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All Textbook Solutions for College Algebra

For the following exercises, use the vertex of the graph of the quadratic function and the direction the graph opensto find the domain and range of the function. 59. Vertex (100,100), opens up.For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function. Contains (1,1) and has shape of f(x)=2x2. Vertex is on the y-axis.For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function. Contains (1,4) and has the shape of f(x)=2x2. Vertex is on the y-axis.For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function. Contains (2,3) and has the shape of f(x)=3x2. Vertex is on the y-axis.For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function. Contains (1,3) and has the shape of f(x)=x2. Vertex is on the y-axis.For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function. Contains (4,3) and has the shape of f(x)=5x2. Vertex is on the y-axis.For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function. Contains (1,6) has the shape of f(x)=3x2. Vertex has x-coordinate of 1.Find the dimensions of the rectangular corralproducing the greatest enclosed area given 200 feetof fencing.Find the dimensions of the rectangular corral splitinto 2 pens of the same size producing the greatestpossible enclosed area given 300 feet of fencing.Find the dimensions of the rectangular corralproducing the greatest enclosed area split into3 pens of the same size given 500 feet of fencing.Among all of the pairs of numbers whose sum is 6,find the pair with the largest product. What is theproduct?Among all of the pairs of numbers whose differenceis 12, find the pair with the smallest product. Whatis the product?Suppose that the price per unit in dollars ofa cell phone production is modeled by p=450.0125x, where x is in thousands ofphones produced, and the revenue represented by thousands of dollars is R=xp. Find theproduction level that will maximize revenue.A rocket is launched in the air. Its height, in metersabove sea level, as a function of time, in seconds,is given by h(t)=4.9t2+229t+234. Find themaximum height the rocket attains.A ball is thrown in the air from the top of a building.Its height, in meters above ground, as a function of time, in seconds, is given by h(t)=4.9t2+24t+8. How long does it take to reach maximum height?A soccer stadium holds 62,000 spectators. With aticket price of$11, the average attendance has been26,000. When the price dropped to $9, the averageattendance rose to 31,000. Assuming that attendanceis linearly related to ticket price, what ticket pricewould maximize revenue?A farmer finds that if she plants 75 trees per acre,each tree will yield 20 bushels of fruit. She estimatesthat for each additional tree planted per acre, theyield of each tree will decrease by 3 bushels. Howmany trees should she plant per acre to maximizeher harvest?Which functions are power functions? f(x)=2x24x3g(x)=x5+5x3h(x)=2x513x2+4 Describe in words and symbols the and behavior of f(x)=5x4.Identify the degree, leading term, and leading coefficient of the polynomial f(x)=4x2x6+2x6.Describe the end behavior, and determine a possible degree of thepolynomial function in Figure 8.Given the function f(x)=0.2(x2)(x+1)(x5), express the function as a polynomial in general form anddetermine the leading term. degree, and end behavior of the function.Given the polynomial function f(x)=2x36x220x, determine the y-and x-intercepts.Without graphing the function, determine the maximum number of x-intercepts and turning points for f(x)=10813x98x4+14x12+2x3 What can we conclude about the polynomial represented by the graph shown in Figure 14 based on its interceptsand turning points?.Given the function f(x)=0.2(x2)(x+1)(x5), determine the local behavior.Explain the difference between the coefficientof a power function and its degree.If a polynomial function is in factored form, whatwould be a good ?rst step in order to determine thedegree of the function?In general, explain the end behavior of a powerfunction with odd degree if the leading coefficient ispositive.What is the relationship between the degree of apolynomial function and the maximum numberof turning points in its graph?What can we conclude if, in general, the graph of apolynomial function exhibits the following endbehavior? As x,f(x) and as x,f(x).For the following exercises, identify the function as a power function, a polynomial function, orneither. 6. f(x)=x5 For the following exercises, identify the function as a power function, a polynomial function, or neither. 7. f(x)=(x2)3 For the following exercises, identify the function as a power function, a polynomial function, orneither. 8. f(x)=xx4 For the following exercises, identify the function as a power function, a polynomial function, or neither. 9. f(x)=x2x21 For the following exercises, identify the function as a power function, a polynomial function, orneither. 10. f(x)=2x(x+2)(x1)2 For the following exercises, identify the function as a power function, a polynomial function, or neither. 11. f(x)=3x+1 For the following exercises, find the degree and leading coefficient for the given polynomial. 3x For the following exercises, find the degree and leading coefficient for the given polynomial. 72x2 For the following exercises, find the degree and leading coefficient for the given polynomial. 2x23x5+x6 For the following exercises, find the degree and leading coefficient for the given polynomial. x(4x2)(2x+1)For the following exercises, find the degree and leading coefficient for the given polynomial. x2(2x3)2 For the following exercises, determine the end behavior of the functions. f(x)=x4 For the following exercises, determine the end behavior of the functions. f(x)=x3 For the following exercises, determine the end behavior of the functions. f(x)=x4 For the following exercises, determine the end behavior of the functions. f(x)=x9 For the following exercises, determine the end behavior of the functions. f(x)=2x43x2+x1 For the following exercises, determine the end behavior of the functions. f(x)=3x2+x2 For the following exercises, determine the end behavior of the functions. f(x)=x2(2x3x+1)For the following exercises, determine the end behavior of the functions. f(x)=(2x)7 For the following exercises, find the intercepts of the functions. f(t)=2(t1)(t+2)(t3)For the following exercises, find the intercepts of the functions. g(n)=2(3n1)(2n+1)For the following exercises, find the intercepts of the functions. f(x)=x416 For the following exercises, find the intercepts of the functions. f(x)=x3+27 For the following exercises, find the intercepts of the functions. f(x)=x(x22x8)For the following exercises, find the intercepts of the functions. f(x)=(x+3)(4x21)For the following exercises, determine the least possible degree of the polynomial function shown.For the following exercises, determine the least possible degree of the polynomial function shown.For the following exercises, determine the least possible degree of the polynomial function shown.For the following exercises, determine the least possible degree of the polynomial function shown.For the following exercises, determine the least possible degree of the polynomial function shown.For the following exercises, determine the least possible degree of the polynomial function shown.For the following exercises, determine the least possible degree of the polynomial function shown.For the following exercises, determine the least possible degree of the polynomial function shown.For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.For the following exercises, make a table to confirm the end behavior of the function. 46. f(x)=x3 For the following exercises, make a table to confirm the end behavior of the function. 47. f(x)=x45x2 For the following exercises, make a table to confirm the end behavior of the function. 48. f(x)=x2(1x)2 For the following exercises, make a table to confirm the end behavior of the function. 49. f(x)=(x1)(x2)(3x)For the following exercises, make a table to confirm the end behavior of the function. 50. f(x)=x510x4 For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determinethe intercepts and the end behavior. 51. f(x)=x3(x2)For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determinethe intercepts and the end behavior. 52. f(x)=x(x3)(x+3)For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determinethe intercepts and the end behavior. 53. f(x)=x(142x)(102x)For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determinethe intercepts and the end behavior. 54. f(x)=x(142x)(102x)2 For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determinethe intercepts and the end behavior. 55. f(x)=x316x For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determinethe intercepts and the end behavior. 56. f(x)=x327 For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determinethe intercepts and the end behavior. 57. f(x)=x481 For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determinethe intercepts and the end behavior. 58. f(x)=x3+x2+2x For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determinethe intercepts and the end behavior. 59. f(x)=x32x215x For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determinethe intercepts and the end behavior. 60. f(x)=x30.01x For the following exercises, use the information about the graph of a polynomial function to determine the function.Assume the leading coefficient is 1 or -1. There may be more than one correct answer. 61. The y-intercept is (0,4). The x-intercepts are (2,0),(2,0). Degree is 2. End behavior: as x,f(x), as x,f(x).For the following exercises, use the information about the graph of a polynomial function to determine the function.Assume the leading coefficient is 1 or -1. There may be more than one correct answer. 62. The y-intercept is (0,9). The x-intercepts are (3,0),(3,0). Degree is 2. End behavior: as x,f(x), as x,f(x).For the following exercises, use the information about the graph of a polynomial function todetermine the function.Assume the leading coef?cient is 1 or 71. There may be more than one correct answer. 63. The y-intercept is (0,0). The x-intercepts are (0,0),(2,0). Degree is 3. End behavior: as x,f(x), as x,f(x).For the following exercises, use the information about the graph of a polynomial function to determine the function.Assume the leading coefficient is 1 or -1. There may be more than one correct answer. 64. The y-intercept is (0,1). The x-intercept is (1,0). Degree is 3. End behavior: as x,f(x), as x,f(x).For the following exercises, use the information about the graph of a polynomial function to determine the function.Assume the leading coefficient is 1 or -1. There may be more than one correct answer. 65. The y-intercept is (0,1). There is no x-intercept.Degree is 4. End behavior: as x,f(x), as x,f(x).For the following exercises, use the written statements to construct a polynomial function that represents the requiredinformation. 66. An oil slick is expanding as a circle. The radius ofthe circle is increasing at the rate of 20 meters perday. Express the area of the circle as a function of d,the number of days elapsed.For the following exercises, use the written statements to construct a polynomial function that represents the requiredinformation. 67. A cube has an edge of 3 feet. The edge is increasingat the rate of 2 feet per minute. Express the volumeof the cube as a function of m, the number ofminutes elapsed.For the following exercises, use the written statements to construct a polynomial function that represents the requiredinformation. 68. A rectangle has a length of 10 inches and a width of6 inches. If the length is increased by x inches andthe width increased by twice that amount, expressthe area of the rectangle as a function of x.For the following exercises, use the written statements to construct a polynomial function that represents the requiredinformation. 69. An open box is to be constructed by cutting outsquare corners ofx-inch sides from a piece of cardboard 8 inches by 8 inches and then foldingup the sides. Express the volume of the box as a function of x.For the following exercises, use the written statements to construct a polynomial function that represents the requiredinformation. 70. A rectangle is twice as long as it is wide. Squares ofside 2 feet are cut out from each corner. Then thesides are folded up to make an open box. Express thevolume of the box as a function of the width (x).Find the y-and x-intercepts of the function f(x)=x419x2+30x.Use the graph of the function of degree 5 in Figure 10 to identify the zeros of the function and their multiplicities.Sketch a graph of f(x)=14x(x1)4(x+3)3.Show that the function f(x)=7x59x4x2 has at least one real zero between x=1 and x=2.Given the graph shown in Figure 20, write a formula for the function shown.Use technology to find the maximum and minimum values on the interval [1,4] of the function f(x)=0.2(x2)3(x+1)2(x4).What is the difference between an x-intercept anda zero of a polynomial function f?If a polynomial function of degree n has n distinctzeros, what do you know about the graph of thefunction?Explain how the Intermediate Value Theorem canassist us in finding a zero of a function.Explain how the factored form of the polynomialhelps us in graphing it.If the graph of a polynomial just touches the x-axisand then changes direction, what can we concludeabout the factored form of the polynomial?For the following exercises, find the x-or t-intercepts of the polynomial functions. 6. C(t)=2(t4)(t+1)(t6)For the following exercises, find the x-or t-intercepts of the polynomial functions. 7. C(t)=3(t+2)(t3)(t+5)For the following exercises, find the x-or t-intercepts of the polynomial functions. 8. C(t)=4t(t2)2(t+1)For the following exercises, find the x-or t-intercepts of the polynomial functions. 9. C(t)=2t(t3)(t+1)2 For the following exercises, find the x-or t-intercepts of the polynomial functions. 10. C(t)=2t48t3+6t2 For the following exercises, find the x-or t-intercepts of the polynomial functions. 11. C(t)=4t4+12t340t2 For the following exercises, find the x-or t-intercepts of the polynomial functions. 12. f(x)=x4x2 For the following exercises, find the x-or t-intercepts of the polynomial functions. 13. f(x)=x3+x220x For the following exercises, find the x-or t-intercepts of the polynomial functions. 14. f(x)=x3+6x27x For the following exercises, find the x-or t-intercepts of the polynomial functions. 15. f(x)=x3+x24x4 For the following exercises, find the x-or t-intercepts of the polynomial functions. 16. f(x)=x3+2x29x18 For the following exercises, find the x-or t-intercepts of the polynomial functions. 17. f(x)=2x3x28x+4 For the following exercises, find the x- or t-intercepts of the polynomial functions. 18. f(x)=x67x38 For the following exercises, find the x-or t-intercepts of the polynomial functions. 19. f(x)=2x4+6x28 For the following exercises, find the x-or t-intercepts of the polynomial functions. 20. f(x)=x33x2x+3 For the following exercises, find the x-or t-intercepts of the polynomial functions. 21. f(x)=x62x43x2 For the following exercises, find the x- or t-intercepts of the polynomial functions. 22. f(x)=x63x44x2 For the following exercises, find the x- or t-intercepts of the polynomial functions. 23. f(x)=x55x3+4x For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval. f(x)=x39x, between x=4 and x=2.For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval. f(x)=x39x, between x=2 and x=4.For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval. f(x)=x52x, between x=1 and x=2.For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval. f(x)=x4+4, between x=1 and x=3.For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval. f(x)=2x3x, between x=1 and x=1.For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval. f(x)=x3100x+2, between x=0.01 and x=0.1 For the following exercises, find the zeros and give the multiplicity of each. f(x)=(x+2)3(x3)2 For the following exercises, find the zeros and give the multiplicity of each. f(x)=x2(2x+3)5(x4)2 For the following exercises, find the zeros and give the multiplicity of each. f(x)=x3(x1)3(x+2)For the following exercises, find the zeros and give the multiplicity of each. f(x)=x2(x2+4x+4)For the following exercises, find the zeros and give the multiplicity of each. f(x)=(2x+1)3(9x26x+1)For the following exercises, find the zeros and give the multiplicity of each. f(x)=(3x+2)5(x210x+25)For the following exercises, find the zeros and give the multiplicity of each. f(x)=x(4x212x+9)(x2+8x+16)For the following exercises, find the zeros and give the multiplicity of each. f(x)=x6x52x4 For the following exercises, find the zeros and give the multiplicity of each. f(x)=3x4+6x3+3x2 For the following exercises, find the zeros and give the multiplicity of each. f(x)=4x512x4+9x3 For the following exercises, find the zeros and give the multiplicity of each. f(x)=2x4(x34x2+4x)For the following exercises, find the zeros and give the multiplicity of each. f(x)=4x4(9x412x3+4x2)For the following exercises, graph the polynomial functions. Note x- and y-intercepts, multiplicity, and end behavior. 42. f(x)=(x+3)2(x2)For the following exercises, graph the polynomial functions. Note x- and y-intercepts, multiplicity,and end behavior. 43. g(x)=(x+4)(x1)2 For the following exercises, graph the polynomial functions. Note x- and y-intercepts, multiplicity, and end behavior. 44. h(x)=(x1)3(x+3)2 For the following exercises, graph the polynomial functions. Note x- and y-intercepts, multiplicity,and end behavior. 45. k(x)=(x3)3(x2)2 For the following exercises, graph the polynomial functions. Note x- and y-intercepts, multiplicity, and end behavior. 46. m(x)=2x(x1)(x+3)For the following exercises, graph the polynomial functions. Note x- and y-intercepts, multiplicity, and end behavior. 47. n(x)=3x(x+2)(x4)For the following exercises, use the graphs to write the formula for a polynomial function of least degree.For the following exercises, use the graphs to write the formula for a polynomial function of least degree.For the following exercises, use the graphs to write the formula for a polynomial function of least degree.For the following exercises, use the graphs to write the formula for a polynomial function of least degree.For the following exercises, use the graphs to write the formula for a polynomial function of least degree.For the following exercises, use the graph to identify zeros and multiplicity.For the following exercises, use the graph to identify zeros and multiplicity.For the following exercises, use the graph to identify zeros and multiplicity.For the following exercises, use the graph to identify zeros and multiplicity.For the following exercises, use the given information about the polynomial graph to write the equation. Degree 3. Zeros at x=2,x=1, and x=3.y-intercept at (0,4).For the following exercises, use the given information about the polynomial graph to write the equation. Degree 3. Zeros at x=5,x=2, and x=1.y-intercept at (0,6)For the following exercises, use the given information about the polynomial graph to write the equation. Degree 5. Roots of multiplicity 2 at x=3 and x=1, and a root of multiplicity l at x=3.y-intercept at (0,9)For the following exercises, use the given information about the polynomial graph to write the equation. Degree 4. Root of multiplicity 2 at x=4, and roots of multiplicity l at x=1 and x=2.y-intercept at (0,3).For the following exercises, use the given information about the polynomial graph to write the equation. Degree 5. Double zero at x=1, and triple zero at x=3. Passes through the point (2,15).For the following exercises, use the given information about the polynomial graph to write the equation. Degree 3. Zeros at x=4,x=3, and x=2.y-intercept at (0,24).For the following exercises, use the given information about the polynomial graph to write the equation. Degree 3. Zeros at x=3,x=2 and x=1.y-intercept at (0,12).For the following exercises, use the given information about the polynomial graph to write the equation. 64. Degree 5. Roots of multiplicity 2 at x=3 and x=2 and a root ofmultiplicity 1 at x=2.y-interceptat (0,4).For the following exercises, use the given information about the polynomial graph to write the equation. Degree 4. Roots of multiplicity 2 at x=12 and roots of multiplicity l at x=6 and x=2.y-intercept at (0,18).For the following exercises, use the given information about the polynomial graph to write the equation. Double zero at x=3 and triple zero at x=0. Passes through the point (1,32).For the following exercises, use a calculator to approximate local minima and maxima at the global minimum andmaximum. 67. f(x)=x3x1 For the following exercises, use a calculator to approximate local minima and maxima at the global minimum andmaximum. 68. f(x)=2x33x1 For the following exercises, use a calculator to approximate local minima and maxima at the global minimum andmaximum. 69. f(x)=x4+x For the following exercises, use a calculator to approximate local minima and maxima at the global minimum andmaximum. 70. f(x)=x4+3x2 For the following exercises, use a calculator to approximate local minima and maxima at the global minimum andmaximum. 71. f(x)=x4x3+1 For the following exercises, use the graphs to write a polynomial function of least degree.For the following exercises, use the graphs to write a polynomial function of least degree.For the following exercises, use the graphs to write a polynomial function of least degree.For the following exercises, write the polynomial function that models the given situation. 75. A rectangle has a length of 10 units and a width of8 units. Squares of x by x units are cut out of eachcorner, and then the sides are folded up to createan open box. Express the volume of the box as apolynomial function in terms ofx.For the following exercises, write the polynomial function that models the given situation. 76. Consider the same rectangle of the precedingproblem. Squares of 2x by 2x units are cut out ofeach corner. Express the volume of the box as apolynomial in terms ofx.For the following exercises, write the polynomial function that models the given situation. 77. A square has sides of 12 units. Squares x+1 By x+1 units are cut out of each corner, and then thesides are folded up to create an open box. Expressthe volume of the box as a function in terms ofx.For the following exercises, write the polynomial function that models the given situation. 78. A cylinder has a radius of x+2 units and a height of3 units greater. Express the volume of the cylinder asa polynomial function.For the following exercises, write the polynomial function that models the given situation. 79. A right circular cone has a radius of 3x+6 and aheight 3 units less. Express the volume of the coneas a polynomial function. The volume of a cone is V=13r2h for radius r and height h.Divide 16x312x2+20x3 by 4x+5.Use synthetic division to divide 3x4+18x33x+40 by x+7.The area ofa rectangle is given by 3x3+14x223x+6. The width of the rectangle is given by x+6. Find an expressionfor the length of the rectangle.If division of a polynomial by a binomial resultsin a remainder of zero, what can be conclude?If a polynomial of degree n is divided by a binomialof degree 1, what is the degree of the quotient?For the following exercises, use long division to divide. Specify the quotient and the remainder. 3. (x2+5x1)(x1)For the following exercises, use long division to divide. Specify the quotient and the remainder. 4. (2x29x5)(x5)For the following exercises, use long division to divide. Specify the quotient and the remainder. 5. (3x2+23x+14)(x+7)For the following exercises, use long division to divide. Specify the quotient and the remainder. 6. (4x210x+6)(4x+2)For the following exercises, use long division to divide. Specify the quotient and the remainder. 7. (6x225x25)(6x+5)For the following exercises, use long division to divide. Specify the quotient and the remainder. 8. (x21)(x+1)For the following exercises, use long division to divide. Specify the quotient and the remainder. 9. (2x23x+2)(x+2)For the following exercises, use long division to divide. Specify the quotient and the remainder. 10. (x3126)(x5)For the following exercises, use long division to divide. Specify the quotient and the remainder. 11. (3x25x+4)(3x+1)For the following exercises, use long division to divide. Specify the quotient and the remainder. 12. (x33x2+5x6)(x2)For the following exercises, use long division to divide. Specify the quotient and the remainder. 13. (2x3+3x24x+15)(x+3)For the following exercises, use synthetic division to find the quotient. (3x32x2+x4)(x+3)For the following exercises, use synthetic division to find the quotient. (2x36x27x+6)(x4)For the following exercises, use synthetic division to find the quotient. (6x310x27x15)(x+1)For the following exercises, use synthetic division to find the quotient. (4x312x25x1)(2x+1)For the following exercises, use synthetic division to find the quotient. (9x39x2+18x+5)(3x1)For the following exercises, use synthetic division to find the quotient. (3x32x2+x4)(x+3)For the following exercises, use synthetic division to find the quotient. (6x3+x24)(2x3)For the following exercises, use synthetic division to find the quotient. (2x3+7x213x3)(2x3)For the following exercises, use synthetic division to find the quotient. (3x35x2+2x+3)(x+2)For the following exercises, use synthetic division to find the quotient. (4x35x2+13)(x+4)For the following exercises, use synthetic division to find the quotient. (x33x+2)(x+2)For the following exercises, use synthetic division to find the quotient. (x321x2+147x343)(x7)For the following exercises, use synthetic division to find the quotient. (x315x2+75x125)(x5)For the following exercises, use synthetic division to find the quotient. (9x3x+2)(3x1)For the following exercises, use synthetic division to find the quotient. (6x3x2+5x+2)(3x+1)For the following exercises, use synthetic division to find the quotient. (x4+x33x22x+1)(x+1)For the following exercises, use synthetic division to find the quotient. (x43x2+1)(x1)For the following exercises, use synthetic division to find the quotient. (x4+2x33x2+2x+6)(x+3)For the following exercises, use synthetic division to find the quotient. (x410x3+37x260x+36)(x2)For the following exercises, use synthetic division to find the quotient. (x48x3+24x232x+16)(x2)For the following exercises, use synthetic division to find the quotient. (x4+5x33x213x+10)(x+5)For the following exercises, use synthetic division to find the quotient. (x412x3+54x2108x+81)(x3)For the following exercises, use synthetic division to find the quotient. (4x42x34x+2)(2x1)For the following exercises, use synthetic division to find the quotient. (4x4+2x34x2+2x+2)(2x+1)For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization. x2,4x33x28x+4 For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization. x2,3x46x35x+10 For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization. x+3,4x3+5x2+8 For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization. x2,4x415x24 For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization. x12,2x4x3+2x1 For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization. x+13,3x4+x33x+1 For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored formof the polynomial suggested by the graph. The leading coefficient is one. 44. Factoris x2x+3.For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored formof the polynomial suggested by the graph. The leading coefficient is one. 45. Factor is x2+2x+4 For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored formof the polynomial suggested by the graph. The leading coefficient is one. 46. Factor is x2+2x+5.For the following exercises, use the graph of the third—degree polynomial and one factor to write the factored formof the polynomial suggested by the graph. The leading coefficient is one. 47. Factor is x2+x+1.For the following exercises, use the graph of the third—degree polynomial and one factor to writethe factored formof the polynomial suggested by the graph. The leading coefficient is one. 48. Factor is x2+2x+2.For the following exercises, use synthetic division to find the quotient and remainder. 4x333x2 For the following exercises, use synthetic division to find the quotient and remainder. 2x3+25x+3 For the following exercises, use synthetic division to find the quotient and remainder. 3x3+2x5x1 For the following exercises, use synthetic division to find the quotient and remainder. 4x3x212x+4 For the following exercises, use synthetic division to find the quotient and remainder. x422x+2 For the following exercises, use a calculator with CAS to answer the questions. 54. Consider xk1x1 with k=1,2,3. What do youexpect the result to be if k=4?For the following exercises, use a calculator with CAS to answer the questions. 55. Consider xk+1x+1 for k=1,3,5. What do you expectthe result to be if k=7?For the following exercises, use a calculator with CAS to answer the questions. 56. Consider x4k4xk for k=1,2,3. What do youexpect the result to be if k=4?For the following exercises, use a calculator with CAS to answer the questions. 57. Consider xkx+1 with k=1,2,3. What do you expectthe result to be if k=4?For the following exercises, use a calculator with CAS to answer the questions. 58. Consider xkx1 with k=1,2,3. What do you expectthe result to be if k=4?For the following exercises, use synthetic division to determine the quotient involving a complex number. 59. x+1xi For the following exercises, use synthetic division to determine the quotient involving a complex number. 60. x2+1xi For the following exercises, use synthetic division to determine the quotient involving a complex number. 61. x+1x+i For the following exercises, use synthetic division to determine the quotient involving a complex number. 62. x2+1x+i For the following exercises, use synthetic division to determine the quotient involving a complex number. 63. x3+1xi For the following exercises, use the given length and area of a rectangle to express the width algebraically. 64. Lengthis x+5, areais 2x2+9x5.For the following exercises, use the given length and area of a rectangle to express the width algebraically. 65. Length is 2x+5, areais 4x3+10x2+6x+15 For the following exercises, use the given length and area of a rectangle to express the width algebraically. 66.Lengthis 3x4, areais 6x48x3+9x29x4.For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically. Volume is 12x3+20x221x36, length is 2x+3, width is 3x4.For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically. Volume is 18x321x240x+48, length is 3x4, width is 3x4.For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically. Volume is 10x3+27x2+2x24, length is 5x4, width is 2x+3.For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically. Volume is 10x3+30x28x24, length is 2, width is x+3.For the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically. Volume is (25x365x229x3), radius is 5x+1.For the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically. Volume is (4x3+12x215x50), radius is 2x+5.For the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically. Volume is (3x4+24x3+46x216x32), radius is x+4.Use the Remainder Theorem to evaluate f(x)=2x53x49x3+8x2+2 at x=3.Use the Factor Theorem to find the zeros of f(x)=x3+4x24x16 given that (x2) is a factor of the polynomial.Use the Rational Zero Theorem to find the rational zeros of f(x)=x35x2+2x+1.Find the zeros of f(x)=2x3+5x211x+4.Find a third degree polynomial with real coefficient that has zeros of 5 and -2i such that f(1)=10.Use Descartes’ Rule of Signs to determine the maximum possible numbers of positive and negative real zeros for f(x)=2x410x3+11x215x+12. Use a graph to verify the numbers of positive and negative real zeros for thefunction.A shipping container in the shape of a rectangular solid must have a volume of 84 cubic meters. The client tells themanufacturer that, because of the contents, the length of the container must be one meter longer than the width, andthe height must be one meter greater than twice the width. What should the dimensions of the container be?Describe a use for the Remainder Theorem.Explain why the Rational Zero Theorem does notguarantee finding zeros of a polynomial function.What is the difference between rational and realzeros?If Descartes’ Rule of Signs reveals a no changeof signs or one sign of changes, what specificconclusion can be drawn?If synthetic division reveals a zero, why should wetry that value again as a possible solution?For the following exercises, use the Remainder Theorem to find the remainder. 6. (x49x2+14)(x2)For the following exercises, use the Remainder theorem to find the remainder. 7. (3x32x2+x4)(x+3)For the following exercises, use the Remainder theorem to find the remainder. 8. (x4+5x34x17)(x+1)For the following exercises, use the Remainder theorem to find the remainder. 9. (3x2+6x+24)(x4)For the following exercises, use the Remainder theorem to find the remainder. 10. (5x54x4+3x32x2+x1)(x+6)For the following exercises, use the Remainder theorem to find the remainder. 11. (x41)(x4)For the following exercises, use the Remainder theorem to find the remainder. 12. (3x3+4x28x+2)(x3)For the following exercises, use the Remainder theorem to find the remainder. 13. (4x3+5x22x+7)(x+2)For the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor. f(x)=2x39x2+13x6;x1 For the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor. f(x)=2x3+x25x+2;x+2 For the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor. f(x)=3x3+x220x+12;x+3 For the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor. f(x)=2x3+3x2+x+6;x+2 For the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor. f(x)=5x3+16x29;x3 For the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor. x3+3x2+4x+12;x+3 For the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor. 4x37x+3;x1 For the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor. 2x3+5x212x30,2x+5 For the following exercises, use the Rational Zero Theorem to find all real zeros. x33x210x+24=0 For the following exercises, use the Rational Zero Theorem to find all real zeros. 2x3+7x210x24=0 For the following exercises, use the Rational Zero Theorem to find all real zeros. x3+2x29x18=0 For the following exercises, use the Rational Zero Theorem to find all real zeros. x3+5x216x80=0 For the following exercises, use the Rational Zero Theorem to find all real zeros. x33x225x+75=0 For the following exercises, use the Rational Zero Theorem to find all real zeros. 2x33x232x15=0 For the following exercises, use the Rational Zero Theorem to find all real zeros. 2x3+x27x6=0 For the following exercises, use the Rational Zero Theorem to find all real zeros. 2x33x2x+1=0 For the following exercises, use the Rational Zero Theorem to find all real zeros. 3x3x211x6=0 For the following exercises, use the Rational Zero Theorem to find all real zeros. 2x35x2+9x9=0 For the following exercises, use the Rational Zero Theorem to find all real zeros. 2x33x2+4x+3=0 For the following exercises, use the Rational Zero Theorem to find all real zeros. x42x37x2+8x+12=0 For the following exercises, use the Rational Zero Theorem to find all real zeros. x4+2x39x22x+8=0 For the following exercises, use the Rational Zero Theorem to find all real zeros. 4x4+4x325x2x+6=0 For the following exercises, use the Rational Zero Theorem to find all real zeros. 36. 2x43x315x2+32x12=0

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