{"id":273,"date":"2026-01-30T23:00:13","date_gmt":"2026-01-30T23:00:13","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/polynomial-functions-get-stronger-key-precalculus-practice-page-answer-keys\/"},"modified":"2026-01-30T23:02:46","modified_gmt":"2026-01-30T23:02:46","slug":"polynomial-functions-get-stronger-key-precalculus-practice-page-answer-keys","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/polynomial-functions-get-stronger-key-precalculus-practice-page-answer-keys\/","title":{"raw":"Polynomial Functions: Get Stronger Key -- Precalculus Practice Page Answer Keys","rendered":"Polynomial Functions: Get Stronger Key — Precalculus Practice Page Answer Keys"},"content":{"raw":"\n\n\t\t
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<\/p>\n\t\t

Polynomial Functions: Get Stronger Key<\/h1>\n\t\t\t\t\t\t\t\t<\/div>\n\t
\n\t\t\t\t

Quadratic Functions Solutions<\/h2>

3. If [latex]a=0[\/latex] then the function becomes a linear function.<\/p>

5. If possible, we can use factoring. Otherwise, we can use the quadratic formula.<\/p>

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

11. [latex]f\\left(x\\right)=3{\\left(x - 1\\right)}^{2}-12[\/latex], Vertex [latex]\\left(1,-12\\right)[\/latex]<\/p>

15. Minimum is [latex]-\\frac{17}{2}[\/latex] and occurs at [latex]\\frac{5}{2}[\/latex]. Axis of symmetry is [latex]x=\\frac{5}{2}[\/latex].<\/p>

17. Minimum is [latex]-\\frac{17}{16}[\/latex] and occurs at [latex]-\\frac{1}{8}[\/latex]. Axis of symmetry is [latex]x=-\\frac{1}{8}[\/latex].<\/p>

21. Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]. Range is [latex]\\left[2,\\infty \\right)[\/latex].<\/p>

23. Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]. Range is [latex]\\left[-5,\\infty \\right)[\/latex].<\/p>

29. [latex]\\left\\{3i\\sqrt{3},-3i\\sqrt{3}\\right\\}[\/latex]<\/p>

31. [latex]\\left\\{2+i,2-i\\right\\}[\/latex]<\/p>

35. [latex]\\left\\{5+i,5-i\\right\\}[\/latex]<\/p>

39. [latex]\\left\\{-\\frac{1}{2}+\\frac{3}{2}i, -\\frac{1}{2}-\\frac{3}{2}i\\right\\}[\/latex]<\/p>

41. [latex]\\left\\{-\\frac{3}{5}+\\frac{1}{5}i, -\\frac{3}{5}-\\frac{1}{5}i\\right\\}[\/latex]<\/p>

53. Vertex [latex]\\left(1,\\text{ }-1\\right)[\/latex], Axis of symmetry is [latex]x=1[\/latex]. Intercepts are [latex]\\left(0,0\\right), \\left(2,0\\right)[\/latex].<\/p>

\"Graph<\/p>

55. Vertex [latex]\\left(\\frac{5}{2},\\frac{-49}{4}\\right)[\/latex], Axis of symmetry is [latex]\\left(0,-6\\right),\\left(-1,0\\right),\\left(6,0\\right)[\/latex].<\/p>

\"Graph<\/p>

57. Vertex [latex]\\left(\\frac{5}{4}, -\\frac{39}{8}\\right)[\/latex], Axis of symmetry is [latex]x=\\frac{5}{4}[\/latex]. Intercepts are [latex]\\left(0, -8\\right)[\/latex].<\/p>

\"Graph<\/p>

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

61. [latex]f\\left(x\\right)=-2{x}^{2}+8x - 1[\/latex]<\/p>

63. [latex]f\\left(x\\right)=\\frac{1}{2}{x}^{2}-3x+\\frac{7}{2}[\/latex]<\/p>

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

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

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

85. 50 feet by 50 feet. Maximize [latex]f\\left(x\\right)=-{x}^{2}+100x[\/latex].<\/p>

91. 2909.56 meters<\/p>

93. $10.70<\/p>

Polynomial Functions Solutions<\/h2>

1. The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function.<\/p>

3. As x<\/em> decreases without bound, so does [latex]f\\left(x\\right)[\/latex]. As x<\/em> increases without bound, so does [latex]f\\left(x\\right)[\/latex].<\/p>

13. Degree = 2, Coefficient = \u20132<\/p>

15. Degree =4, Coefficient = \u20132<\/p>

17. [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>

21. [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>

23. [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>

25. y<\/em>-intercept is [latex]\\left(0,12\\right)[\/latex], t<\/em>-intercepts are [latex]\\left(1,0\\right);\\left(-2,0\\right);\\text{and }\\left(3,0\\right)[\/latex].<\/p>

27. y<\/em>-intercept is [latex]\\left(0,-16\\right)[\/latex]. x<\/em>-intercepts are [latex]\\left(2,0\\right)[\/latex] and [latex]\\left(-2,0\\right)[\/latex].<\/p>

29. y<\/em>-intercept is [latex]\\left(0,0\\right)[\/latex].i x-intercepts are [latex]\\left(0,0\\right),\\left(4,0\\right)[\/latex], and [latex]\\left(-2, 0\\right)[\/latex].<\/p>

31. 3<\/p>

33. 5<\/p>

47. [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>
x<\/em><\/th> f<\/em>(x<\/em>)<\/th> <\/tr> <\/thead>
10<\/td> 9,500<\/td> <\/tr>
100<\/td> 99,950,000<\/td> <\/tr>
\u201310<\/td> 9,500<\/td> <\/tr>
\u2013100<\/td> 99,950,000<\/td> <\/tr> <\/tbody> <\/table>

49. [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>
x<\/em><\/th> f<\/em>(x<\/em>)<\/th> <\/tr> <\/thead>
10<\/td> \u2013504<\/td> <\/tr>
100<\/td> \u2013941,094<\/td> <\/tr>
\u201310<\/td> 1,716<\/td> <\/tr>
\u2013100<\/td> 1,061,106<\/td> <\/tr> <\/tbody> <\/table>

51. The y<\/em>-intercept is [latex]\\left(0, 0\\right)[\/latex]. The x<\/em>-intercepts are [latex]\\left(0, 0\\right),\\text{ }\\left(2, 0\\right)[\/latex]. [latex]\\text{As }x\\to -\\infty ,f\\left(x\\right)\\to \\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty[\/latex]
\"Graph<\/p>

57. The y<\/em>-intercept is [latex]\\left(0, -81\\right)[\/latex]. The x<\/em>-intercept are [latex]\\left(3, 0\\right),\\text{ }\\left(-3, 0\\right)[\/latex]. [latex]\\text{As }x\\to -\\infty ,f\\left(x\\right)\\to \\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty[\/latex]
\"Graph<\/p>

59. The y<\/em>-intercept is [latex]\\left(0, 0\\right)[\/latex]. The x<\/em>-intercepts are [latex]\\left(-3, 0\\right),\\text{ }\\left(0, 0\\right),\\text{ }\\left(5, 0\\right)[\/latex]. [latex]\\text{As }x\\to -\\infty ,f\\left(x\\right)\\to -\\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty[\/latex]
\"Graph<\/p>

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

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

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

69. [latex]V\\left(x\\right)=4{x}^{3}-32{x}^{2}+64x[\/latex]<\/p>

Graphs of Polynomial Functions Solutions<\/h2>

1. The x-<\/em>intercept is where the graph of the function crosses the x<\/em>-axis, and the zero of the function is the input value for which [latex]f\\left(x\\right)=0[\/latex].<\/p>

7. [latex]\\left(-2,0\\right),\\left(3,0\\right),\\left(-5,0\\right)[\/latex]<\/p>

9. [latex]\\left(3,0\\right),\\left(-1,0\\right),\\left(0,0\\right)[\/latex]<\/p>

13. [latex]\\left(0,0\\right),\\text{ }\\left(-5,0\\right),\\text{ }\\left(4,0\\right)[\/latex]<\/p>

17. [latex]\\left(-2,0\\right),\\left(2,0\\right),\\left(\\frac{1}{2},0\\right)[\/latex]<\/p>

19. [latex]\\left(1,0\\right),\\text{ }\\left(-1,0\\right)[\/latex]<\/p>

21. [latex]\\left(0,0\\right),\\left(\\sqrt{3},0\\right),\\left(-\\sqrt{3},0\\right)[\/latex]<\/p>

25. [latex]f\\left(2\\right)=-10[\/latex] and [latex]f\\left(4\\right)=28[\/latex]. Sign change confirms.<\/p>

27. [latex]f\\left(1\\right)=3[\/latex] and [latex]f\\left(3\\right)=-77[\/latex]. Sign change confirms.<\/p>

31. 0 with multiplicity 2, [latex]-\\frac{3}{2}[\/latex] with multiplicity 5, 4 with multiplicity 2<\/p>

33. 0 with multiplicity 2, \u20132 with multiplicity 2<\/p>

37. [latex]\\text{0}\\text{ with multiplicity }4\\text{,}2\\text{ with multiplicity }1\\text{,}-\\text{1}\\text{ with multiplicity }1[\/latex]<\/p>

39. [latex]\\frac{3}{2}[\/latex] with multiplicity 2, 0 with multiplicity 3<\/p>

43. x<\/em>-intercepts, [latex]\\left(1, 0\\right)[\/latex] with multiplicity 2, [latex]\\left(-4, 0\\right)[\/latex] with multiplicity 1, y-<\/em>intercept [latex]\\left(0, 4\\right)[\/latex]. As [latex]x\\to -\\infty[\/latex] , [latex]f\\left(x\\right)\\to -\\infty[\/latex] , as [latex]x\\to \\infty[\/latex] , [latex]f\\left(x\\right)\\to \\infty[\/latex] .
\"Graph<\/p>

45. x<\/em>-intercepts [latex]\\left(3,0\\right)[\/latex] with multiplicity 3, [latex]\\left(2,0\\right)[\/latex] with multiplicity 2, y<\/em>-intercept [latex]\\left(0,-108\\right)[\/latex] . As [latex]x\\to -\\infty[\/latex], [latex]f\\left(x\\right)\\to -\\infty[\/latex] , as [latex]x\\to \\infty[\/latex] , [latex]f\\left(x\\right)\\to \\infty[\/latex].
\"Graph<\/p>

47. x-intercepts [latex]\\left(0, 0\\right),\\left(-2, 0\\right),\\left(4, 0\\right)[\/latex] with multiplicity 1, y<\/em>-intercept [latex]\\left(0, 0\\right)[\/latex]. As [latex]x\\to -\\infty[\/latex] , [latex]f\\left(x\\right)\\to \\infty[\/latex] , as [latex]x\\to \\infty[\/latex] , [latex]f\\left(x\\right)\\to -\\infty[\/latex].
\"Graph<\/p>

49. [latex]f\\left(x\\right)=-\\frac{2}{9}\\left(x - 3\\right)\\left(x+1\\right)\\left(x+3\\right)[\/latex]<\/p>

53. [latex]f\\left(x\\right)=-\\frac{1}{8}\\left(x +4\\right)\\left(x+2\\right)\\left(x-1\\right)\\left(x-3\\right)[\/latex]<\/p>

55. [latex]f\\left(x\\right)=\\frac{1}{12}\\left(x +2\\right)^2\\left(x+3\\right)^2[\/latex]<\/p>

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

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

65. [latex]f\\left(x\\right)=-\\frac{3}{2}{\\left(2x - 1\\right)}^{2}\\left(x - 6\\right)\\left(x+2\\right)[\/latex]<\/p>

75. [latex]f\\left(x\\right)=4{x}^{3}-36{x}^{2}+80x[\/latex]<\/p>

77. [latex]f\\left(x\\right)=4{x}^{3}-36{x}^{2}+60x+100[\/latex]<\/p>

79. [latex]f\\left(x\\right)=\\pi \\left(9{x}^{3}+45{x}^{2}+72x+36\\right)[\/latex]<\/p> \n\t<\/div>\n\t\t\t\n\t\t\t<\/div>\n\n\t\t<\/div>\n\t\n","rendered":"

\n
\n
\n

\n

Polynomial Functions: Get Stronger Key<\/h1>\n<\/p><\/div>\n
\n

Quadratic Functions Solutions<\/h2>\n

3. If [latex]a=0[\/latex] then the function becomes a linear function.<\/p>\n

5. If possible, we can use factoring. Otherwise, we can use the quadratic formula.<\/p>\n

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

11. [latex]f\\left(x\\right)=3{\\left(x - 1\\right)}^{2}-12[\/latex], Vertex [latex]\\left(1,-12\\right)[\/latex]<\/p>\n

15. Minimum is [latex]-\\frac{17}{2}[\/latex] and occurs at [latex]\\frac{5}{2}[\/latex]. Axis of symmetry is [latex]x=\\frac{5}{2}[\/latex].<\/p>\n

17. Minimum is [latex]-\\frac{17}{16}[\/latex] and occurs at [latex]-\\frac{1}{8}[\/latex]. Axis of symmetry is [latex]x=-\\frac{1}{8}[\/latex].<\/p>\n

21. Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]. Range is [latex]\\left[2,\\infty \\right)[\/latex].<\/p>\n

23. Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]. Range is [latex]\\left[-5,\\infty \\right)[\/latex].<\/p>\n

29. [latex]\\left\\{3i\\sqrt{3},-3i\\sqrt{3}\\right\\}[\/latex]<\/p>\n

31. [latex]\\left\\{2+i,2-i\\right\\}[\/latex]<\/p>\n

35. [latex]\\left\\{5+i,5-i\\right\\}[\/latex]<\/p>\n

39. [latex]\\left\\{-\\frac{1}{2}+\\frac{3}{2}i, -\\frac{1}{2}-\\frac{3}{2}i\\right\\}[\/latex]<\/p>\n

41. [latex]\\left\\{-\\frac{3}{5}+\\frac{1}{5}i, -\\frac{3}{5}-\\frac{1}{5}i\\right\\}[\/latex]<\/p>\n

53. Vertex [latex]\\left(1,\\text{ }-1\\right)[\/latex], Axis of symmetry is [latex]x=1[\/latex]. Intercepts are [latex]\\left(0,0\\right), \\left(2,0\\right)[\/latex].<\/p>\n

\"Graph<\/p>\n

55. Vertex [latex]\\left(\\frac{5}{2},\\frac{-49}{4}\\right)[\/latex], Axis of symmetry is [latex]\\left(0,-6\\right),\\left(-1,0\\right),\\left(6,0\\right)[\/latex].<\/p>\n

\"Graph<\/p>\n

57. Vertex [latex]\\left(\\frac{5}{4}, -\\frac{39}{8}\\right)[\/latex], Axis of symmetry is [latex]x=\\frac{5}{4}[\/latex]. Intercepts are [latex]\\left(0, -8\\right)[\/latex].<\/p>\n

\"Graph<\/p>\n

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

61. [latex]f\\left(x\\right)=-2{x}^{2}+8x - 1[\/latex]<\/p>\n

63. [latex]f\\left(x\\right)=\\frac{1}{2}{x}^{2}-3x+\\frac{7}{2}[\/latex]<\/p>\n

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

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

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

85. 50 feet by 50 feet. Maximize [latex]f\\left(x\\right)=-{x}^{2}+100x[\/latex].<\/p>\n

91. 2909.56 meters<\/p>\n

93. $10.70<\/p>\n

Polynomial Functions Solutions<\/h2>\n

1. The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function.<\/p>\n

3. As x<\/em> decreases without bound, so does [latex]f\\left(x\\right)[\/latex]. As x<\/em> increases without bound, so does [latex]f\\left(x\\right)[\/latex].<\/p>\n

13. Degree = 2, Coefficient = \u20132<\/p>\n

15. Degree =4, Coefficient = \u20132<\/p>\n

17. [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

21. [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

23. [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

25. y<\/em>-intercept is [latex]\\left(0,12\\right)[\/latex], t<\/em>-intercepts are [latex]\\left(1,0\\right);\\left(-2,0\\right);\\text{and }\\left(3,0\\right)[\/latex].<\/p>\n

27. y<\/em>-intercept is [latex]\\left(0,-16\\right)[\/latex]. x<\/em>-intercepts are [latex]\\left(2,0\\right)[\/latex] and [latex]\\left(-2,0\\right)[\/latex].<\/p>\n

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

31. 3<\/p>\n

33. 5<\/p>\n

47. [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\n\n\n\n\n\n\n\n
x<\/em><\/th>\nf<\/em>(x<\/em>)<\/th>\n<\/tr>\n<\/thead>\n
10<\/td>\n9,500<\/td>\n<\/tr>\n
100<\/td>\n99,950,000<\/td>\n<\/tr>\n
\u201310<\/td>\n9,500<\/td>\n<\/tr>\n
\u2013100<\/td>\n99,950,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

49. [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\n\n\n\n\n\n\n\n
x<\/em><\/th>\nf<\/em>(x<\/em>)<\/th>\n<\/tr>\n<\/thead>\n
10<\/td>\n\u2013504<\/td>\n<\/tr>\n
100<\/td>\n\u2013941,094<\/td>\n<\/tr>\n
\u201310<\/td>\n1,716<\/td>\n<\/tr>\n
\u2013100<\/td>\n1,061,106<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

51. The y<\/em>-intercept is [latex]\\left(0, 0\\right)[\/latex]. The x<\/em>-intercepts are [latex]\\left(0, 0\\right),\\text{ }\\left(2, 0\\right)[\/latex]. [latex]\\text{As }x\\to -\\infty ,f\\left(x\\right)\\to \\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty[\/latex]
\"Graph<\/p>\n

57. The y<\/em>-intercept is [latex]\\left(0, -81\\right)[\/latex]. The x<\/em>-intercept are [latex]\\left(3, 0\\right),\\text{ }\\left(-3, 0\\right)[\/latex]. [latex]\\text{As }x\\to -\\infty ,f\\left(x\\right)\\to \\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty[\/latex]
\"Graph<\/p>\n

59. The y<\/em>-intercept is [latex]\\left(0, 0\\right)[\/latex]. The x<\/em>-intercepts are [latex]\\left(-3, 0\\right),\\text{ }\\left(0, 0\\right),\\text{ }\\left(5, 0\\right)[\/latex]. [latex]\\text{As }x\\to -\\infty ,f\\left(x\\right)\\to -\\infty ,\\text{ as }x\\to \\infty ,f\\left(x\\right)\\to \\infty[\/latex]
\"Graph<\/p>\n

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

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

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

69. [latex]V\\left(x\\right)=4{x}^{3}-32{x}^{2}+64x[\/latex]<\/p>\n

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

1. The x-<\/em>intercept is where the graph of the function crosses the x<\/em>-axis, and the zero of the function is the input value for which [latex]f\\left(x\\right)=0[\/latex].<\/p>\n

7. [latex]\\left(-2,0\\right),\\left(3,0\\right),\\left(-5,0\\right)[\/latex]<\/p>\n

9. [latex]\\left(3,0\\right),\\left(-1,0\\right),\\left(0,0\\right)[\/latex]<\/p>\n

13. [latex]\\left(0,0\\right),\\text{ }\\left(-5,0\\right),\\text{ }\\left(4,0\\right)[\/latex]<\/p>\n

17. [latex]\\left(-2,0\\right),\\left(2,0\\right),\\left(\\frac{1}{2},0\\right)[\/latex]<\/p>\n

19. [latex]\\left(1,0\\right),\\text{ }\\left(-1,0\\right)[\/latex]<\/p>\n

21. [latex]\\left(0,0\\right),\\left(\\sqrt{3},0\\right),\\left(-\\sqrt{3},0\\right)[\/latex]<\/p>\n

25. [latex]f\\left(2\\right)=-10[\/latex] and [latex]f\\left(4\\right)=28[\/latex]. Sign change confirms.<\/p>\n

27. [latex]f\\left(1\\right)=3[\/latex] and [latex]f\\left(3\\right)=-77[\/latex]. Sign change confirms.<\/p>\n

31. 0 with multiplicity 2, [latex]-\\frac{3}{2}[\/latex] with multiplicity 5, 4 with multiplicity 2<\/p>\n

33. 0 with multiplicity 2, \u20132 with multiplicity 2<\/p>\n

37. [latex]\\text{0}\\text{ with multiplicity }4\\text{,}2\\text{ with multiplicity }1\\text{,}-\\text{1}\\text{ with multiplicity }1[\/latex]<\/p>\n

39. [latex]\\frac{3}{2}[\/latex] with multiplicity 2, 0 with multiplicity 3<\/p>\n

43. x<\/em>-intercepts, [latex]\\left(1, 0\\right)[\/latex] with multiplicity 2, [latex]\\left(-4, 0\\right)[\/latex] with multiplicity 1, y-<\/em>intercept [latex]\\left(0, 4\\right)[\/latex]. As [latex]x\\to -\\infty[\/latex] , [latex]f\\left(x\\right)\\to -\\infty[\/latex] , as [latex]x\\to \\infty[\/latex] , [latex]f\\left(x\\right)\\to \\infty[\/latex] .
\"Graph<\/p>\n

45. x<\/em>-intercepts [latex]\\left(3,0\\right)[\/latex] with multiplicity 3, [latex]\\left(2,0\\right)[\/latex] with multiplicity 2, y<\/em>-intercept [latex]\\left(0,-108\\right)[\/latex] . As [latex]x\\to -\\infty[\/latex], [latex]f\\left(x\\right)\\to -\\infty[\/latex] , as [latex]x\\to \\infty[\/latex] , [latex]f\\left(x\\right)\\to \\infty[\/latex].
\"Graph<\/p>\n

47. x-intercepts [latex]\\left(0, 0\\right),\\left(-2, 0\\right),\\left(4, 0\\right)[\/latex] with multiplicity 1, y<\/em>-intercept [latex]\\left(0, 0\\right)[\/latex]. As [latex]x\\to -\\infty[\/latex] , [latex]f\\left(x\\right)\\to \\infty[\/latex] , as [latex]x\\to \\infty[\/latex] , [latex]f\\left(x\\right)\\to -\\infty[\/latex].
\"Graph<\/p>\n

49. [latex]f\\left(x\\right)=-\\frac{2}{9}\\left(x - 3\\right)\\left(x+1\\right)\\left(x+3\\right)[\/latex]<\/p>\n

53. [latex]f\\left(x\\right)=-\\frac{1}{8}\\left(x +4\\right)\\left(x+2\\right)\\left(x-1\\right)\\left(x-3\\right)[\/latex]<\/p>\n

55. [latex]f\\left(x\\right)=\\frac{1}{12}\\left(x +2\\right)^2\\left(x+3\\right)^2[\/latex]<\/p>\n

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

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

65. [latex]f\\left(x\\right)=-\\frac{3}{2}{\\left(2x - 1\\right)}^{2}\\left(x - 6\\right)\\left(x+2\\right)[\/latex]<\/p>\n

75. [latex]f\\left(x\\right)=4{x}^{3}-36{x}^{2}+80x[\/latex]<\/p>\n

77. [latex]f\\left(x\\right)=4{x}^{3}-36{x}^{2}+60x+100[\/latex]<\/p>\n

79. [latex]f\\left(x\\right)=\\pi \\left(9{x}^{3}+45{x}^{2}+72x+36\\right)[\/latex]<\/p>\n<\/p><\/div>\n<\/p><\/div>\n<\/p><\/div>\n","protected":false},"author":13,"menu_order":4,"template":"","meta":{"_candela_citation":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":224,"module-header":"","content_attributions":null,"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\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/273"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/users\/13"}],"version-history":[{"count":1,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/273\/revisions"}],"predecessor-version":[{"id":296,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/273\/revisions\/296"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/parts\/224"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/273\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/media?parent=273"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapter-type?post=273"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/contributor?post=273"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/license?post=273"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}