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All Textbook Solutions for Precalculus: Mathematics for Calculus (Standalone Book)

69E 70E 71E 72E 73E 74E 75E Families of Polynomials Graph the family of polynomials in the same viewing rectangle, using the given values of c. Explain how changing the value of c affects the graph. 76. P(x) = x3 + cx; c = 2, 0, 2, 4 77E 78E 79E Power Functions Portions of the graphs of y = x2, y = x3, y = x4, y = x5, and y = x6 are plotted in the figures. Determine which function belongs to each graph.81E 82E 83E Local Extrema These exercises involve local maxima and minima of polynomial functions. 84. (a) Graph the function P(x) = (x 2)(x 4)(x 5) and determine how many local extrema it has. (b) If a b c, explain why the function P(x)=(xa)(xb)(xc)must have two local extrema.Local Extrema These exercises involve local maxima and minima of polynomial functions. 85. Maximum Number of Local Extrema What is the smallest possible degree that the polynomial whose graph is shown can have? Explain.86E Market Research A market analyst working for a small-appliance manufacturer finds that if the firm produces and sells x blenders annually, the total profit (in dollars) is P(x)=8x+0.3x20.0013x3372 Graph the function P in an appropriate viewing rectangle and use the graph to answer the following questions. (a) When just a few blenders are manufactured, the firm loses money (profit is negative). (For example, P(10) = 263.3, so the firm loses 263.30 if it produces and sells only 10 blenders.) How many blenders must the firm produce to break even? (b) Does profit increase indefinitely as more blenders are produced and sold? If not, what is the largest possible profit the firm could have?Population Change The rabbit population on a small island is observed to be given by the function P(t)=120t0.4t4+1000 where t is the time (in months) since observations of the island began. (a) When is the maximum population attained, and what is that maximum population? (b) When does the rabbit population disappear from the island?Volume of a Box An open box is to be constructed from a piece of cardboard 20 cm by 40 cm by cutting squares of side length x from each comer and folding up the sides, as shown in the figure. (a) Express the volume V of the box as a function of x. (b) What is the domain of V? (Use the fact that length and volume must be positive.) (c) Draw a graph of the function V, and use it to estimate the maximum volume for such a box.Volume of a Box A cardboard box has a square base, with each edge of the base having length x inches, as shown in the figure. The total length of all 12 edges of the box is 144 in. (a) Show that the volume of the box is given by the function V(x)=2x2(18x). (b) What is the domain of V? (Use the fact that length and volume must be positive.) (c) Draw a graph of the function V and use it to estimate the maximum volume for such a box.91E DISCUSS DISCOVER: Possible Number of Local Extrema Is it possible for a third-degree polynomial to have exactly one local extremum? Can a fourth-degree polynomial have exactly two local extrema? How many local extrema can polynomials of third, fourth, fifth, and sixth degree have? (Think about the end behavior of such polynomials.) Now give an example of a polynomial that has six local extrema.If we divide the polynomial P by the factor x c and we obtain the equation P(x) = (x c)Q(x) + R(x), then we say that x c is the divisor, Q(x) is the __________, and R(x) is the __________.(a) If we divide the polynomial P(x) by the factor x c and we obtain a remainder of 0, then we know that c is a __________ of P. (b) If we divide the polynomial P(x) by the factor x c and we obtain a remainder of k, then we know that P(c) = __________.Division of Polynomials Two polynomials P and D are given. Use either synthetic or long division to divide P(x) by D(x), and express the quotient P(x)/D(x) in the form P(x)D(x)=Q(x)+R(x)D(x) 3. P(x) = 2x2 5x 7, D(x) = x 2 Division of Polynomials Two polynomials P and D are given. Use either synthetic or long division to divide P(x) by D(x), and express the quotient P(x)/D(x) in the form P(x)D(x)=Q(x)+R(x)D(x) 4. P(x) = 3x3 + 9x2 5x 1, D(x) = x + 4 Division of Polynomials Two polynomials P and D are given. Use either synthetic or long division to divide P(x) by D(x), and express the quotient P(x)/D(x) in the form P(x)D(x)=Q(x)+R(x)D(x) 5. P(x) = 4x2 3x 7, D(x) = 2x 1 Division of Polynomials Two polynomials P and D are given. Use either synthetic or long division to divide P(x) by D(x), and express the quotient P(x)/D(x) in the form P(x)D(x)=Q(x)+R(x)D(x) 6. P(x) = 6x3 + x2 12x + 5, D(x) = 3x 4 Division of Polynomials Two polynomials P and D are given. Use either synthetic or long division to divide P(x) by D(x), and express the quotient P(x)/D(x) in the form P(x)D(x)=Q(x)+R(x)D(x) 7. P(x) = 2x4 x3 + 9x2, D(x) = x2 + 4 8E 9E 10E 11E 12E 13E 14E 15E 16E 17E 18E 19E 20E 21E 22E 23E 24E 25E 26E 27E 28E 29E 30E 31E 32E 33E 34E 35E 36E 37E 38E 39E 40E 41E 42E 43E 44E 45E 46E 47E Remainder Theorem Use synthetic division and the Remainder Theorem to evaluate P(c). 48. P(x) = 2x6 + 7x5 + 40x4 7x2 + 10x + 112, c = 3 49E 50E 51E 52E 53E Factor Theorem Use the Factor Theorem to show that x c is a factor of P(x) for the given value(s) of c. 54. P(x) = x3 + 2x2 3x 10, c = 2 Factor Theorem Use the Factor Theorem to show that x c is a factor of P(x) for the given value(s) of c. 55. P(x) = 2x3 + 7x2 + 6x 5, c=12 56E 57E 58E 59E 60E Factor Theorem Show that the given value(s) of c are zeros of P(x), and find all other zeros of P(x). 61. P(x) = 3x4 8x3 14x2 + 31x + 6, c = 2, 3 62E Finding a Polynomial with Specified Zeros Find a polynomial of the specified degree that has the given zeros. 63. Degree 3; zeros 1, 1, 3 Finding a Polynomial with Specified Zeros Find a polynomial of the specified degree that has the given zeros. 64. Degree 4; zeros 2, 0, 2, 4 Finding a Polynomial with Specified Zeros Find a polynomial of the specified degree that has the given zeros. 65. Degree 4; zeros 1, 1, 3, 5 66E Polynomials with Specified Zeros Find a polynomial of the specified degree that satisfies the given conditions. 67. Degree 4; zeros2, 0, 1, 3; coefficient of x3 is 4 Polynomials with Specified Zeros Find a polynomial of the specified degree that satisfies the given conditions. 68. Degree 4; zeros1, 0, 2, 12; coefficient of x3 is 3 Polynomials with Specified Zeros Find a polynomial of the specified degree that satisfies the given conditions. 69. Degree 4; zeros1, 1, 2; integer coefficients and constant term 6 70E Finding a Polynomial from a Graph Find the polynomial of the specified degree whose graph is shown. 71. Degree 3 Finding a Polynomial from a Graph Find the polynomial of the specified degree whose graph is shown. 72. Degree 3 Finding a Polynomial from a Graph Find the polynomial of the specified degree whose graph is shown. 73. Degree 4 74E DISCUSS: Impossible Division? Suppose you were asked to solve the following two problems on a test: A. Find the remainder when 6x1000 17x562 + 12x + 26 is divided by x + 1. B. Is x 1 a factor of x567 3x400 + x9 + 2? Obviously, its impossible to solve these problems by dividing, because the polynomials are of such large degree. Use one or more of the theorems in this section to solve these problems without actually dividing.76E If the polynomial function P(x)=anxn+an1xn1++a1x+a0 has integer coefficients, then the only numbers that could possibly be rational zeros of P are all of the form pq, where p is a factor of __________ and q is a factor of __________ The possible rational zeros of P(x) = 6x3 + 5x2 19x 10 are _______________.Using Descartes Rule of Signs, we can tell that the polynomial P(x) = x5 3x4 + 2x3 x2 + 8x 8 has __________, __________, or __________ positive real zeros and __________ negative real zeros.True or False? If c is a real zero of the polynomial P, then all the other zeros of P are zeros of P(x)/(x c).True or False? If a is an upper bound for the real zeros of the polynomial P, then a is necessarily a lower bound for the real zeros of P.5E 6E Possible Rational Zeros List all possible rational zeros given by the Rational Zeros Theorem (but dont check to see which actually are zeros). 7. R(x) = 2x5 + 3x3 + 4x2 8 8E 9E 10E 11E 12E Possible Rational Zeros A polynomial function P and its graph are given. (a) List all possible rational zeros of P given by the Rational Zeros Theorem. (b) From the graph, determine which of the possible rational zeros actually turn out to be zeros. 13. P(x) = 2x4 9x3 + 9x2 + x 3 Possible Rational Zeros A polynomial function P and its graph are given. (a) List all possible rational zeros of P given by the Rational Zeros Theorem. (b) From the graph, determine which of the possible rational zeros actually turn out to be zeros. 14. P(x) = 4x4 x3 4x + 1 15E Integer Zeros All the real zeros of the given polynomial are integers. Find the zeros, and write the polynomial in factored form. 16. P(x) = x3 4x2 19x 14 17E 18E 19E Integer Zeros All the real zeros of the given polynomial are integers. Find the zeros, and write the polynomial in factored form. 20. P(x) = x3 + 12x2 + 48x + 64 21E 22E 23E 24E 25E 26E 27E 28E 29E 30E 31E 32E 33E 34E 35E 36E 37E 38E 39E 40E 41E 42E 43E 44E 45E 46E Real Zeros of a Polynomial Find all the real zeros of the polynomial. Use the Quadratic Formula if necessary, as in Example 3(a). 47. P(x) = x4 6x3 + 4x2 + 15x + 4 Real Zeros of a Polynomial Find all the real zeros of the polynomial. Use the Quadratic Formula if necessary, as in Example 3(a). 48. P(x) = x4 + 2x3 2x2 3x + 2 49E 50E 51E 52E 53E 54E 55E 56E 57E 58E 59E 60E 61E 62E Descartes Rule of Signs Use Descartes Rule of Signs to determine how many positive and how many negative real zeros the polynomial can have. Then determine the possible total number of real zeros. 63. P(x) = x3 x2 x 3 Descartes Rule of Signs Use Descartes Rule of Signs to determine how many positive and how many negative real zeros the polynomial can have. Then determine the possible total number of real zeros. 64. P(x) = 2x3 x2 + 4x 7 65E 66E 67E 68E 69E 70E 71E 72E 73E 74E 75E 76E Upper and Lower Bounds Find integers that are upper and lower bounds for the real zeros of the polynomial. 77. P(x) = x3 3x2 + 4 78E 79E 80E 81E 82E 83E 84E 85E 86E 87E 88E 89E Polynomials With No Rational Zeros Show that the polynomial does not have any rational zeros. 90. P(x) = x50 5x25 + x2 1 91E 92E 93E 94E 95E 96E 97E 98E Volume of a Silo A grain silo consists of a cylindrical main section and a hemispherical roof. If the total volume of the silo (including the part inside the roof section) is 15,000 ft3 and the cylindrical part is 30 ft tall, what is the radius of the silo, rounded to the nearest tenth of a foot?Dimensions of a Lot A rectangular parcel of land has an area of 5000 ft2. A diagonal between opposite comers is measured to be 10 ft longer than one side of the parcel. What are the dimensions of the land, rounded to the nearest foot?Depth of Snowfall Snow began falling at noon on Sunday. The amount of snow on the ground at a certain location at time t was given by the function h(t)=11.60t12.41t2+6.20t31.58t4+0.20t50.01t6 where f is measured in days from the start of the snowfall and h(t) is the depth of snow in inches. Draw a graph of this function, and use your graph to answer the following questions. (a) What happened shortly after noon on Tuesday? (b) Was there ever more than 5 in. of snow on the ground? If so, on what day(s)? (c) On what day and at what time (to the nearest hour) did the snow disappear completely?Volume of a Box An open box with a volume of 1500 cm3 is to be constructed by taking a piece of cardboard 20 cm by 40 cm, cutting squares of side length x cm from each corner, and folding up the sides. Show that this can be done in two different ways, and find the exact dimensions of the box in each case.Volume of a Rocket A rocket consists of a right circular cylinder of height 20 m surmounted by a cone whose height and diameter are equal and whose radius is the same as that of the cylindrical section. What should this radius be (rounded to two decimal places) if the total volume is to be 500/3 m3?Volume of a Box A rectangular box with a volume of 22ft3 has a square base as shown below. The diagonal of the box (between a pair of opposite comers) is 1 ft longer than each side of the base. (a) If the base has sides of length x feet, show that x62x5x4+8=0 (b) Show that two different boxes satisfy the given conditions. Find the dimensions in each case, rounded to the nearest hundredth of a foot.Girth of a Box A box with a square base has length plus girth of 108 in. (Girth is the distance around the box.) What is the length of the box if its volume is 2200 in3?DISCUSS DISCOVER: How Many Real Zeros Can a Polynomial Have? Give examples of polynomials that have the following properties, or explain why it is impossible to find such a polynomial. (a) A polynomial of degree 3 that has no real zeros (b) A polynomial of degree 4 that has no real zeros (c) A polynomial of degree 3 that has three real zeros, only one of which is rational (d) A polynomial of degree 4 that has four real zeros, none of which is rational (a) What must be true about the degree of a polynomial with integer coefficients if it has no real zeros?107E 108E PROVE: Upper and Lower Bounds Theorem Let P(x) be a polynomial with real coefficients, and let b 0. Use the Division Algorithm to write P(x)=(xb)Q(x)+r Suppose that r 0 and that all the coefficients in Q(x) are nonnegative. Let z b. (a) Show that P(z) 0. (b) Prove the first part of the Upper and Lower Bounds Theorem. (c) Use the first part of the Upper and Lower Bounds Theorem to prove the second part. [Hint: Show that if P(x) satisfies the second part of the theorem, then P(x) satisfies the first part.]110E The polynomial P(x) = 5x2(x 4)3(x + 7) has degree ________. It has zeros 0, 4, and ________. The zero 0 has multiplicity _______, and the zero 4 has multiplicity _______.(a) If a is a zero of the polynomial P, then _________ must be a factor of P(x). (b) If a is a zero of multiplicity m of the polynomial P, then __________ must be a factor of P(x) when we factor P completely.A polynomial of degree n 1 has exactly ________ zeros if a zero of multiplicity m is counted m times.If the polynomial function P has real coefficients and if a + bi is a zero of P, then ________ is also a zero of P. So if 3 + i is a zero of P, then _______is also a zero of P.True or False? If False, give a reason. 5. Let P(x) = x4 + 1. (a) The polynomial P has four complex zeros. (b) The polynomial P can be factored into linear factors with complex coefficients. (c) Some of the zeros of P are real.True or False? If False, give a reason. 6. Let P(x) = x3 + x. (a) The polynomial P has three real zeros. (b) The polynomial P has at least one real zero. (c) The polynomial P can be factored into linear factors with real coefficients.Complete Factorization A polynomial P is given. (a) Find all zeros of P, real and complex. (b) Factor P completely. 7. P(x) = x4 + 4x2 8E 9E 10E 11E 12E 13E 14E 15E 16E 17E 18E 19E 20E 21E 22E 23E 24E 25E 26E 27E 28E 29E 30E 31E 32E 33E 34E 35E Complete Factorization Factor the polynomial completely, and find all its zeros. State the multiplicity of each zero. 36. P(x) = x6 + 16x3 + 64 37E 38E 39E 40E 41E Finding a Polynomial with Specified Zeros Find a polynomial with integer coefficients that satisfies the given conditions. 42. Q has degree 3 and zeros 3 and 1 + i.Finding a Polynomial with Specified Zeros Find a polynomial with integer coefficients that satisfies the given conditions. 43. R has degree 4 and zeros 1 2i and 1, with 1 a zero of multiplicity 2.44E 45E 46E 47E 48E 49E 50E 51E 52E 53E 54E 55E 56E 57E 58E 59E 60E 61E Finding Complex Zeros Find all zeros of the polynomial. 62. P(x) = 4x4 + 2x3 2x2 3x 1 Finding Complex Zeros Find all zeros of the polynomial. 63. P(x) = x5 3x4 + 12x3 28x2 + 27x 9 64E 65E 66E 67E 68E 69E 70E 71E 72E (a) Show that 2i and 1 i are both solutions of the equation x2(1+i)x+(2+2i)=0but that their complex conjugates 2i and 1 + i are not. (b) Explain why the result of part (a) does not violate the Conjugate Zeros Theorem.(a) Find the polynomial with real coefficients of the smallest possible degree for which i and 1 + i are zeros and in which the coefficient of the highest power is 1. (b) Find the polynomial with complex coefficients of the smallest possible degree for which i and 1 + i are zeros and in which the coefficient of the highest power is 1.DISCUSS: Polynomials of Odd Degree The Conjugate Zeros Theorem says that the complex zeros of a polynomial with real coefficients occur in complex conjugate pairs. Explain how this fact proves that a polynomial with real coefficients and odd degree has at least one real zero.76E If the rational function y = r(x) has the vertical asymptote x = 2, then as x 2+, either y _____ or y _____.If the rational function y = r(x) has the horizontal asymptote y = 2, then y _____ as x .3E 4E 5E 6E 7E True or False? 8. The graph of a rational function may cross a horizontal asymptote.9E Table of Values A rational function is given. (a) Complete each table for the function. (b) Describe the behavior of the function near its vertical asymptote, based on Tables 1 and 2. (c) Determine the horizontal asymptote, based on Tables 3 and 4. TABLE 1 x r(x) 1.5 1.9 1.99 1.999 TABLE 2 x r(x) 2.5 2.1 2.01 2.001 TABLE 3 x r(x) 10 50 100 1000 TABLE 4 x r(x) 10 50 100 1000 10. r(x)=4x+1x2 11E 12E Graphing Rational Functions Using Transformations Use transformations of the graph of y = 1/x to graph the rational function, and state the domain and range, as in Example 2. 13. r(x)=1x1 14E

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