x 3. So, the, 2, 2, 3, and 3,. Test interva... |
on one side and zero on the other). Whenever this is not the case, you should begin the solution process by writing the inequality in general form. Example 2 Solving a Polynomial Inequality Solve 2x3 3x2 32x > 48. Solution Begin by writing the inequality in general form. 2x 3 3x2 32x > 48 Write original inequality. 2x... |
3x2 32x β₯ 48 the solution would have consisted of the closed interval 4,. 4, 3 2 and the interval 333202_0207.qxd 12/7/05 9:40 AM Page 200 200 Chapter 2 Polynomial and Rational Functions Each of the polynomial inequalities in Examples 1 and 2 has a solution set that consists of a single interval or the union of two in... |
for which its denominator is zero). These two types of numbers make up the critical numbers of a rational inequality. When solving a rational inequality, begin by writing the inequality in general form with the rational expression on the left and zero on the right. x x Example 4 Solving a Rational Inequality Solve 2x ... |
of producing calculator plus a development cost of $2,500,000. So, the total cost is x calculators is $10 per C 10x 2,500,000. Cost equation What price should the company charge per calculator to obtain a profit of at least $190,000,000? Solution Verbal Model: Equation: Profit Revenue Cost P R C P 100x 0.00001x2 10x 2... |
interval 4, 4. is the interval 4, 4. So, the domain of the expression 64 4x2 Graphical Solution Begin by sketching the graph of the equation y 64 4x2, as shown in Figure 2.58. From the graph, you can determine that the -values 4 and 4). So, extend from to 4 (including the domain of the expression is the interval 4 64 ... |
4. x 2 x 4 β₯ 3 3x2 x2 4 < 1 Values (a) (c) (a) (c) (a) (c) (a) (cb) (d) (b) (d) (b) (d) (b) (d In Exercises 5β8, find the critical numbers of the expression. 5. 7. 2x2 x 6 2 3 x 5 6. 8. 9x3 25x 2 2 x x 2 x 1 27. 29. 31. 4x3 6x2 < 0 x3 4x β₯ 0 x 12x 23 β₯ 0 28. 30. 32. 4x3 12x 2 > 0 2x3 x 4 β€ 0 x4x 3 β€ 0 Graphical Analys... |
x 5 1 x 3 x2 2x x2 9 x2 x 6 x > β€ β€ 0 1 2x 3 9 4x 3 β₯ 0 5 x 1 3x x 1 2x x 1 x x 4 β€ < 1 3 333202_0207.qxd 12/7/05 9:40 AM Page 205 Graphical Analysis In Exercises 51β54, use a graphing utility to graph the equation. Use the graph to approximate that satisfy each inequality. the values of x 71. Cost, Revenue, and Profi... |
Geometry A rectangular playing field with a perimeter of 100 meters is to have an area of at least 500 square meters. Within what bounds must the length of the rectangle lie? 70. Geometry A rectangular parking lot with a perimeter of 440 feet is to have an area of at least 8000 square feet. Within what bounds must the... |
d 6, d 8, d 10, and d 12. Use the results to create a bar graph. (b) Determine the minimum depth of the beam that will safely support a load of 2000 pounds. 75. Resistors When two resistors of resistances R1 and R2 connected in parallel (see figure), the total resistance satisfies the equation are R Synthesis True or ... |
4 54x t 14 16 18 In Exercises 89 and 90, write an expression for the area of the region. 8 10 6 Year (0 β 1990) 12 (a) According to the model, during what year did the number of masterβs degrees earned by women exceed 220,000? (b) Use the graph to verify the result of part (a). (c) According to the model, during what ... |
the Upper and Lower Bound Rules (p. 177), to find zeros of polynomials. Section 2.6 Find the domains of rational functions (p. 184). Find the horizontal and vertical asymptotes of graphs of rational functions (p. 185). Analyze and sketch graphs of rational functions (p. 187). Sketch graphs of rational functions that h... |
18, write the standard form of the equation of the parabola that has the indicated vertex and whose graph passes through the given point. 15. y 16. y 2 β2 β4 β6 17. Vertex: 18. Vertex: (4, 1) 4 (2, β1) x 8 6 2 (0, 3) (2, 2) β2 2 4 6 x 1, 4; 2, 3; point: 2, 3 point: 1, 6 19. Geometry The perimeter of a rectangle is 200 ... |
x x 35 f x 1 2x5 3 333202_020R.qxd 12/7/05 9:43 AM Page 209 In Exercises 29β32, describe the right-hand and left-hand behavior of the graph of the polynomial function. 29. 30. 31. 32. f x x 2 6x 9 f x 1 2 x3 2x x4 3x 2 2 gx 3 4 hx x5 7x 2 10x In Exercises 33β38, find all the real zeros of the polynomial function. Dete... |
13x 2 5x 2 2x2 1 Review Exercises 209 In Exercises 53β56, use synthetic division to divide. 53. 54. 55. 56. 6x4 4x3 27x 2 18x x 2 0.1x3 0.3x 2 0.5 x 5 2x3 19x 2 38x 24 x 4 3x3 20x2 29x 12 x 3 In Exercises 57 and 58, use synthetic division to determine x whether the given values of are zeros of the function. 57. 58. x ... |
R.qxd 12/7/05 9:43 AM Page 210 210 Chapter 2 Polynomial and Rational Functions In Exercises 75 and 76, write the quotient in standard form. 75. 6 i 4 i 76. 3 2i 5 i In Exercises 77 and 78, perform the operation and write the result in standard form. 77. 4 2 3i 2 1 i 78. 1 2 i 5 1 4i In Exercises 79β 82, find all soluti... |
eros of the function and write the polynomial as a product of linear factors. 103. 104. 105. 106. f x x3 4x2 5x gx x3 7x2 36 gx x4 4x3 3x2 40x 208 f x x4 8x3 8x2 72x 153 In Exercises 107 and 108, use Descartesβs Rule of Signs to determine the possible numbers of positive and negative zeros of the function. 107. 108. gx... |
131β134, (a) state the domain of the function, (b) identify all intercepts, (c) identify any vertical and slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. f x x2 1 x 1 f x 2x3 x2 1 131. 132. 133. 134. f x 3x3 2x2 3x 2 3x2 x 4 f x 3x3 4x2 12x 16 3x2 5x 2 ... |
amount of uptake. CO2 2.7 In Exercises 139β146, solve the inequality. 139. 141. 143. 145. 6x2 5x < 4 x3 16x β₯ 0 3 2 x 1 x 1 x2 7x 12 x β€ β₯ 0 140. 142. 144. 146. 2x2 x β₯ 15 12x3 20x2 < > 147. Investment P dollars invested at interest rate r compounded annually increases to an amount A P1 r2 in 2 years. An investment of... |
y is the height (in feet) of is the horizontal distance (in feet) from where the ball was thrown. (a) Find the maximum height of the ball. (b) Which number determines the height at which the ball was thrown? Does changing this value change the coordinates of the maximum height of the ball? Explain. 4. Determine the ri... |
by x k, f x the remainder is Proof From the Division Algorithm, you have f x x kqx rx and because either rx you know that you have at x k, rx 0 must be a constant. That is, or the degree of rx is less than the degree of rx r. Now, by evaluating x k, f x f k k kqk r 0qk r r. To be successful in algebra, it is important... |
Fundamental Theorem of Algebra is Carl Friedrich Gauss, who published the proof in his doctoral thesis in 1799. f x Linear Factorization Theorem (p. 169) n > 0, is a polynomial of degree where If linear factors f x an n, x c1 c1, c2,..., cn... x cn x c2 are complex numbers. where then has precisely f n Proof Using the... |
, then using long division. In where other words, verify the Remainder Theorem for a thirddegree polynomial function. 2. In 2000 B.C., the Babylonians solved polynomial equations by referring to tables of values. One such table gave the values of To be able to use this table, the Babylonians sometimes had to manipulate... |
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