{"id":2397,"date":"2019-05-14T13:47:23","date_gmt":"2019-05-14T13:47:23","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/?post_type=chapter&p=2397"},"modified":"2019-05-14T13:47:23","modified_gmt":"2019-05-14T13:47:23","slug":"4-2-section-exercises","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/4-2-section-exercises\/","title":{"raw":"4.2 Section Exercises","rendered":"4.2 Section Exercises"},"content":{"raw":"
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4.2 Section Exercises<\/h3>\r\n
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Verbal<\/h4>\r\n
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1. What is the difference between an[latex]\\text{ }x\\text{-}[\/latex]intercept and a zero of a polynomial function[latex]\\text{ }f?\\text{ }[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135311581\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135311581\"]\r\n

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

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2. If a polynomial function of degree[latex]\\text{ }n\\text{ }[\/latex] has[latex]\\text{ }n\\text{ }[\/latex] distinct zeros, what do you know about the graph of the function?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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3. Explain how the Intermediate Value Theorem can assist us in finding a zero of a function.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135314696\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135314696\"]\r\n

If we evaluate the function at[latex]\\text{ }a\\text{ }[\/latex] and at[latex]\\text{ }b\\text{ }[\/latex] and the sign of the function value changes, then we know a zero exists between[latex]\\text{ }a\\text{ }[\/latex] and[latex]\\text{ }b.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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4. Explain how the factored form of the polynomial helps us in graphing it.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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5. If the graph of a polynomial just touches the[latex]\\text{ }x\\text{-}[\/latex]axis and then changes direction, what can we conclude about the factored form of the polynomial?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135621855\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135621855\"]\r\n

There will be a factor raised to an even power.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Algebraic<\/h4>\r\n

For the following exercises, find the[latex]\\text{ }x\\text{-}[\/latex] or t<\/em>-intercepts of the polynomial functions.<\/p>\r\n\r\n

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6. [latex]\\text{ }C\\left(t\\right)=2\\left(t-4\\right)\\left(t+1\\right)\\left(t-6\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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

[reveal-answer q=\"fs-id1165135347433\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135347433\"][latex]\\left(-2,0\\right),\\left(3,0\\right),\\left(-5,0\\right)[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n
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8. [latex]\\text{ }C\\left(t\\right)=4t{\\left(t-2\\right)}^{2}\\left(t+1\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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

[reveal-answer q=\"fs-id1165132941732\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165132941732\"]\r\n

[latex]\\text{ }\\left(3,0\\right),\\left(-1,0\\right),\\left(0,0\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n10. [latex]\\text{ }C\\left(t\\right)=2{t}^{4}-8{t}^{3}+6{t}^{2}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
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11. [latex]\\text{ }C\\left(t\\right)=4{t}^{4}+12{t}^{3}-40{t}^{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134149980\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134149980\"]\r\n

[latex]\\left(0,0\\right),\\text{ }\\left(-5,0\\right),\\text{ }\\left(2,0\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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

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

[reveal-answer q=\"fs-id1165134331988\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134331988\"]\r\n

[latex]\\left(0,0\\right),\\text{ }\\left(-5,0\\right),\\text{ }\\left(4,0\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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14. [latex]f\\left(x\\right)={x}^{3}+6{x}^{2}-7x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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

[reveal-answer q=\"fs-id1165134103103\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134103103\"]\r\n

[latex]\\left(2,0\\right),\\text{ }\\left(-2,0\\right),\\text{ }\\left(-1,0\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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16. [latex]f\\left(x\\right)={x}^{3}+2{x}^{2}-9x-18[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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

[reveal-answer q=\"fs-id1165134478959\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134478959\"]\r\n

[latex]\\left(-2,0\\right),\\text{ }\\left(2,0\\right),\\text{ }\\left(\\frac{1}{2},0\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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18. [latex]f\\left(x\\right)={x}^{6}-7{x}^{3}-8[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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

[reveal-answer q=\"fs-id1165134340053\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134340053\"]\r\n

[latex]\\left(1,0\\right),\\text{ }\\left(-1,0\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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

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

[reveal-answer q=\"fs-id1165133104635\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165133104635\"]\r\n

[latex]\\left(0,0\\right),\\text{ }\\left(\\sqrt{3},0\\right),\\text{ }\\left(-\\sqrt{3},0\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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

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

[reveal-answer q=\"fs-id1165135376471\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135376471\"]\r\n

[latex]\\left(0,0\\right),\\text{ }\\left(1,0\\right)\\text{, }\\left(-1,0\\right),\\text{ }\\left(2,0\\right),\\text{ }\\left(-2,0\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\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>\r\n\r\n

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24. [latex]f\\left(x\\right)={x}^{3}-9x,\\text{ }[\/latex] between[latex]\\text{ }x=-4\\text{ }[\/latex] and[latex]\\text{ }x=-2.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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25. [latex]f\\left(x\\right)={x}^{3}-9x,\\text{ }[\/latex] between[latex]\\text{ }x=2\\text{ }[\/latex] and[latex]\\text{ }x=4.[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135621876\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135621876\"]\r\n

[latex]f\\left(2\\right)=\u201310\\text{ }[\/latex] and[latex]\\text{ }f\\left(4\\right)=28.[\/latex] Sign change confirms.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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26. [latex]f\\left(x\\right)={x}^{5}-2x,\\text{ }[\/latex] between[latex]\\text{ }x=1\\text{ }[\/latex] and[latex]\\text{ }x=2.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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27. [latex]f\\left(x\\right)=-{x}^{4}+4,\\text{ }[\/latex] between[latex]\\text{ }x=1\\text{ }[\/latex] and[latex]\\text{ }x=3[\/latex] .<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165133073928\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165133073928\"]\r\n

[latex]f\\left(1\\right)=3\\text{ }[\/latex] and[latex]\\text{ }f\\left(3\\right)=\u201377.\\text{ }[\/latex] Sign change confirms.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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28. [latex]f\\left(x\\right)=-2{x}^{3}-x,\\text{ }[\/latex] between[latex]\\text{ }x=\u20131\\text{ }[\/latex] and[latex]\\text{ }x=1.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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29. [latex]f\\left(x\\right)={x}^{3}-100x+2,\\text{ }[\/latex] between[latex]\\text{ }x=0.01\\text{ }[\/latex] and[latex]\\text{ }x=0.1[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135524657\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135524657\"]\r\n

[latex]f\\left(0.01\\right)=1.000001\\text{ }[\/latex] and[latex]\\text{ }f\\left(0.1\\right)=\u20137.999.\\text{ }[\/latex] Sign change confirms.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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

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

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

[reveal-answer q=\"fs-id1165134534140\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134534140\"]\r\n

0 with multiplicity 2,[latex]\\text{ }-\\frac{3}{2}\\text{ }[\/latex] with multiplicity 5, 4 with multiplicity 2<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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

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

[reveal-answer q=\"fs-id1165134357524\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134357524\"]\r\n

0 with multiplicity 2, \u20132 with multiplicity 2<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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34. [latex]f\\left(x\\right)={\\left(2x+1\\right)}^{3}\\left(9{x}^{2}-6x+1\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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35. [latex]f\\left(x\\right)={\\left(3x+2\\right)}^{5}\\left({x}^{2}-10x+25\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165133277602\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165133277602\"]\r\n

[latex]-\\frac{2}{3}\\text{ }\\text{with}\\text{ }\\text{multiplicity}\\text{ }5\\text{,}\\text{ }5\\text{ }\\text{with}\\text{ }\\text{multiplicity}\\text{ }\\text{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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36. [latex]f\\left(x\\right)=x\\left(4{x}^{2}-12x+9\\right)\\left({x}^{2}+8x+16\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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

[reveal-answer q=\"fs-id1165135349287\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135349287\"]\r\n

[latex]\\text{0}\\text{ }\\text{with}\\text{ }\\text{multiplicity}\\text{ }4\\text{,}\\text{ }2\\text{ }\\text{with}\\text{ }\\text{multiplicity}\\text{ }1\\text{,}\\text{ }\u2013\\text{1}\\text{ }\\text{with}\\text{ }\\text{multiplicity}\\text{ }1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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

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

[reveal-answer q=\"fs-id1165133035945\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165133035945\"]\r\n

[latex]\\frac{3}{2}\\text{ }[\/latex] with multiplicity 2, 0 with multiplicity 3<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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

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41. [latex]f\\left(x\\right)=4{x}^{4}\\left(9{x}^{4}-12{x}^{3}+4{x}^{2}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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42. [latex]\\text{0}\\text{ }\\text{with}\\text{ }\\text{multiplicity}\\text{ }6\\text{,}\\text{ }\\frac{2}{3}\\text{ }\\text{with}\\text{ }\\text{multiplicity}\\text{ }2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Graphical<\/h4>\r\n

For the following exercises, graph the polynomial functions. Note[latex]\\text{ }x\\text{-}[\/latex] and[latex]\\text{ }y\\text{-}[\/latex]intercepts, multiplicity, and end behavior.<\/p>\r\n\r\n

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

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

[reveal-answer q=\"fs-id1165137701835\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137701835\"]\r\n

x<\/em>-intercepts, [latex]\\left(1, 0\\right)[\/latex] with multiplicity 2, [latex]\\left(\u20134, 0\\right)[\/latex] with multiplicity 1, [latex]y\\text{-}[\/latex] 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].<\/p>\r\n\"Graph<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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

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

[reveal-answer q=\"fs-id1165134040448\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134040448\"]\r\n

x<\/em>-intercepts [latex]\\left(3,0\\right)[\/latex] with multiplicity 3, [latex]\\left(2,0\\right)[\/latex] with multiplicity 2, [latex]y\\text{-}[\/latex] intercept [latex]\\left(0,\u2013108\\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]<\/p>\r\n\"Graph<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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47. [latex]m\\left(x\\right)=-2x\\left(x-1\\right)\\left(x+3\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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

[reveal-answer q=\"fs-id1165134547321\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134547321\"]\r\n

x<\/em>-intercepts [latex]\\left(0, 0\\right),\\phantom{\\rule{0.2em}{0ex}}\\left(\u20132, 0\\right),\\phantom{\\rule{0.2em}{0ex}}\\left(4, 0\\right)[\/latex] with multiplicity 1, [latex]y[\/latex]-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]<\/p>\r\n\"Graph<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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

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49.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n
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50.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165134087659\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134087659\"]\r\n

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

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51.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n
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52.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165133104571\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165133104571\"]\r\n

[latex]f\\left(x\\right)=\\frac{1}{4}{\\left(x+2\\right)}^{2}\\left(x-3\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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53.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n

For the following exercises, use the graph to identify zeros and multiplicity.<\/p>\r\n\r\n

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54.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165134156011\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134156011\"]\r\n

\u20134, \u20132, 1, 3 with multiplicity 1<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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55.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n
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56.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165134347476\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134347476\"]\r\n

\u20132, 3 each with multiplicity 2<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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57.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n

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

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58. Degree 3. Zeros at[latex]\\text{ }x=\u20132,[\/latex] [latex]\\text{ }x=1,\\text{ }[\/latex]and[latex]\\text{ }x=3.\\text{ }[\/latex]y<\/em>-intercept at[latex]\\text{ }\\left(0,\u20134\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137733543\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137733543\"]\r\n

[latex]f\\left(x\\right)=-\\frac{2}{3}\\left(x+2\\right)\\left(x-1\\right)\\left(x-3\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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59. Degree 3. Zeros at[latex]\\text{ }x=\\text{\u20135,}[\/latex] [latex]\\text{ }x=\u20132,[\/latex]and[latex]\\text{ }x=1.\\text{ }[\/latex]y<\/em>-intercept at[latex]\\text{ }\\left(0,6\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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60. Degree 5. Roots of multiplicity 2 at[latex]\\text{ }x=3\\text{ }[\/latex] and[latex]\\text{ }x=1\\text{ }[\/latex] , and a root of multiplicity 1 at[latex]\\text{ }x=\u20133.\\text{ }[\/latex] y<\/em>-intercept at[latex]\\text{ }\\left(0,9\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135459825\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135459825\"]\r\n

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

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61. Degree 4. Root of multiplicity 2 at[latex]\\text{ }x=4,\\text{ }[\/latex]and a roots of multiplicity 1 at[latex]\\text{ }x=1\\text{ }[\/latex]and[latex]\\text{ }x=\u20132.\\text{ }[\/latex]y<\/em>-intercept at[latex]\\text{ }\\left(0,\\text{\u2013}3\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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62. Degree 5. Double zero at[latex]\\text{ }x=1,\\text{ }[\/latex]and triple zero at[latex]\\text{ }x=3.\\text{ }[\/latex] Passes through the point[latex]\\text{ }\\left(2,15\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134277296\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134277296\"]\r\n

[latex]f\\left(x\\right)=-15{\\left(x-1\\right)}^{2}{\\left(x-3\\right)}^{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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63. Degree 3. Zeros at[latex]\\text{ }x=4,[\/latex][latex]\\text{ }x=3,[\/latex]and[latex]\\text{ }x=2.\\text{ }[\/latex]y<\/em>-intercept at[latex]\\text{ }\\left(0,-24\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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64. Degree 3. Zeros at[latex]\\text{ }x=-3,[\/latex] [latex]\\text{ }x=-2\\text{ }[\/latex] and[latex]\\text{ }x=1.\\text{ }[\/latex] y<\/em>-intercept at[latex]\\text{ }\\left(0,12\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135263639\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135263639\"]\r\n

[latex]f\\left(x\\right)=-2\\left(x+3\\right)\\left(x+2\\right)\\left(x-1\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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65. Degree 5. Roots of multiplicity 2 at[latex]\\text{ }x=-3\\text{ }[\/latex] and[latex]\\text{ }x=2\\text{ }[\/latex] and a root of multiplicity 1 at[latex]\\text{ }x=-2.[\/latex]<\/p>\r\n

y<\/em>-intercept at[latex]\\text{ }\\left(0, 4\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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66. Degree 4. Roots of multiplicity 2 at[latex]\\text{ }x=\\frac{1}{2}\\text{ }[\/latex]and roots of multiplicity 1 at[latex]\\text{ }x=6\\text{ }[\/latex]and[latex]\\text{ }x=-2.[\/latex]<\/p>\r\n

y<\/em>-intercept at[latex]\\text{ }\\left(0,18\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134332683\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134332683\"]\r\n

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

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67. Double zero at[latex]\\text{ }x=-3\\text{ }[\/latex] and triple zero at[latex]\\text{ }x=0.\\text{ }[\/latex] Passes through the point[latex]\\text{ }\\left(1,32\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Technology<\/h4>\r\n

For the following exercises, use a calculator to approximate local minima and maxima or the global minimum and maximum.<\/p>\r\n\r\n

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68. [latex]f\\left(x\\right)={x}^{3}-x-1[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165133348575\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165133348575\"]\r\n

local max[latex]\\text{ }\\left(\u2013\\text{.58, \u2013}.62\\right),\\text{ }[\/latex] local min[latex]\\text{ }\\left(\\text{.58, \u20131}\\text{.38}\\right)\\text{ }[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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

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

[reveal-answer q=\"fs-id1165135314791\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135314791\"]\r\n

global min[latex]\\text{ }\\left(\u2013\\text{.63, \u2013}\\text{.47}\\right)\\text{ }[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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

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

[reveal-answer q=\"fs-id1165134402645\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134402645\"]\r\n

global min[latex]\\text{ }\\text{(}\\text{.75, }\\text{.89)}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Extensions<\/h4>\r\n

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

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73.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n
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74.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165134402716\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134402716\"]\r\n

[latex]f\\left(x\\right)={\\left(x-500\\right)}^{2}\\left(x+200\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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75.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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Real-World Applications<\/h4>\r\n

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

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76. A rectangle has a length of 10 units and a width of 8 units. Squares of[latex]\\text{ }x\\text{ }[\/latex] by[latex]\\text{ }x\\text{ }[\/latex] units 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[latex]\\text{ }x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165133309788\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165133309788\"]\r\n

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

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77. Consider the same rectangle of the preceding problem. Squares of[latex]\\text{ }2x\\text{ }[\/latex] by[latex]\\text{ }2x\\text{ }[\/latex] units are cut out of each corner. Express the volume of the box as a polynomial in terms of[latex]\\text{ }x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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78. A square has sides of 12 units. Squares[latex]\\text{ }x\\text{ }+1\\text{ }[\/latex] by[latex]\\text{ }x\\text{ }+1\\text{ }[\/latex] units 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[latex]\\text{ }x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135546142\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135546142\"]\r\n

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

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79. A cylinder has a radius of[latex]\\text{ }x+2\\text{ }[\/latex] units and a height of 3 units greater. Express the volume of the cylinder as a polynomial function.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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80. A right circular cone has a radius of[latex]\\text{ }3x+6\\text{ }[\/latex] and a height 3 units less. Express the volume of the cone as a polynomial function. The volume of a cone is[latex]\\text{ }V=\\frac{1}{3}\\pi {r}^{2}h\\text{ }[\/latex] for radius[latex]\\text{ }r\\text{ }[\/latex] and height[latex]\\text{ }h.[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134094539\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134094539\"]\r\n

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

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4.2 Section Exercises<\/h3>\n
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Verbal<\/h4>\n
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1. What is the difference between an[latex]\\text{ }x\\text{-}[\/latex]intercept and a zero of a polynomial function[latex]\\text{ }f?\\text{ }[\/latex]<\/p>\n<\/div>\n

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

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2. If a polynomial function of degree[latex]\\text{ }n\\text{ }[\/latex] has[latex]\\text{ }n\\text{ }[\/latex] distinct zeros, what do you know about the graph of the function?<\/p>\n<\/div>\n<\/div>\n

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3. Explain how the Intermediate Value Theorem can assist us in finding a zero of a function.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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If we evaluate the function at[latex]\\text{ }a\\text{ }[\/latex] and at[latex]\\text{ }b\\text{ }[\/latex] and the sign of the function value changes, then we know a zero exists between[latex]\\text{ }a\\text{ }[\/latex] and[latex]\\text{ }b.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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4. Explain how the factored form of the polynomial helps us in graphing it.<\/p>\n<\/div>\n<\/div>\n

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5. If the graph of a polynomial just touches the[latex]\\text{ }x\\text{-}[\/latex]axis and then changes direction, what can we conclude about the factored form of the polynomial?<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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There will be a factor raised to an even power.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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Algebraic<\/h4>\n

For the following exercises, find the[latex]\\text{ }x\\text{-}[\/latex] or t<\/em>-intercepts of the polynomial functions.<\/p>\n

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6. [latex]\\text{ }C\\left(t\\right)=2\\left(t-4\\right)\\left(t+1\\right)\\left(t-6\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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

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Show Solution<\/span><\/p>\n
[latex]\\left(-2,0\\right),\\left(3,0\\right),\\left(-5,0\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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8. [latex]\\text{ }C\\left(t\\right)=4t{\\left(t-2\\right)}^{2}\\left(t+1\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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

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Show Solution<\/span><\/p>\n
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[latex]\\text{ }\\left(3,0\\right),\\left(-1,0\\right),\\left(0,0\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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10. [latex]\\text{ }C\\left(t\\right)=2{t}^{4}-8{t}^{3}+6{t}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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11. [latex]\\text{ }C\\left(t\\right)=4{t}^{4}+12{t}^{3}-40{t}^{2}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]\\left(0,0\\right),\\text{ }\\left(-5,0\\right),\\text{ }\\left(2,0\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

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

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Show Solution<\/span><\/p>\n
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[latex]\\left(0,0\\right),\\text{ }\\left(-5,0\\right),\\text{ }\\left(4,0\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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14. [latex]f\\left(x\\right)={x}^{3}+6{x}^{2}-7x[\/latex]<\/p>\n<\/div>\n<\/div>\n

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

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Show Solution<\/span><\/p>\n
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[latex]\\left(2,0\\right),\\text{ }\\left(-2,0\\right),\\text{ }\\left(-1,0\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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16. [latex]f\\left(x\\right)={x}^{3}+2{x}^{2}-9x-18[\/latex]<\/p>\n<\/div>\n<\/div>\n

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

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Show Solution<\/span><\/p>\n
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[latex]\\left(-2,0\\right),\\text{ }\\left(2,0\\right),\\text{ }\\left(\\frac{1}{2},0\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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18. [latex]f\\left(x\\right)={x}^{6}-7{x}^{3}-8[\/latex]<\/p>\n<\/div>\n<\/div>\n

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

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Show Solution<\/span><\/p>\n
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[latex]\\left(1,0\\right),\\text{ }\\left(-1,0\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

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

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Show Solution<\/span><\/p>\n
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[latex]\\left(0,0\\right),\\text{ }\\left(\\sqrt{3},0\\right),\\text{ }\\left(-\\sqrt{3},0\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

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

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Show Solution<\/span><\/p>\n
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[latex]\\left(0,0\\right),\\text{ }\\left(1,0\\right)\\text{, }\\left(-1,0\\right),\\text{ }\\left(2,0\\right),\\text{ }\\left(-2,0\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\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

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24. [latex]f\\left(x\\right)={x}^{3}-9x,\\text{ }[\/latex] between[latex]\\text{ }x=-4\\text{ }[\/latex] and[latex]\\text{ }x=-2.[\/latex]<\/p>\n<\/div>\n<\/div>\n

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

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Show Solution<\/span><\/p>\n
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[latex]f\\left(2\\right)=\u201310\\text{ }[\/latex] and[latex]\\text{ }f\\left(4\\right)=28.[\/latex] Sign change confirms.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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26. [latex]f\\left(x\\right)={x}^{5}-2x,\\text{ }[\/latex] between[latex]\\text{ }x=1\\text{ }[\/latex] and[latex]\\text{ }x=2.[\/latex]<\/p>\n<\/div>\n<\/div>\n

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

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Show Solution<\/span><\/p>\n
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[latex]f\\left(1\\right)=3\\text{ }[\/latex] and[latex]\\text{ }f\\left(3\\right)=\u201377.\\text{ }[\/latex] Sign change confirms.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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28. [latex]f\\left(x\\right)=-2{x}^{3}-x,\\text{ }[\/latex] between[latex]\\text{ }x=\u20131\\text{ }[\/latex] and[latex]\\text{ }x=1.[\/latex]<\/p>\n<\/div>\n<\/div>\n

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29. [latex]f\\left(x\\right)={x}^{3}-100x+2,\\text{ }[\/latex] between[latex]\\text{ }x=0.01\\text{ }[\/latex] and[latex]\\text{ }x=0.1[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]f\\left(0.01\\right)=1.000001\\text{ }[\/latex] and[latex]\\text{ }f\\left(0.1\\right)=\u20137.999.\\text{ }[\/latex] Sign change confirms.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

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

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

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Show Solution<\/span><\/p>\n
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0 with multiplicity 2,[latex]\\text{ }-\\frac{3}{2}\\text{ }[\/latex] with multiplicity 5, 4 with multiplicity 2<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

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

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Show Solution<\/span><\/p>\n
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0 with multiplicity 2, \u20132 with multiplicity 2<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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34. [latex]f\\left(x\\right)={\\left(2x+1\\right)}^{3}\\left(9{x}^{2}-6x+1\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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35. [latex]f\\left(x\\right)={\\left(3x+2\\right)}^{5}\\left({x}^{2}-10x+25\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\n

[latex]-\\frac{2}{3}\\text{ }\\text{with}\\text{ }\\text{multiplicity}\\text{ }5\\text{,}\\text{ }5\\text{ }\\text{with}\\text{ }\\text{multiplicity}\\text{ }\\text{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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36. [latex]f\\left(x\\right)=x\\left(4{x}^{2}-12x+9\\right)\\left({x}^{2}+8x+16\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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

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Show Solution<\/span><\/p>\n
\n

[latex]\\text{0}\\text{ }\\text{with}\\text{ }\\text{multiplicity}\\text{ }4\\text{,}\\text{ }2\\text{ }\\text{with}\\text{ }\\text{multiplicity}\\text{ }1\\text{,}\\text{ }\u2013\\text{1}\\text{ }\\text{with}\\text{ }\\text{multiplicity}\\text{ }1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

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

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Show Solution<\/span><\/p>\n
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[latex]\\frac{3}{2}\\text{ }[\/latex] with multiplicity 2, 0 with multiplicity 3<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

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41. [latex]f\\left(x\\right)=4{x}^{4}\\left(9{x}^{4}-12{x}^{3}+4{x}^{2}\\right)[\/latex]<\/p>\n<\/div>\n

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42. [latex]\\text{0}\\text{ }\\text{with}\\text{ }\\text{multiplicity}\\text{ }6\\text{,}\\text{ }\\frac{2}{3}\\text{ }\\text{with}\\text{ }\\text{multiplicity}\\text{ }2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

\n

Graphical<\/h4>\n

For the following exercises, graph the polynomial functions. Note[latex]\\text{ }x\\text{-}[\/latex] and[latex]\\text{ }y\\text{-}[\/latex]intercepts, multiplicity, and end behavior.<\/p>\n

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

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

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Show Solution<\/span><\/p>\n
\n

x<\/em>-intercepts, [latex]\\left(1, 0\\right)[\/latex] with multiplicity 2, [latex]\\left(\u20134, 0\\right)[\/latex] with multiplicity 1, [latex]y\\text{-}[\/latex] 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].<\/p>\n

\"Graph<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

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

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Show Solution<\/span><\/p>\n
\n

x<\/em>-intercepts [latex]\\left(3,0\\right)[\/latex] with multiplicity 3, [latex]\\left(2,0\\right)[\/latex] with multiplicity 2, [latex]y\\text{-}[\/latex] intercept [latex]\\left(0,\u2013108\\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]<\/p>\n

\"Graph<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

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

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Show Solution<\/span><\/p>\n
\n

x<\/em>-intercepts [latex]\\left(0, 0\\right),\\phantom{\\rule{0.2em}{0ex}}\\left(\u20132, 0\\right),\\phantom{\\rule{0.2em}{0ex}}\\left(4, 0\\right)[\/latex] with multiplicity 1, [latex]y[\/latex]-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]<\/p>\n

\"Graph<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

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49.\u00a0\"Graph<\/span><\/div>\n<\/div>\n
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50.\u00a0\"Graph<\/span><\/div>\n
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Show Solution<\/span><\/p>\n
\n

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

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51.\u00a0\"Graph<\/span><\/div>\n<\/div>\n
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52.\u00a0\"Graph<\/span><\/div>\n
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Show Solution<\/span><\/p>\n
\n

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

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53.\u00a0\"Graph<\/span><\/div>\n<\/div>\n

For the following exercises, use the graph to identify zeros and multiplicity.<\/p>\n

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54.\u00a0\"Graph<\/span><\/div>\n
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Show Solution<\/span><\/p>\n
\n

\u20134, \u20132, 1, 3 with multiplicity 1<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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55.\u00a0\"Graph<\/span><\/div>\n<\/div>\n
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56.\u00a0\"Graph<\/span><\/div>\n
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Show Solution<\/span><\/p>\n
\n

\u20132, 3 each with multiplicity 2<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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57.\u00a0\"Graph<\/span><\/div>\n<\/div>\n

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

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58. Degree 3. Zeros at[latex]\\text{ }x=\u20132,[\/latex] [latex]\\text{ }x=1,\\text{ }[\/latex]and[latex]\\text{ }x=3.\\text{ }[\/latex]y<\/em>-intercept at[latex]\\text{ }\\left(0,\u20134\\right).[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\n

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

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59. Degree 3. Zeros at[latex]\\text{ }x=\\text{\u20135,}[\/latex] [latex]\\text{ }x=\u20132,[\/latex]and[latex]\\text{ }x=1.\\text{ }[\/latex]y<\/em>-intercept at[latex]\\text{ }\\left(0,6\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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60. Degree 5. Roots of multiplicity 2 at[latex]\\text{ }x=3\\text{ }[\/latex] and[latex]\\text{ }x=1\\text{ }[\/latex] , and a root of multiplicity 1 at[latex]\\text{ }x=\u20133.\\text{ }[\/latex] y<\/em>-intercept at[latex]\\text{ }\\left(0,9\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\n

[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<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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61. Degree 4. Root of multiplicity 2 at[latex]\\text{ }x=4,\\text{ }[\/latex]and a roots of multiplicity 1 at[latex]\\text{ }x=1\\text{ }[\/latex]and[latex]\\text{ }x=\u20132.\\text{ }[\/latex]y<\/em>-intercept at[latex]\\text{ }\\left(0,\\text{\u2013}3\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n

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62. Degree 5. Double zero at[latex]\\text{ }x=1,\\text{ }[\/latex]and triple zero at[latex]\\text{ }x=3.\\text{ }[\/latex] Passes through the point[latex]\\text{ }\\left(2,15\\right).[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\n

[latex]f\\left(x\\right)=-15{\\left(x-1\\right)}^{2}{\\left(x-3\\right)}^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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63. Degree 3. Zeros at[latex]\\text{ }x=4,[\/latex][latex]\\text{ }x=3,[\/latex]and[latex]\\text{ }x=2.\\text{ }[\/latex]y<\/em>-intercept at[latex]\\text{ }\\left(0,-24\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n

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64. Degree 3. Zeros at[latex]\\text{ }x=-3,[\/latex] [latex]\\text{ }x=-2\\text{ }[\/latex] and[latex]\\text{ }x=1.\\text{ }[\/latex] y<\/em>-intercept at[latex]\\text{ }\\left(0,12\\right).[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\n

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

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65. Degree 5. Roots of multiplicity 2 at[latex]\\text{ }x=-3\\text{ }[\/latex] and[latex]\\text{ }x=2\\text{ }[\/latex] and a root of multiplicity 1 at[latex]\\text{ }x=-2.[\/latex]<\/p>\n

y<\/em>-intercept at[latex]\\text{ }\\left(0, 4\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n

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66. Degree 4. Roots of multiplicity 2 at[latex]\\text{ }x=\\frac{1}{2}\\text{ }[\/latex]and roots of multiplicity 1 at[latex]\\text{ }x=6\\text{ }[\/latex]and[latex]\\text{ }x=-2.[\/latex]<\/p>\n

y<\/em>-intercept at[latex]\\text{ }\\left(0,18\\right).[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\n

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

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67. Double zero at[latex]\\text{ }x=-3\\text{ }[\/latex] and triple zero at[latex]\\text{ }x=0.\\text{ }[\/latex] Passes through the point[latex]\\text{ }\\left(1,32\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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Technology<\/h4>\n

For the following exercises, use a calculator to approximate local minima and maxima or the global minimum and maximum.<\/p>\n

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

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Show Solution<\/span><\/p>\n
\n

local max[latex]\\text{ }\\left(\u2013\\text{.58, \u2013}.62\\right),\\text{ }[\/latex] local min[latex]\\text{ }\\left(\\text{.58, \u20131}\\text{.38}\\right)\\text{ }[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

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

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Show Solution<\/span><\/p>\n
\n

global min[latex]\\text{ }\\left(\u2013\\text{.63, \u2013}\\text{.47}\\right)\\text{ }[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

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

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Show Solution<\/span><\/p>\n
\n

global min[latex]\\text{ }\\text{(}\\text{.75, }\\text{.89)}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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Extensions<\/h4>\n

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

\n
73.\u00a0\"Graph<\/span><\/div>\n<\/div>\n
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74.\u00a0\"Graph<\/span><\/div>\n
\n
Show Solution<\/span><\/p>\n
\n

[latex]f\\left(x\\right)={\\left(x-500\\right)}^{2}\\left(x+200\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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75.\u00a0\"Graph<\/span><\/div>\n<\/div>\n<\/div>\n
\n

Real-World Applications<\/h4>\n

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

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76. A rectangle has a length of 10 units and a width of 8 units. Squares of[latex]\\text{ }x\\text{ }[\/latex] by[latex]\\text{ }x\\text{ }[\/latex] units 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[latex]\\text{ }x.[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\n

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

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77. Consider the same rectangle of the preceding problem. Squares of[latex]\\text{ }2x\\text{ }[\/latex] by[latex]\\text{ }2x\\text{ }[\/latex] units are cut out of each corner. Express the volume of the box as a polynomial in terms of[latex]\\text{ }x.[\/latex]<\/p>\n<\/div>\n<\/div>\n

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78. A square has sides of 12 units. Squares[latex]\\text{ }x\\text{ }+1\\text{ }[\/latex] by[latex]\\text{ }x\\text{ }+1\\text{ }[\/latex] units 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[latex]\\text{ }x.[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\n

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

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79. A cylinder has a radius of[latex]\\text{ }x+2\\text{ }[\/latex] units and a height of 3 units greater. Express the volume of the cylinder as a polynomial function.<\/p>\n<\/div>\n<\/div>\n

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80. A right circular cone has a radius of[latex]\\text{ }3x+6\\text{ }[\/latex] and a height 3 units less. Express the volume of the cone as a polynomial function. The volume of a cone is[latex]\\text{ }V=\\frac{1}{3}\\pi {r}^{2}h\\text{ }[\/latex] for radius[latex]\\text{ }r\\text{ }[\/latex] and height[latex]\\text{ }h.[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\n

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

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