{"id":2085,"date":"2025-08-04T16:44:22","date_gmt":"2025-08-04T16:44:22","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=2085"},"modified":"2026-01-05T16:15:08","modified_gmt":"2026-01-05T16:15:08","slug":"polynomial-functions-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/polynomial-functions-get-stronger\/","title":{"raw":"Polynomial Functions: Get Stronger","rendered":"Polynomial Functions: Get Stronger"},"content":{"raw":"

Quadratic Functions<\/h2>\r\n3. Explain why the condition of [latex]a\\ne 0[\/latex] is imposed in the definition of the quadratic function.\r\n\r\n5. What two algebraic methods can be used to find the horizontal intercepts of a quadratic function?\r\n\r\nFor the following exercises, rewrite the quadratic functions in standard form and give the vertex.\r\n\r\n7. [latex]g\\left(x\\right)={x}^{2}+2x - 3[\/latex]\r\n\r\n11. [latex]k\\left(x\\right)=3{x}^{2}-6x - 9[\/latex]\r\n\r\nFor the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.\r\n\r\n15. [latex]f\\left(x\\right)=2{x}^{2}-10x+4[\/latex]\r\n\r\n17. [latex]f\\left(x\\right)=4{x}^{2}+x - 1[\/latex]\r\n\r\nFor the following exercises, determine the domain and range of the quadratic function.\r\n\r\n21. [latex]f\\left(x\\right)={\\left(x - 3\\right)}^{2}+2[\/latex]\r\n\r\n23. [latex]f\\left(x\\right)={x}^{2}+6x+4[\/latex]\r\n\r\nFor the following exercises, solve the equations over the complex numbers.\r\n\r\n29. [latex]{x}^{2}+27=0[\/latex]\r\n\r\n31. [latex]{x}^{2}-4x+5=0[\/latex]\r\n\r\n35. [latex]{x}^{2}-10x+26=0[\/latex]\r\n\r\n39. [latex]2{x}^{2}+2x+5=0[\/latex]\r\n\r\n41. [latex]5{x}^{2}+6x+2=0[\/latex]\r\n\r\nFor the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts.\r\n\r\n53. [latex]f\\left(x\\right)={x}^{2}-2x[\/latex]\r\n\r\n55. [latex]f\\left(x\\right)={x}^{2}-5x - 6[\/latex]\r\n\r\n57. [latex]f\\left(x\\right)=-2{x}^{2}+5x - 8[\/latex]\r\n\r\nFor the following exercises, write the equation for the graphed function.\r\n\r\n59.\r\n\"Graph\r\n\r\n61.\r\n\"Graph\r\n\r\n63.\r\n\"Graph\r\n\r\nFor the following exercises, use the table of values that represent points on the graph of a quadratic function. Find the general form of the equation of the quadratic function.\r\n\r\n65.\r\n\r\n\r\n\r\n\r\n
x<\/strong><\/em><\/td>\r\n\u20132<\/td>\r\n\u20131<\/td>\r\n0<\/td>\r\n1<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
y<\/strong><\/em><\/td>\r\n5<\/td>\r\n2<\/td>\r\n1<\/td>\r\n2<\/td>\r\n5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n67.<\/span>\r\n\r\n\r\n\r\n\r\n
x<\/strong><\/em><\/td>\r\n\u20132<\/td>\r\n\u20131<\/td>\r\n0<\/td>\r\n1<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
y<\/strong><\/em><\/td>\r\n\u20132<\/td>\r\n1<\/td>\r\n2<\/td>\r\n1<\/td>\r\n\u20132<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n69.<\/span>\r\n\r\n\r\n\r\n\r\n
x<\/strong><\/em><\/td>\r\n\u20132<\/td>\r\n\u20131<\/td>\r\n0<\/td>\r\n1<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
y<\/strong><\/em><\/td>\r\n8<\/td>\r\n2<\/td>\r\n0<\/td>\r\n2<\/td>\r\n8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\n85.\u00a0Find the dimensions of the rectangular corral producing the greatest enclosed area given 200 feet of fencing.\r\n\r\n91. A rocket is launched in the air. Its height, in meters above sea level, as a function of time, in seconds, is given by [latex]h\\left(t\\right)=-4.9{t}^{2}+229t+234[\/latex]. Find the maximum height the rocket attains.\r\n\r\n93. A soccer stadium holds 62,000 spectators. With a ticket price of $11, the average attendance has been 26,000. When the price dropped to $9, the average attendance rose to 31,000. Assuming that attendance is linearly related to ticket price, what ticket price would maximize revenue?\r\n

Polynomial Functions<\/h2>\r\n1. Explain the difference between the coefficient of a power function and its degree.\r\n\r\n3. Explain the end behavior of a power function with odd degree if the leading coefficient is positive.\r\n\r\nFor the following exercises, find the degree and leading coefficient for the given polynomial.\r\n\r\n13. [latex]7 - 2{x}^{2}[\/latex]\r\n\r\n15. [latex]x\\left(4-{x}^{2}\\right)\\left(2x+1\\right)[\/latex]\r\n\r\nFor the following exercises, determine the end behavior of the functions.\r\n\r\n17. [latex]f\\left(x\\right)={x}^{4}[\/latex]\r\n\r\n21. [latex]f\\left(x\\right)=-2{x}^{4}- 3{x}^{2}+ x - 1[\/latex]\r\n\r\n23. [latex]f\\left(x\\right)={x}^{2}\\left(2{x}^{3}-x+1\\right)[\/latex]\r\n\r\nFor the following exercises, find the intercepts of the functions.\r\n\r\n25. [latex]f\\left(t\\right)=2\\left(t - 1\\right)\\left(t+2\\right)\\left(t - 3\\right)[\/latex]\r\n\r\n27. [latex]f\\left(x\\right)={x}^{4}-16[\/latex]\r\n\r\n29. [latex]f\\left(x\\right)=x\\left({x}^{2}-2x - 8\\right)[\/latex]\r\n\r\nFor the following exercises, determine the least possible degree of the polynomial function shown.\r\n\r\n31.\r\n\"Graph\r\n\r\n33.\r\n\"Graph\r\n\r\nFor the following exercises, describe the end behavior of the function.\r\n\r\n47. [latex]f\\left(x\\right)={x}^{4}-5{x}^{2}[\/latex]\r\n\r\n49. [latex]f\\left(x\\right)=\\left(x - 1\\right)\\left(x - 2\\right)\\left(3-x\\right)[\/latex]\r\n\r\nFor the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.\r\n\r\n51. [latex]f\\left(x\\right)={x}^{3}\\left(x - 2\\right)[\/latex]\r\n\r\n57. [latex]f\\left(x\\right)={x}^{4}-81[\/latex]\r\n\r\n59. [latex]f\\left(x\\right)={x}^{3}-2{x}^{2}-15x[\/latex]\r\n\r\nFor the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or \u20131. There may be more than one correct answer.\r\n\r\n61. The y<\/em>-intercept is [latex]\\left(0,-4\\right)[\/latex]. The x<\/em>-intercepts are [latex]\\left(-2,0\\right),\\left(2,0\\right)[\/latex]. Degree is 2.\r\n\r\nEnd behavior: [latex]\\text{as }x\\to -\\infty ,f\\left(x\\right)\\to \\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty [\/latex].\r\n\r\n63. The y<\/em>-intercept is [latex]\\left(0,0\\right)[\/latex]. The x<\/em>-intercepts are [latex]\\left(0,0\\right),\\left(2,0\\right)[\/latex]. Degree is 3.\r\n\r\nEnd behavior: [latex]\\text{as }x\\to -\\infty ,f\\left(x\\right)\\to -\\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty [\/latex].\r\n\r\n65. The y<\/em>-intercept is [latex]\\left(0,1\\right)[\/latex]. There is no x<\/em>-intercept. Degree is 4.\r\n\r\nEnd behavior: [latex]\\text{as }x\\to -\\infty ,f\\left(x\\right)\\to \\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty[\/latex].\r\n\r\n69. An open box is to be constructed by cutting out square corners of x<\/em>-inch sides from a piece of cardboard 8 inches by 8 inches and then folding up the sides. Express the volume of the box as a function of x<\/em>.\r\n

Graphs of Polynomial Functions<\/h2>\r\n
\r\n\r\n1. What is the difference between an x<\/em>-intercept and a zero of a polynomial function <\/span>f<\/em>?<\/span>\r\n\r\n<\/div>\r\nFor the following exercises, find the x<\/em>-\u00a0or t<\/em>-intercepts of the polynomial functions.\r\n\r\n7. [latex]C\\left(t\\right)=3\\left(t+2\\right)\\left(t - 3\\right)\\left(t+5\\right)[\/latex]\r\n\r\n9. [latex]C\\left(t\\right)=2t\\left(t - 3\\right){\\left(t+1\\right)}^{2}[\/latex]\r\n\r\n13. [latex]f\\left(x\\right)={x}^{3}+{x}^{2}-20x[\/latex]\r\n\r\n17. [latex]f\\left(x\\right)=2{x}^{3}-{x}^{2}-8x+4[\/latex]\r\n\r\n19. [latex]f\\left(x\\right)=2{x}^{4}+6{x}^{2}-8[\/latex]\r\n\r\n21. [latex]f\\left(x\\right)={x}^{6}-2{x}^{4}-3{x}^{2}[\/latex]\r\n\r\nFor the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval.\r\n\r\n25. [latex]f\\left(x\\right)={x}^{3}-9x[\/latex],\u00a0between [latex]x=2[\/latex]\u00a0and [latex]x=4[\/latex].\r\n\r\n27. [latex]f\\left(x\\right)=-{x}^{4}+4[\/latex],\u00a0between [latex]x=1[\/latex]\u00a0and [latex]x=3[\/latex].\r\n\r\nFor the following exercises, find the zeros and give the multiplicity of each.\r\n\r\n31. [latex]f\\left(x\\right)={x}^{2}{\\left(2x+3\\right)}^{5}{\\left(x - 4\\right)}^{2}[\/latex]\r\n\r\n33. [latex]f\\left(x\\right)={x}^{2}\\left({x}^{2}+4x+4\\right)[\/latex]\r\n\r\n37. [latex]f\\left(x\\right)={x}^{6}-{x}^{5}-2{x}^{4}[\/latex]\r\n\r\n39. [latex]f\\left(x\\right)=4{x}^{5}-12{x}^{4}+9{x}^{3}[\/latex]\r\n\r\nFor the following exercises, graph the polynomial functions. Note x-<\/em>\u00a0and y<\/em>-intercepts, multiplicity, and end behavior.\r\n\r\n43. [latex]g\\left(x\\right)=\\left(x+4\\right){\\left(x - 1\\right)}^{2}[\/latex]\r\n\r\n45. [latex]k\\left(x\\right)={\\left(x - 3\\right)}^{3}{\\left(x - 2\\right)}^{2}[\/latex]\r\n\r\n47. [latex]n\\left(x\\right)=-3x\\left(x+2\\right)\\left(x - 4\\right)[\/latex]\r\n\r\nFor the following exercises, use the graphs to write the formula for a polynomial function of least degree.\r\n\r\n49.\r\n\"Graph\r\n\r\n53.\r\n\"Graph\r\n\r\n55.\r\n\"Graph\r\n\r\nFor the following exercises, use the given information about the polynomial graph to write the equation.\r\n\r\n59. Degree 5. Roots of multiplicity 2 at [latex]x=3[\/latex]\u00a0and [latex]x=1[\/latex], and a root of multiplicity 1 at [latex]x=-3[\/latex]. y<\/em>-intercept at [latex]\\left(0,9\\right)[\/latex]\r\n\r\n63. Degree 3. Zeros at [latex]x=-3[\/latex], [latex]x=-2[\/latex]\u00a0and [latex]x=1[\/latex]. y<\/em>-intercept at [latex]\\left(0,12\\right)[\/latex].\r\n\r\n65. Degree 4. Roots of multiplicity 2 at [latex]x=\\frac{1}{2}[\/latex] and roots of multiplicity 1 at [latex]x=6[\/latex] and [latex]x=-2[\/latex].\u00a0y<\/em>-intercept at [latex]\\left(0,18\\right)[\/latex].\r\n\r\nFor the following exercises, write the polynomial function that models the given situation.\r\n\r\n75. A rectangle has a length of 10 units and a width of 8 units. Squares of x<\/em>\u00a0by x<\/em>\u00a0units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a polynomial function in terms of x<\/em>.\r\n\r\n77. A square has sides of 12 units. Squares [latex]x+1[\/latex]\u00a0by [latex]x+1[\/latex]\u00a0units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a function in terms of x<\/em>.\r\n\r\n79. A right circular cone has a radius of [latex]3x+6[\/latex]\u00a0and a height 3 units less. Express the volume of the cone as a polynomial function. The volume of a cone is [latex]V=\\frac{1}{3}\\pi {r}^{2}h[\/latex]\u00a0for radius r<\/em>\u00a0and height h<\/em>.","rendered":"

Quadratic Functions<\/h2>\n

3. Explain why the condition of [latex]a\\ne 0[\/latex] is imposed in the definition of the quadratic function.<\/p>\n

5. What two algebraic methods can be used to find the horizontal intercepts of a quadratic function?<\/p>\n

For the following exercises, rewrite the quadratic functions in standard form and give the vertex.<\/p>\n

7. [latex]g\\left(x\\right)={x}^{2}+2x - 3[\/latex]<\/p>\n

11. [latex]k\\left(x\\right)=3{x}^{2}-6x - 9[\/latex]<\/p>\n

For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.<\/p>\n

15. [latex]f\\left(x\\right)=2{x}^{2}-10x+4[\/latex]<\/p>\n

17. [latex]f\\left(x\\right)=4{x}^{2}+x - 1[\/latex]<\/p>\n

For the following exercises, determine the domain and range of the quadratic function.<\/p>\n

21. [latex]f\\left(x\\right)={\\left(x - 3\\right)}^{2}+2[\/latex]<\/p>\n

23. [latex]f\\left(x\\right)={x}^{2}+6x+4[\/latex]<\/p>\n

For the following exercises, solve the equations over the complex numbers.<\/p>\n

29. [latex]{x}^{2}+27=0[\/latex]<\/p>\n

31. [latex]{x}^{2}-4x+5=0[\/latex]<\/p>\n

35. [latex]{x}^{2}-10x+26=0[\/latex]<\/p>\n

39. [latex]2{x}^{2}+2x+5=0[\/latex]<\/p>\n

41. [latex]5{x}^{2}+6x+2=0[\/latex]<\/p>\n

For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts.<\/p>\n

53. [latex]f\\left(x\\right)={x}^{2}-2x[\/latex]<\/p>\n

55. [latex]f\\left(x\\right)={x}^{2}-5x - 6[\/latex]<\/p>\n

57. [latex]f\\left(x\\right)=-2{x}^{2}+5x - 8[\/latex]<\/p>\n

For the following exercises, write the equation for the graphed function.<\/p>\n

59.
\n\"Graph<\/p>\n

61.
\n\"Graph<\/p>\n

63.
\n\"Graph<\/p>\n

For the following exercises, use the table of values that represent points on the graph of a quadratic function. Find the general form of the equation of the quadratic function.<\/p>\n

65.<\/p>\n\n\n\n\n
x<\/strong><\/em><\/td>\n\u20132<\/td>\n\u20131<\/td>\n0<\/td>\n1<\/td>\n2<\/td>\n<\/tr>\n
y<\/strong><\/em><\/td>\n5<\/td>\n2<\/td>\n1<\/td>\n2<\/td>\n5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

67.<\/span><\/p>\n\n\n\n\n
x<\/strong><\/em><\/td>\n\u20132<\/td>\n\u20131<\/td>\n0<\/td>\n1<\/td>\n2<\/td>\n<\/tr>\n
y<\/strong><\/em><\/td>\n\u20132<\/td>\n1<\/td>\n2<\/td>\n1<\/td>\n\u20132<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

69.<\/span><\/p>\n\n\n\n\n
x<\/strong><\/em><\/td>\n\u20132<\/td>\n\u20131<\/td>\n0<\/td>\n1<\/td>\n2<\/td>\n<\/tr>\n
y<\/strong><\/em><\/td>\n8<\/td>\n2<\/td>\n0<\/td>\n2<\/td>\n8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

85.\u00a0Find the dimensions of the rectangular corral producing the greatest enclosed area given 200 feet of fencing.<\/p>\n

91. A rocket is launched in the air. Its height, in meters above sea level, as a function of time, in seconds, is given by [latex]h\\left(t\\right)=-4.9{t}^{2}+229t+234[\/latex]. Find the maximum height the rocket attains.<\/p>\n

93. A soccer stadium holds 62,000 spectators. With a ticket price of $11, the average attendance has been 26,000. When the price dropped to $9, the average attendance rose to 31,000. Assuming that attendance is linearly related to ticket price, what ticket price would maximize revenue?<\/p>\n

Polynomial Functions<\/h2>\n

1. Explain the difference between the coefficient of a power function and its degree.<\/p>\n

3. Explain the end behavior of a power function with odd degree if the leading coefficient is positive.<\/p>\n

For the following exercises, find the degree and leading coefficient for the given polynomial.<\/p>\n

13. [latex]7 - 2{x}^{2}[\/latex]<\/p>\n

15. [latex]x\\left(4-{x}^{2}\\right)\\left(2x+1\\right)[\/latex]<\/p>\n

For the following exercises, determine the end behavior of the functions.<\/p>\n

17. [latex]f\\left(x\\right)={x}^{4}[\/latex]<\/p>\n

21. [latex]f\\left(x\\right)=-2{x}^{4}- 3{x}^{2}+ x - 1[\/latex]<\/p>\n

23. [latex]f\\left(x\\right)={x}^{2}\\left(2{x}^{3}-x+1\\right)[\/latex]<\/p>\n

For the following exercises, find the intercepts of the functions.<\/p>\n

25. [latex]f\\left(t\\right)=2\\left(t - 1\\right)\\left(t+2\\right)\\left(t - 3\\right)[\/latex]<\/p>\n

27. [latex]f\\left(x\\right)={x}^{4}-16[\/latex]<\/p>\n

29. [latex]f\\left(x\\right)=x\\left({x}^{2}-2x - 8\\right)[\/latex]<\/p>\n

For the following exercises, determine the least possible degree of the polynomial function shown.<\/p>\n

31.
\n\"Graph<\/p>\n

33.
\n\"Graph<\/p>\n

For the following exercises, describe the end behavior of the function.<\/p>\n

47. [latex]f\\left(x\\right)={x}^{4}-5{x}^{2}[\/latex]<\/p>\n

49. [latex]f\\left(x\\right)=\\left(x - 1\\right)\\left(x - 2\\right)\\left(3-x\\right)[\/latex]<\/p>\n

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.<\/p>\n

51. [latex]f\\left(x\\right)={x}^{3}\\left(x - 2\\right)[\/latex]<\/p>\n

57. [latex]f\\left(x\\right)={x}^{4}-81[\/latex]<\/p>\n

59. [latex]f\\left(x\\right)={x}^{3}-2{x}^{2}-15x[\/latex]<\/p>\n

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 \u20131. There may be more than one correct answer.<\/p>\n

61. The y<\/em>-intercept is [latex]\\left(0,-4\\right)[\/latex]. The x<\/em>-intercepts are [latex]\\left(-2,0\\right),\\left(2,0\\right)[\/latex]. Degree is 2.<\/p>\n

End behavior: [latex]\\text{as }x\\to -\\infty ,f\\left(x\\right)\\to \\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty[\/latex].<\/p>\n

63. The y<\/em>-intercept is [latex]\\left(0,0\\right)[\/latex]. The x<\/em>-intercepts are [latex]\\left(0,0\\right),\\left(2,0\\right)[\/latex]. Degree is 3.<\/p>\n

End behavior: [latex]\\text{as }x\\to -\\infty ,f\\left(x\\right)\\to -\\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty[\/latex].<\/p>\n

65. The y<\/em>-intercept is [latex]\\left(0,1\\right)[\/latex]. There is no x<\/em>-intercept. Degree is 4.<\/p>\n

End behavior: [latex]\\text{as }x\\to -\\infty ,f\\left(x\\right)\\to \\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty[\/latex].<\/p>\n

69. An open box is to be constructed by cutting out square corners of x<\/em>-inch sides from a piece of cardboard 8 inches by 8 inches and then folding up the sides. Express the volume of the box as a function of x<\/em>.<\/p>\n

Graphs of Polynomial Functions<\/h2>\n
\n

1. What is the difference between an x<\/em>-intercept and a zero of a polynomial function <\/span>f<\/em>?<\/span><\/p>\n<\/div>\n

For the following exercises, find the x<\/em>–\u00a0or t<\/em>-intercepts of the polynomial functions.<\/p>\n

7. [latex]C\\left(t\\right)=3\\left(t+2\\right)\\left(t - 3\\right)\\left(t+5\\right)[\/latex]<\/p>\n

9. [latex]C\\left(t\\right)=2t\\left(t - 3\\right){\\left(t+1\\right)}^{2}[\/latex]<\/p>\n

13. [latex]f\\left(x\\right)={x}^{3}+{x}^{2}-20x[\/latex]<\/p>\n

17. [latex]f\\left(x\\right)=2{x}^{3}-{x}^{2}-8x+4[\/latex]<\/p>\n

19. [latex]f\\left(x\\right)=2{x}^{4}+6{x}^{2}-8[\/latex]<\/p>\n

21. [latex]f\\left(x\\right)={x}^{6}-2{x}^{4}-3{x}^{2}[\/latex]<\/p>\n

For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval.<\/p>\n

25. [latex]f\\left(x\\right)={x}^{3}-9x[\/latex],\u00a0between [latex]x=2[\/latex]\u00a0and [latex]x=4[\/latex].<\/p>\n

27. [latex]f\\left(x\\right)=-{x}^{4}+4[\/latex],\u00a0between [latex]x=1[\/latex]\u00a0and [latex]x=3[\/latex].<\/p>\n

For the following exercises, find the zeros and give the multiplicity of each.<\/p>\n

31. [latex]f\\left(x\\right)={x}^{2}{\\left(2x+3\\right)}^{5}{\\left(x - 4\\right)}^{2}[\/latex]<\/p>\n

33. [latex]f\\left(x\\right)={x}^{2}\\left({x}^{2}+4x+4\\right)[\/latex]<\/p>\n

37. [latex]f\\left(x\\right)={x}^{6}-{x}^{5}-2{x}^{4}[\/latex]<\/p>\n

39. [latex]f\\left(x\\right)=4{x}^{5}-12{x}^{4}+9{x}^{3}[\/latex]<\/p>\n

For the following exercises, graph the polynomial functions. Note x-<\/em>\u00a0and y<\/em>-intercepts, multiplicity, and end behavior.<\/p>\n

43. [latex]g\\left(x\\right)=\\left(x+4\\right){\\left(x - 1\\right)}^{2}[\/latex]<\/p>\n

45. [latex]k\\left(x\\right)={\\left(x - 3\\right)}^{3}{\\left(x - 2\\right)}^{2}[\/latex]<\/p>\n

47. [latex]n\\left(x\\right)=-3x\\left(x+2\\right)\\left(x - 4\\right)[\/latex]<\/p>\n

For the following exercises, use the graphs to write the formula for a polynomial function of least degree.<\/p>\n

49.
\n\"Graph<\/p>\n

53.
\n\"Graph<\/p>\n

55.
\n\"Graph<\/p>\n

For the following exercises, use the given information about the polynomial graph to write the equation.<\/p>\n

59. Degree 5. Roots of multiplicity 2 at [latex]x=3[\/latex]\u00a0and [latex]x=1[\/latex], and a root of multiplicity 1 at [latex]x=-3[\/latex]. y<\/em>-intercept at [latex]\\left(0,9\\right)[\/latex]<\/p>\n

63. Degree 3. Zeros at [latex]x=-3[\/latex], [latex]x=-2[\/latex]\u00a0and [latex]x=1[\/latex]. y<\/em>-intercept at [latex]\\left(0,12\\right)[\/latex].<\/p>\n

65. Degree 4. Roots of multiplicity 2 at [latex]x=\\frac{1}{2}[\/latex] and roots of multiplicity 1 at [latex]x=6[\/latex] and [latex]x=-2[\/latex].\u00a0y<\/em>-intercept at [latex]\\left(0,18\\right)[\/latex].<\/p>\n

For the following exercises, write the polynomial function that models the given situation.<\/p>\n

75. A rectangle has a length of 10 units and a width of 8 units. Squares of x<\/em>\u00a0by x<\/em>\u00a0units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a polynomial function in terms of x<\/em>.<\/p>\n

77. A square has sides of 12 units. Squares [latex]x+1[\/latex]\u00a0by [latex]x+1[\/latex]\u00a0units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a function in terms of x<\/em>.<\/p>\n

79. A right circular cone has a radius of [latex]3x+6[\/latex]\u00a0and a height 3 units less. Express the volume of the cone as a polynomial function. The volume of a cone is [latex]V=\\frac{1}{3}\\pi {r}^{2}h[\/latex]\u00a0for radius r<\/em>\u00a0and height h<\/em>.<\/p>\n","protected":false},"author":67,"menu_order":24,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":74,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2085"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2085\/revisions"}],"predecessor-version":[{"id":5177,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2085\/revisions\/5177"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/74"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2085\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2085"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2085"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2085"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2085"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}