{"id":16622,"date":"2020-04-07T05:05:53","date_gmt":"2020-04-07T05:05:53","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/?post_type=chapter&p=16622"},"modified":"2020-05-21T04:45:11","modified_gmt":"2020-05-21T04:45:11","slug":"chapter-3-solutions-to-odd-problems","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/chapter\/chapter-3-solutions-to-odd-problems\/","title":{"raw":"Chapter 2 Solutions to Odd-Numbered Problems","rendered":"Chapter 2 Solutions to Odd-Numbered Problems"},"content":{"raw":"

Section 2.1 Solutions<\/h2>\r\n1. It is a second-degree equation (the highest variable exponent is 2).\r\n\r\n3. We want to take advantage of the zero property of multiplication in the fact that if [latex]a\\cdot b=0[\/latex] then it must follow that each factor separately offers a solution to the product being zero [latex]a=0\\text{ or }b=0[\/latex].\r\n\r\n5. One, when no linear term is present (no [latex]x[\/latex] term), such as [latex]x^{2}=16[\/latex]. Two, when the equation is already in the form [latex](ax+b)^{2}=d[\/latex].\r\n\r\n7. [latex]\\text{ }x=6,x=3[\/latex]\r\n\r\n9. [latex]\\text{ }x=-\\dfrac{5}{2},x=-\\dfrac{1}{3}[\/latex]\r\n\r\n11. [latex]\\text{ }x=5,x=-5[\/latex]\r\n\r\n13. [latex]\\text{ }x=-\\dfrac{3}{2},x=\\dfrac{3}{2}[\/latex]\r\n\r\n15. [latex]\\text{ }x=-2,x=3[\/latex]\r\n\r\n17. [latex]\\text{ }x=0,x=-\\dfrac{3}{7}[\/latex]\r\n\r\n19. [latex]\\text{ }x=6,x=-6[\/latex]\r\n\r\n21. [latex]\\text{ }x=6,x=-4[\/latex]\r\n\r\n23. [latex]\\text{ }x=1,x=-2[\/latex]\r\n\r\n25. [latex]\\text{ }x=-2,x=11[\/latex]\r\n\r\n27. [latex]\\text{ }x=3\\pm\\sqrt{22}[\/latex]\r\n\r\n29. [latex]\\text{ }z=\\dfrac{2}{3},z=-\\dfrac{1}{2}[\/latex]\r\n\r\n31. [latex]\\text{ }x=\\dfrac{3\\pm\\sqrt{17}}{4}[\/latex]\r\n\r\n33. Not real\r\n\r\n35. One rational\r\n\r\n37. Two real; rational\r\n\r\n39. [latex]\\text{ }x=\\dfrac{-1\\pm\\sqrt{17}}{2}[\/latex]\r\n\r\n41. [latex]\\text{ }x=\\dfrac{5\\pm\\sqrt{13}}{6}[\/latex]\r\n\r\n43. [latex]\\text{ }x=\\dfrac{-1\\pm\\sqrt{17}}{8}[\/latex]\r\n\r\n45. [latex]ax^{2}+bx+c=0 \\\\ x^2+\\dfrac{b}{a}x=-\\dfrac{c}{a} \\\\ x^{2}+\\dfrac{b}{a}x+\\dfrac{b^{2}}{4a^{2}}=-\\dfrac{c}{a}+\\dfrac{b}{4a^{2}} \\\\ \\left(x+\\dfrac{b}{2a}\\right)^{2}=\\dfrac{b^{2}-4ac}{4a^{2}} \\\\ x+\\dfrac{b}{2a}=\\pm \\sqrt{\\dfrac{b^{2}-4ac}{4a^{2}}} \\\\ x=\\dfrac{-b \\pm \\sqrt{b^{2}-4ac}}{2a}[\/latex]\r\n\r\n47. [latex]\\text{ }x(x+10)=119[\/latex]; [latex]7\\text{ }ft[\/latex] and [latex]9\\text{ }ft[\/latex]\r\n\r\n49. Maximum at [latex]x=70[\/latex]\r\n\r\n51. The quadratic equation would be [latex]\\left(100x-0.5x^{2}\\right)-(60x+300)=300[\/latex]. The two values of [latex]x[\/latex] are 20 and 60.\r\n\r\n53. [latex]\\text{ }3\\text{ }ft[\/latex]\r\n

Section 2.2 Solutions<\/h2>\r\n1.\u00a0Add the real parts together and the imaginary parts together.\r\n\r\n3.\u00a0i<\/em>\u00a0times i<\/em>\u00a0equals \u20131, which is not imaginary. (answers vary)\r\n\r\n5.\u00a0[latex]-8+2i[\/latex]\r\n\r\n7.\u00a0[latex]14+7i[\/latex]\r\n\r\n9.\u00a0[latex]-\\frac{23}{29}+\\frac{15}{29}i[\/latex]\r\n\r\n11.\u00a02 real and 0 nonreal\r\n\r\n13.\r\n\"Graph\r\n\r\n15.\r\n\"Graph\r\n\r\n17.\u00a0[latex]8-i[\/latex]\r\n\r\n19.\u00a0[latex]-11+4i[\/latex]\r\n\r\n21.\u00a0[latex]2 - 5i[\/latex]\r\n\r\n23.\u00a0[latex]6+15i[\/latex]\r\n\r\n25.\u00a0[latex]-16+32i[\/latex]\r\n\r\n27.\u00a0[latex]-4 - 7i[\/latex]\r\n\r\n29.\u00a025\r\n\r\n31.\u00a0[latex]2-\\frac{2}{3}i[\/latex]\r\n\r\n33.\u00a0[latex]4 - 6i[\/latex]\r\n\r\n35.\u00a0[latex]\\frac{2}{5}+\\frac{11}{5}i[\/latex]\r\n\r\n37. 15i<\/em>\r\n\r\n39.\u00a0[latex]1+i\\sqrt{3}[\/latex]\r\n\r\n41. 1\r\n\r\n43. \u20131\r\n\r\n45.\u00a0128i<\/em>\r\n\r\n47.\u00a0[latex]{\\left(\\frac{\\sqrt{3}}{2}+\\frac{1}{2}i\\right)}^{6}=-1[\/latex]\r\n\r\n49. 3i<\/em>\r\n\r\n51.\u00a00\r\n\r\n53.\u00a05 \u2013 5i<\/em>\r\n\r\n55. \u20132i<\/em>\r\n\r\n57.\u00a0[latex]\\frac{9}{2}-\\frac{9}{2}i[\/latex]\r\n

Section 2.3 Solutions<\/h2>\r\n1.\u00a0When written in that form, the vertex can be easily identified.\r\n\r\n3.\u00a0If [latex]a=0[\/latex] then the function becomes a linear function.\r\n\r\n5.\u00a0If possible, we can use factoring. Otherwise, we can use the quadratic formula.\r\n\r\n7.\u00a0[latex]f\\left(x\\right)={\\left(x+1\\right)}^{2}-2[\/latex], Vertex [latex]\\left(-1,-4\\right)[\/latex]\r\n\r\n9.\u00a0[latex]f\\left(x\\right)={\\left(x+\\frac{5}{2}\\right)}^{2}-\\frac{33}{4}[\/latex], Vertex [latex]\\left(-\\frac{5}{2},-\\frac{33}{4}\\right)[\/latex]\r\n\r\n11.\u00a0[latex]f\\left(x\\right)=3{\\left(x - 1\\right)}^{2}-12[\/latex], Vertex [latex]\\left(1,-12\\right)[\/latex]\r\n\r\n13.\u00a0[latex]f\\left(x\\right)=3{\\left(x-\\frac{5}{6}\\right)}^{2}-\\frac{37}{12}[\/latex], Vertex [latex]\\left(\\frac{5}{6},-\\frac{37}{12}\\right)[\/latex]\r\n\r\n15.\u00a0Minimum is [latex]-\\frac{17}{2}[\/latex] and occurs at [latex]\\frac{5}{2}[\/latex]. Axis of symmetry is [latex]x=\\frac{5}{2}[\/latex].\r\n\r\n17.\u00a0Minimum is [latex]-\\frac{17}{16}[\/latex] and occurs at [latex]-\\frac{1}{8}[\/latex]. Axis of symmetry is [latex]x=-\\frac{1}{8}[\/latex].\r\n\r\n19.\u00a0Minimum is [latex]-\\frac{7}{2}[\/latex] and occurs at \u20133. Axis of symmetry is [latex]x=-3[\/latex].\r\n\r\n21.\u00a0Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]. Range is [latex]\\left[2,\\infty \\right)[\/latex].\r\n\r\n23.\u00a0Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]. Range is [latex]\\left[-5,\\infty \\right)[\/latex].\r\n\r\n25.\u00a0Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]. Range is [latex]\\left[-12,\\infty \\right)[\/latex].\r\n\r\n27.\u00a0[latex]\\left\\{2i\\sqrt{2},-2i\\sqrt{2}\\right\\}[\/latex]\r\n\r\n29.\u00a0[latex]\\left\\{3i\\sqrt{3},-3i\\sqrt{3}\\right\\}[\/latex]\r\n\r\n31.\u00a0[latex]\\left\\{2+i,2-i\\right\\}[\/latex]\r\n\r\n33.\u00a0[latex]\\left\\{2+3i,2 - 3i\\right\\}[\/latex]\r\n\r\n35.\u00a0[latex]\\left\\{5+i,5-i\\right\\}[\/latex]\r\n\r\n37.\u00a0[latex]\\left\\{2+2\\sqrt{6}, 2 - 2\\sqrt{6}\\right\\}[\/latex]\r\n\r\n39.\u00a0[latex]\\left\\{-\\frac{1}{2}+\\frac{3}{2}i, -\\frac{1}{2}-\\frac{3}{2}i\\right\\}[\/latex]\r\n\r\n41.\u00a0[latex]\\left\\{-\\frac{3}{5}+\\frac{1}{5}i, -\\frac{3}{5}-\\frac{1}{5}i\\right\\}[\/latex]\r\n\r\n43.\u00a0[latex]\\left\\{-\\frac{1}{2}+\\frac{1}{2}i\\sqrt{7}, -\\frac{1}{2}-\\frac{1}{2}i\\sqrt{7}\\right\\}[\/latex]\r\n\r\n45.\u00a0[latex]f\\left(x\\right)={x}^{2}-4x+4[\/latex]\r\n\r\n47.\u00a0[latex]f\\left(x\\right)={x}^{2}+1[\/latex]\r\n\r\n49.\u00a0[latex]f\\left(x\\right)=\\frac{6}{49}{x}^{2}+\\frac{60}{49}x+\\frac{297}{49}[\/latex]\r\n\r\n51.\u00a0[latex]f\\left(x\\right)=-{x}^{2}+1[\/latex]\r\n\r\n53.\u00a0Vertex [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].\r\n\r\n\"Graph\r\n\r\n55.\u00a0Vertex [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].\r\n\r\n\"Graph\r\n\r\n57.\u00a0Vertex [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].\r\n\r\n\"Graph\r\n\r\n59.\u00a0[latex]f\\left(x\\right)={x}^{2}-4x+1[\/latex]\r\n\r\n61.\u00a0[latex]f\\left(x\\right)=-2{x}^{2}+8x - 1[\/latex]\r\n\r\n63.\u00a0[latex]f\\left(x\\right)=\\frac{1}{2}{x}^{2}-3x+\\frac{7}{2}[\/latex]\r\n\r\n65.\u00a0[latex]f\\left(x\\right)={x}^{2}+1[\/latex]\r\n\r\n67.\u00a0[latex]f\\left(x\\right)=2-{x}^{2}[\/latex]\r\n\r\n69.\u00a0[latex]f\\left(x\\right)=2{x}^{2}[\/latex]\r\n\r\n71.\u00a0The graph is shifted up or down (a vertical shift).\r\n\r\n73.\u00a050 feet\r\n\r\n75.\u00a0Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]. Range is [latex]\\left[-2,\\infty \\right)[\/latex].\r\n\r\n77.\u00a0Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex] Range is [latex]\\left(-\\infty ,11\\right][\/latex].\r\n\r\n79.\u00a0[latex]f\\left(x\\right)=2{x}^{2}-1[\/latex]\r\n\r\n81.\u00a0[latex]f\\left(x\\right)=3{x}^{2}-9[\/latex]\r\n\r\n83.\u00a0[latex]f\\left(x\\right)=5{x}^{2}-77[\/latex]\r\n\r\n85.\u00a050 feet by 50 feet. Maximize [latex]f\\left(x\\right)=-{x}^{2}+100x[\/latex].\r\n\r\n87.\u00a0125 feet by 62.5 feet. Maximize [latex]f\\left(x\\right)=-2{x}^{2}+250x[\/latex].\r\n\r\n89. 6 and \u20136; product is \u201336; maximize [latex]f\\left(x\\right)={x}^{2}+12x[\/latex].\r\n\r\n91.\u00a02909.56 meters\r\n\r\n93.\u00a0$10.70\r\n

Section 2.4 Solutions<\/h2>\r\n1.\u00a0The 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].\r\n\r\n3. The maximum number of turning points is always 1 less than the degree.\r\n\r\n5. The factored form is used to find the zeros.\u00a0 The behavior at each zero can be found from the factored form, and this aids in graphing.\r\n\r\n7. Degree 5 polynomial; [latex]a_{n}=6[\/latex]\r\n\r\n9. Not a polynomial\r\n\r\n11. Degree 4 polynomial; [latex]a_{n}=2[\/latex]\r\n\r\n13. Not a polynomial\r\n\r\n15. 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].\r\n\r\n17.\u00a0y<\/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].\r\n\r\n19.\u00a0y<\/em>-intercept is [latex]\\left(0,0\\right)[\/latex]. x-intercepts are [latex]\\left(0,0\\right),\\left(4,0\\right)[\/latex], and [latex]\\left(-2, 0\\right)[\/latex].\r\n\r\n21.\u00a0 2\r\n\r\n23.\u00a0 1\r\n\r\n25.\u00a0 4\r\n\r\n27.\u00a0Yes. Number of turning points is 2. Least possible degree is 3.\r\n\r\n29.\u00a0Yes. Number of turning points is 1. Least possible degree is 2.\r\n\r\n31.\u00a0Yes. Number of turning points is 0. Least possible degree is 1.\r\n\r\n33.\u00a0Yes. Number of turning points is 0. Least possible degree is 1.\r\n\r\n33.\u00a0 0 with multiplicity 2 and behavior: [latex]4x^{2}[\/latex]\r\n[latex]\u20132[\/latex] with multiplicity 2 and behavior: [latex]4(x+2)^{2}[\/latex]\r\n\r\n35.[latex]\\text{ }-\\frac{2}{3}[\/latex] with multiplicity 5 and behavior: [latex]\\frac{289}{9}(3x+2)^3[\/latex]\r\n5 with multiplicity 2 and behavior: [latex]4913(x-5)^{2}[\/latex]\r\n\r\n37.\u00a00 with multiplicity 4 and behavior: [latex]-2x^{4}[\/latex]\r\n2 with multiplicity 1 and behavior: [latex]48(x-2)[\/latex]\r\n-1 with multiplicity 1 and behavior: [latex]-(x+1)[\/latex]\r\n\r\n39.\u00a0[latex]\\text{ }\\frac{3}{2}[\/latex] with multiplicity 2 and behavior: [latex]\\frac{27}{8}(2x-3)^2[\/latex]\r\n0 with multiplicity 3 and behavior: [latex]9x^{3}[\/latex]\r\n\r\n41.\u00a00 with multiplicity 6 and behavior: [latex]16x^{6}[\/latex]\r\n[latex]\\frac{2}{3}[\/latex] with multiplicity 2 and behavior: [latex]\\frac{729}{16}(3x-2)^{2}[\/latex]\r\n\r\n43.\u00a0zeros [latex]\\left(1, 0\\right)[\/latex]\u00a0with multiplicity 2, [latex]\\left(-4, 0\\right)[\/latex] with multiplicity 1, y-<\/em>intercept [latex]\\left(0, 4\\right)[\/latex]. As\u00a0[latex]x\\to -\\infty [\/latex] ,\u00a0[latex]f\\left(x\\right)\\to -\\infty [\/latex] , as\u00a0[latex]x\\to \\infty [\/latex] ,\u00a0[latex]f\\left(x\\right)\\to \\infty [\/latex].\r\nAt [latex]x=-4[\/latex] the behavior is: [latex]25(x+4)[\/latex]\r\nAt [latex]x=1[\/latex] the behavior is: [latex]5(x-1)^{2}[\/latex].\r\n\"Graph\r\n\r\n45.\u00a0zeros\u00a0[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\u00a0[latex]x\\to -\\infty [\/latex],\u00a0[latex]f\\left(x\\right)\\to -\\infty [\/latex] , as [latex]x\\to \\infty [\/latex] ,\u00a0[latex]f\\left(x\\right)\\to \\infty [\/latex].\r\nAt [latex]x=2[\/latex] the behavior is: [latex]-(x-2)^{2}[\/latex].\r\nAt [latex]x=3[\/latex] the behavior is: [latex](x-3)^{3}[\/latex].\r\n\"Graph\r\n\r\n47.\u00a0x-intercepts [latex]\\left(0, 0\\right),\\left(-2, 0\\right),\\left(4, 0\\right)[\/latex]\u00a0with multiplicity 1, y<\/em>-intercept [latex]\\left(0, 0\\right)[\/latex]. As\u00a0[latex]x\\to -\\infty [\/latex] ,\u00a0[latex]f\\left(x\\right)\\to \\infty [\/latex] , as\u00a0[latex]x\\to \\infty [\/latex] ,\u00a0[latex]f\\left(x\\right)\\to -\\infty [\/latex].\r\nAt [latex]x=0[\/latex] the behavior is: [latex]24x[\/latex]\r\nAt [latex]x=-2[\/latex] the behavior is: [latex]-36(x+2)[\/latex]\r\nAt [latex]x=4[\/latex] the behavior is: [latex]-72(x-4)[\/latex].\r\n\"Graph\r\n\r\n49.\u00a0The <\/span>y<\/em>-intercept is [latex]\\left(0, 0\\right)[\/latex].\u00a0The\u00a0zero<\/span>s 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]<\/span>\r\nAt [latex]x=0[\/latex] the behavior is: [latex]-2x^3[\/latex]\r\nAt [latex]x=2[\/latex] the behavior is: [latex]8(x-2)[\/latex].\r\n\"Graph\r\n\r\n51.\u00a0[latex]f\\left(x\\right)=x^{2}+x-6[\/latex]\r\n\r\n53.\u00a0[latex]f\\left(x\\right)=x(x-2)^{2}[\/latex] or\u00a0[latex]f\\left(x\\right)=x^{2}(x-2)[\/latex]\r\n\r\n55. [latex]f\\left(x\\right)=x^{4}+1[\/latex]\r\n\r\n57. [latex]f\\left(x\\right)=-\\frac{2}{9}(x+3)(x+1)(x-3)[\/latex]\r\n\r\n59. [latex]f\\left(x\\right)=\\frac{1}{4}(x+2)^{2}(x-3)[\/latex]\r\n\r\n61. [latex]f\\left(x\\right)=-\\frac{1}{8}(x+4)(x+2)(x-1)(x-3)[\/latex]\r\n\r\n63. [latex]f\\left(x\\right)=\\frac{1}\/{6}(x+3)(x+2)(x-1)^3[\/latex]\r\n\r\n65.\u00a0[latex]f\\left(2\\right)=-10[\/latex]\u00a0and [latex]f\\left(4\\right)=28[\/latex].\u00a0Sign change confirms.\r\n\r\n67.\u00a0[latex]f\\left(1\\right)=3[\/latex]\u00a0and [latex]f\\left(3\\right)=-77[\/latex].\u00a0Sign change confirms.\r\n\r\n69.\u00a0[latex]f\\left(0.01\\right)=1.000001[\/latex]\u00a0and [latex]f\\left(0.1\\right)=-7.999[\/latex].\u00a0Sign change confirms.\r\n

Section 2.5 Solutions<\/h2>\r\n1.\u00a0The binomial is a factor of the polynomial.\r\n\r\n3.\u00a0[latex]x+6+\\frac{5}{x - 1}\\text{,}\\text{quotient:}x+6\\text{,}\\text{remainder:}\\text{5}[\/latex]\r\n\r\n5.\u00a0[latex]3x+2\\text{,}\\text{ quotient: }3x+2\\text{, }\\text{remainder: 0}[\/latex]\r\n\r\n7.\u00a0[latex]x - 5\\text{,}\\text{quotient: }x - 5\\text{,}\\text{remainder: }\\text{0}[\/latex]\r\n\r\n9.\u00a0[latex]2x - 7+\\frac{16}{x+2}\\text{,}\\text{quotient: }\\text{ }2x - 7\\text{,}\\text{remainder: }\\text{16}[\/latex]\r\n\r\n11.\u00a0[latex]x - 2+\\frac{6}{3x+1}\\text{,}\\text{quotient: }x - 2\\text{,}\\text{remainder: }\\text{6}[\/latex]\r\n\r\n13.\u00a0[latex]2{x}^{2}-3x+5\\text{,}\\text{quotient:}2{x}^{2}-3x+5\\text{,}\\text{remainder: }\\text{0}[\/latex]\r\n\r\n15.\u00a0[latex]2{x}^{2}+2x+1+\\frac{10}{x - 4}[\/latex]\r\n\r\n17.\u00a0[latex]2{x}^{2}-7x+1-\\frac{2}{2x+1}[\/latex]\r\n\r\n19.\u00a0[latex]3{x}^{2}-11x+34-\\frac{106}{x+3}[\/latex]\r\n\r\n21.\u00a0[latex]{x}^{2}+5x+1[\/latex]\r\n\r\n23.\u00a0[latex]4{x}^{2}-21x+84-\\frac{323}{x+4}[\/latex]\r\n\r\n25.\u00a0[latex]{x}^{2}-14x+49[\/latex]\r\n\r\n27.\u00a0[latex]3{x}^{2}+x+\\frac{2}{3x - 1}[\/latex]\r\n\r\n29.\u00a0[latex]{x}^{3}-3x+1[\/latex]\r\n\r\n31.\u00a0[latex]{x}^{3}-{x}^{2}+2[\/latex]\r\n\r\n33.\u00a0[latex]{x}^{3}-6{x}^{2}+12x - 8[\/latex]\r\n\r\n35.\u00a0[latex]{x}^{3}-9{x}^{2}+27x - 27[\/latex]\r\n\r\n37.\u00a0[latex]2{x}^{3}-2x+2[\/latex]\r\n\r\n39.\u00a0[latex]\\left(x - 1\\right)\\left({x}^{2}+2x+4\\right)[\/latex]\r\n\r\n41.\u00a0[latex]\\left(x - 5\\right)\\left({x}^{2}+x+1\\right)[\/latex]\r\n\r\n43.\u00a0[latex]\\text{Quotient: }4{x}^{2}+8x+16\\text{,}\\text{remainder: }-1[\/latex]\r\n\r\n45.\u00a0[latex]\\text{Quotient: }3{x}^{2}+3x+5\\text{,}\\text{remainder: }0[\/latex]\r\n\r\n47.\u00a0[latex]\\text{Quotient: }{x}^{3}-2{x}^{2}+4x - 8\\text{,}\\text{remainder: }-6[\/latex]\r\n\r\n49.\u00a0[latex]{x}^{6}-{x}^{5}+{x}^{4}-{x}^{3}+{x}^{2}-x+1[\/latex]\r\n\r\n51.\u00a0[latex]{x}^{3}-{x}^{2}+x - 1+\\frac{1}{x+1}[\/latex]\r\n\r\n53.\u00a0[latex]1+\\frac{1+i}{x-i}[\/latex]\r\n\r\n55.\u00a0[latex]1+\\frac{1-i}{x+i}[\/latex]\r\n\r\n57.\u00a0[latex]{x}^{2}-ix - 1+\\frac{1-i}{x-i}[\/latex]\r\n\r\n59.\u00a0[latex]2{x}^{2}+3[\/latex]\r\n\r\n61.\u00a0[latex]2x+3[\/latex]\r\n\r\n63.\u00a0[latex]x+2[\/latex]\r\n\r\n65.\u00a0[latex]x - 3[\/latex]\r\n\r\n67.\u00a0[latex]3{x}^{2}-2[\/latex]\r\n

Section 2.6 Solutions<\/h2>\r\n1. The theorem can be used to evaluate a polynomial.\r\n\r\n3.\u00a0Rational zeros can be expressed as fractions whereas real zeros include irrational numbers.\r\n\r\n5.\u00a0Polynomial functions can have repeated zeros, so the fact that number is a zero doesn\u2019t preclude it being a zero again.\r\n\r\n7. \u2013106\r\n\r\n9.\u00a00\r\n\r\n11.\u00a0255\r\n\r\n13. \u20131\r\n\r\n15. \u20132, 1, [latex]\\frac{1}{2}[\/latex]\r\n\r\n17. \u20132\r\n\r\n19. \u20133\r\n\r\n21.\u00a0[latex]-\\frac{5}{2}, \\sqrt{6}, -\\sqrt{6}[\/latex]\r\n\r\n23.\u00a0[latex]2, -4, -\\frac{3}{2}[\/latex]\r\n\r\n25. 4, \u20134, \u20135\r\n\r\n27.\u00a0[latex]5, -3, -\\frac{1}{2}[\/latex]\r\n\r\n29.\u00a0[latex]\\frac{1}{2}, \\frac{1+\\sqrt{5}}{2}, \\frac{1-\\sqrt{5}}{2}[\/latex]\r\n\r\n31.\u00a0[latex]\\frac{3}{2}[\/latex]\r\n\r\n33. 2, 3, \u20131, \u20132\r\n\r\n35.\u00a0[latex]\\frac{1}{2}, -\\frac{1}{2}, 2, -3[\/latex]\r\n\r\n37.\u00a0[latex]-1, -1, \\sqrt{5}, -\\sqrt{5}[\/latex]\r\n\r\n39.\u00a0[latex]-\\frac{3}{4}, -\\frac{1}{2}[\/latex]\r\n\r\n41.\u00a0[latex]2, 3+2i, 3 - 2i[\/latex]\r\n\r\n43.\u00a0[latex]-\\frac{2}{3}, 1+2i, 1 - 2i[\/latex]\r\n\r\n45.\u00a0[latex]-\\frac{1}{2}, 1+4i, 1 - 4i[\/latex]\r\n\r\n47.\u00a01 positive, 1 negative\r\n\"Graph\r\n\r\n49.\u00a03 or 1 positive, 0 negative\r\n\"Graph\r\n\r\n51.\u00a00 positive, 3 or 1 negative\r\n\"Graph\r\n\r\n53.\u00a02 or 0 positive, 2 or 0 negative\r\n\"Graph\r\n\r\n55.\u00a02 or 0 positive, 2 or 0 negative\r\n\"Graph\r\n\r\n57.\u00a0[latex]\\pm 5, \\pm 1, \\pm \\frac{5}{2}[\/latex]\r\n\r\n59.\u00a0[latex]\\pm 1, \\pm \\frac{1}{2}, \\pm \\frac{1}{3}, \\pm \\frac{1}{6}[\/latex]\r\n\r\n61.\u00a0[latex]1, \\frac{1}{2}, -\\frac{1}{3}[\/latex]\r\n\r\n63.\u00a0[latex]2, \\frac{1}{4}, -\\frac{3}{2}[\/latex]\r\n\r\n65.\u00a0[latex]\\frac{5}{4}[\/latex]\r\n\r\n67.\u00a0[latex]f\\left(x\\right)=\\frac{4}{9}\\left({x}^{3}+{x}^{2}-x - 1\\right)[\/latex]\r\n\r\n69.\u00a0[latex]f\\left(x\\right)=-\\frac{1}{5}\\left(4{x}^{3}-x\\right)[\/latex]\r\n\r\n71.\u00a08 by 4 by 6 inches\r\n\r\n73.\u00a05.5 by 4.5 by 3.5 inches\r\n\r\n75.\u00a08 by 5 by 3 inches\r\n\r\n77.\u00a0Radius = 6 meters, Height = 2 meters\r\n\r\n79.\u00a0Radius = 2.5 meters, Height = 4.5 meters\r\n

Section 2.7 Solutions<\/h2>\r\n1. Critical points are where the polynomial is equal to zero. It allows you to create the different intervals on the number line for testing.\r\n\r\n3. [latex]\\left(-3,4\\right)[\/latex]\r\n\r\n5. [latex]\\left(-\\infty,1\\right]\\cup\\left[\\frac{4}{3},\\infty\\right)[\/latex]\r\n\r\n7. [latex]\\left(-4,-1\\right)[\/latex]\r\n\r\n9.\u00a0[latex]\\left[-8,-2\\right]\\cup\\left[3,\\infty\\right)[\/latex]\r\n\r\n11.\u00a0[latex]\\left(-\\infty,-8\\right)\\cup\\left(-8,-7\\right)\\cup\\left(-5,\\infty\\right)[\/latex]\r\n\r\n13.\u00a0[latex]\\left(-\\infty,-4\\right]\\cup\\left[-3,\\frac{1}{2}\\right)[\/latex]\r\n\r\n15. [latex]\\left(-5,-1\\right)\\cup\\left(1,2\\right)[\/latex]\r\n\r\n17. [latex]\\left(1,7\\right)[\/latex]\r\n\r\n19. [latex]\\left[-\\infty,-4\\right]\\cup\\left[\\frac{3}{2},\\infty\\right)[\/latex]\r\n\r\n21.\u00a0[latex]\\left(-6,7\\right][\/latex]\r\n\r\n23.\u00a0[latex]\\left[-5,-3\\right]\\cup\\left(-2,\\infty\\right)[\/latex]\r\n\r\n25.\u00a0[latex]\\left(-\\infty,-\\frac{2}{3}\\right)\\cup\\left[2,7\\right][\/latex]\r\n\r\n27.\u00a0[latex]\\left(-\\infty,-6\\right]\\cup\\left(-3,8\\right)[\/latex]\r\n\r\n29.\u00a0[latex]\\left(-7,3\\right)[\/latex]\r\n\r\n31.\u00a0[latex]\\left(-3,-2\\right)\\cup\\left(\\frac{1}{2},\\infty\\right)[\/latex]\r\n

Section 2.8 Solutions<\/h2>\r\n1.\u00a0The rational function will be represented by a quotient of polynomial functions.\r\n\r\n3.\u00a0The numerator and denominator must have a common factor.\r\n\r\n5.\u00a0Yes. The numerator of the formula of the functions would have only complex roots and\/or factors common to both the numerator and denominator.\r\n\r\n7.\u00a0[latex]\\text{All reals }x\\ne -1, 1[\/latex]\r\n\r\n9.\u00a0[latex]\\text{All reals }x\\ne -1, -2, 1, 2[\/latex]\r\n\r\n11.\u00a0V.A. at [latex]x=-\\frac{2}{5}[\/latex]; H.A. at [latex]y=0[\/latex]; Domain is all reals [latex]x\\ne -\\frac{2}{5}[\/latex]\r\n\r\n13.\u00a0V.A. at [latex]x=4, -9[\/latex]; H.A. at [latex]y=0[\/latex]; Domain is all reals [latex]x\\ne 4, -9[\/latex]\r\n\r\n15.\u00a0V.A. at [latex]x=0, 4, -4[\/latex]; H.A. at [latex]y=0[\/latex]; Domain is all reals [latex]x\\ne 0,4, -4[\/latex]\r\n\r\n17.\u00a0V.A. at [latex]x=-5[\/latex]; H.A. at [latex]y=0[\/latex]; Domain is all reals [latex]x\\ne 5,-5[\/latex]\r\n\r\n19.\u00a0V.A. at [latex]x=\\frac{1}{3}[\/latex]; H.A. at [latex]y=-\\frac{2}{3}[\/latex]; Domain is all reals [latex]x\\ne \\frac{1}{3}[\/latex].\r\n\r\n21.\u00a0none\r\n\r\n23.\u00a0[latex]x\\text{-intercepts none, }y\\text{-intercept }\\left(0,\\frac{1}{4}\\right)[\/latex]\r\n\r\n25.\u00a0Local behavior: [latex]x\\to -{\\frac{1}{2}}^{+},f\\left(x\\right)\\to -\\infty ,x\\to -{\\frac{1}{2}}^{-},f\\left(x\\right)\\to \\infty [\/latex]\r\n\r\nEnd behavior: [latex]x\\to \\pm \\infty ,f\\left(x\\right)\\to \\frac{1}{2}[\/latex]\r\n\r\n27.\u00a0Local behavior: [latex]x\\to {6}^{+},f\\left(x\\right)\\to -\\infty ,x\\to {6}^{-},f\\left(x\\right)\\to \\infty [\/latex], End behavior: [latex]x\\to \\pm \\infty ,f\\left(x\\right)\\to -2[\/latex]\r\n\r\n29.\u00a0Local behavior: [latex]x\\to -{\\frac{1}{3}}^{+},f\\left(x\\right)\\to \\infty ,x\\to -{\\frac{1}{3}}^{-}[\/latex], [latex]f\\left(x\\right)\\to -\\infty ,x\\to {\\frac{5}{2}}^{-},f\\left(x\\right)\\to \\infty ,x\\to {\\frac{5}{2}}^{+}[\/latex] ,\u00a0[latex]f\\left(x\\right)\\to -\\infty [\/latex]\r\n\r\nEnd behavior: [latex]x\\to \\pm \\infty\\\\ [\/latex], [latex]f\\left(x\\right)\\to \\frac{1}{3}[\/latex]\r\n\r\n31.\u00a0[latex]y=2x+4[\/latex]\r\n\r\n33.\u00a0[latex]y=2x[\/latex]\r\n\r\n35.\u00a0[latex]V.A.\\text{ }x=0,H.A.\\text{ }y=2[\/latex]\r\n\"Graph\r\n\r\n37.\u00a0[latex]V.A.\\text{ }x=2,\\text{ }H.A.\\text{ }y=0[\/latex]\r\n\"Graph\r\n\r\n39.\u00a0[latex]V.A.\\text{ }x=-4,\\text{ }H.A.\\text{ }y=2;\\left(\\frac{3}{2},0\\right);\\left(0,-\\frac{3}{4}\\right)[\/latex]\r\n\"Graph\r\n\r\n41.\u00a0[latex]V.A.\\text{ }x=2,\\text{ }H.A.\\text{ }y=0,\\text{ }\\left(0,1\\right)[\/latex]\r\n\"Graph\r\n\r\n43.\u00a0[latex]V.A.\\text{ }x=-4,\\text{ }x=\\frac{4}{3},\\text{ }H.A.\\text{ }y=1;\\left(5,0\\right);\\left(-\\frac{1}{3},0\\right);\\left(0,\\frac{5}{16}\\right)[\/latex]\r\n\r\n45.\u00a0[latex]V.A.\\text{ }x=-1,\\text{ }H.A.\\text{ }y=1;\\left(-3,0\\right);\\left(0,3\\right)[\/latex]\r\n\"Graph\r\n\r\n47.\u00a0[latex]V.A.\\text{ }x=4,\\text{ }S.A.\\text{ }y=2x+9;\\left(-1,0\\right);\\left(\\frac{1}{2},0\\right);\\left(0,\\frac{1}{4}\\right)[\/latex]\r\n\"Graph\r\n\r\n49.\u00a0[latex]V.A.\\text{ }x=-2,\\text{ }x=4,\\text{ }H.A.\\text{ }y=1,\\left(1,0\\right);\\left(5,0\\right);\\left(-3,0\\right);\\left(0,-\\frac{15}{16}\\right)[\/latex]\r\n\"Graph\r\n\r\n51.\u00a0[latex]y=50\\frac{{x}^{2}-x - 2}{{x}^{2}-25}[\/latex]\r\n\r\n53.\u00a0[latex]y=7\\frac{{x}^{2}+2x - 24}{{x}^{2}+9x+20}[\/latex]\r\n\r\n55.\u00a0[latex]y=\\frac{1}{2}\\frac{{x}^{2}-4x+4}{x+1}[\/latex]\r\n\r\n57.\u00a0[latex]y=4\\frac{x - 3}{{x}^{2}-x - 12}[\/latex]\r\n\r\n59.\u00a0[latex]y=-9\\frac{x - 2}{{x}^{2}-9}[\/latex]\r\n\r\n61.\u00a0[latex]y=\\frac{1}{3}\\frac{{x}^{2}+x - 6}{x - 1}[\/latex]\r\n\r\n63.\u00a0[latex]y=-6\\frac{{\\left(x - 1\\right)}^{2}}{\\left(x+3\\right){\\left(x - 2\\right)}^{2}}[\/latex]\r\n\r\n65.\r\n\r\n\r\n\r\n\r\n
x<\/em><\/td>\r\n2.01<\/td>\r\n2.001<\/td>\r\n2.0001<\/td>\r\n1.99<\/td>\r\n1.999<\/td>\r\n<\/tr>\r\n
y<\/em><\/td>\r\n100<\/td>\r\n1,000<\/td>\r\n10,000<\/td>\r\n\u2013100<\/td>\r\n\u20131,000<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n\r\n
x<\/em><\/td>\r\n10<\/td>\r\n100<\/td>\r\n1,000<\/td>\r\n10,000<\/td>\r\n100,000<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n
y<\/em><\/td>\r\n.125<\/td>\r\n.0102<\/td>\r\n.001<\/td>\r\n.0001<\/td>\r\n.00001<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nVertical asymptote [latex]x=2[\/latex], Horizontal asymptote [latex]y=0[\/latex]\r\n\r\n67.\r\n\r\n\r\n\r\n\r\n
x<\/em><\/td>\r\n\u20134.1<\/td>\r\n\u20134.01<\/td>\r\n\u20134.001<\/td>\r\n\u20133.99<\/td>\r\n\u20133.999<\/td>\r\n<\/tr>\r\n
y<\/em><\/td>\r\n82<\/td>\r\n802<\/td>\r\n8,002<\/td>\r\n\u2013798<\/td>\r\n\u20137998<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n
x<\/em><\/td>\r\n10<\/td>\r\n100<\/td>\r\n1,000<\/td>\r\n10,000<\/td>\r\n100,000<\/td>\r\n<\/tr>\r\n
y<\/em><\/td>\r\n1.4286<\/td>\r\n1.9331<\/td>\r\n1.992<\/td>\r\n1.9992<\/td>\r\n1.999992<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Vertical asymptote [latex]x=-4[\/latex], Horizontal asymptote [latex]y=2[\/latex]<\/p>\r\n69.\r\n\r\n\r\n\r\n\r\n
x<\/em><\/td>\r\n\u2013.9<\/td>\r\n\u2013.99<\/td>\r\n\u2013.999<\/td>\r\n\u20131.1<\/td>\r\n\u20131.01<\/td>\r\n<\/tr>\r\n
y<\/em><\/td>\r\n81<\/td>\r\n9,801<\/td>\r\n998,001<\/td>\r\n121<\/td>\r\n10,201<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n
x<\/em><\/td>\r\n10<\/td>\r\n100<\/td>\r\n1,000<\/td>\r\n10,000<\/td>\r\n100,000<\/td>\r\n<\/tr>\r\n
y<\/i><\/td>\r\n.82645<\/td>\r\n.9803<\/td>\r\n.998<\/td>\r\n.9998<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nVertical asymptote [latex]x=-1[\/latex], Horizontal asymptote [latex]y=1[\/latex]\r\n\r\n71.\u00a0[latex]\\left(\\frac{3}{2},\\infty \\right)[\/latex]\r\n\"Graph\r\n\r\n73.\u00a0[latex]\\left(-2,1\\right)\\cup \\left(4,\\infty \\right)[\/latex]\r\n\"Graph\r\n\r\n75.\u00a0[latex]\\left(2,4\\right)[\/latex]\r\n\r\n77.\u00a0[latex]\\left(2,5\\right)[\/latex]\r\n\r\n79.\u00a0[latex]\\left(-1,\\text{1}\\right)[\/latex]\r\n\r\n81.\u00a0[latex]C\\left(t\\right)=\\frac{8+2t}{300+20t}[\/latex]\r\n\r\n83.\u00a0After about 6.12 hours.\r\n\r\n85.\u00a0[latex]A\\left(x\\right)=50{x}^{2}+\\frac{800}{x}[\/latex]. 2 by 2 by 5 feet.\r\n\r\n87.\u00a0[latex]A\\left(x\\right)=\\pi {x}^{2}+\\frac{100}{x}[\/latex]. Radius = 2.52 meters.\r\n\r\n ","rendered":"

Section 2.1 Solutions<\/h2>\n

1. It is a second-degree equation (the highest variable exponent is 2).<\/p>\n

3. We want to take advantage of the zero property of multiplication in the fact that if [latex]a\\cdot b=0[\/latex] then it must follow that each factor separately offers a solution to the product being zero [latex]a=0\\text{ or }b=0[\/latex].<\/p>\n

5. One, when no linear term is present (no [latex]x[\/latex] term), such as [latex]x^{2}=16[\/latex]. Two, when the equation is already in the form [latex](ax+b)^{2}=d[\/latex].<\/p>\n

7. [latex]\\text{ }x=6,x=3[\/latex]<\/p>\n

9. [latex]\\text{ }x=-\\dfrac{5}{2},x=-\\dfrac{1}{3}[\/latex]<\/p>\n

11. [latex]\\text{ }x=5,x=-5[\/latex]<\/p>\n

13. [latex]\\text{ }x=-\\dfrac{3}{2},x=\\dfrac{3}{2}[\/latex]<\/p>\n

15. [latex]\\text{ }x=-2,x=3[\/latex]<\/p>\n

17. [latex]\\text{ }x=0,x=-\\dfrac{3}{7}[\/latex]<\/p>\n

19. [latex]\\text{ }x=6,x=-6[\/latex]<\/p>\n

21. [latex]\\text{ }x=6,x=-4[\/latex]<\/p>\n

23. [latex]\\text{ }x=1,x=-2[\/latex]<\/p>\n

25. [latex]\\text{ }x=-2,x=11[\/latex]<\/p>\n

27. [latex]\\text{ }x=3\\pm\\sqrt{22}[\/latex]<\/p>\n

29. [latex]\\text{ }z=\\dfrac{2}{3},z=-\\dfrac{1}{2}[\/latex]<\/p>\n

31. [latex]\\text{ }x=\\dfrac{3\\pm\\sqrt{17}}{4}[\/latex]<\/p>\n

33. Not real<\/p>\n

35. One rational<\/p>\n

37. Two real; rational<\/p>\n

39. [latex]\\text{ }x=\\dfrac{-1\\pm\\sqrt{17}}{2}[\/latex]<\/p>\n

41. [latex]\\text{ }x=\\dfrac{5\\pm\\sqrt{13}}{6}[\/latex]<\/p>\n

43. [latex]\\text{ }x=\\dfrac{-1\\pm\\sqrt{17}}{8}[\/latex]<\/p>\n

45. [latex]ax^{2}+bx+c=0 \\\\ x^2+\\dfrac{b}{a}x=-\\dfrac{c}{a} \\\\ x^{2}+\\dfrac{b}{a}x+\\dfrac{b^{2}}{4a^{2}}=-\\dfrac{c}{a}+\\dfrac{b}{4a^{2}} \\\\ \\left(x+\\dfrac{b}{2a}\\right)^{2}=\\dfrac{b^{2}-4ac}{4a^{2}} \\\\ x+\\dfrac{b}{2a}=\\pm \\sqrt{\\dfrac{b^{2}-4ac}{4a^{2}}} \\\\ x=\\dfrac{-b \\pm \\sqrt{b^{2}-4ac}}{2a}[\/latex]<\/p>\n

47. [latex]\\text{ }x(x+10)=119[\/latex]; [latex]7\\text{ }ft[\/latex] and [latex]9\\text{ }ft[\/latex]<\/p>\n

49. Maximum at [latex]x=70[\/latex]<\/p>\n

51. The quadratic equation would be [latex]\\left(100x-0.5x^{2}\\right)-(60x+300)=300[\/latex]. The two values of [latex]x[\/latex] are 20 and 60.<\/p>\n

53. [latex]\\text{ }3\\text{ }ft[\/latex]<\/p>\n

Section 2.2 Solutions<\/h2>\n

1.\u00a0Add the real parts together and the imaginary parts together.<\/p>\n

3.\u00a0i<\/em>\u00a0times i<\/em>\u00a0equals \u20131, which is not imaginary. (answers vary)<\/p>\n

5.\u00a0[latex]-8+2i[\/latex]<\/p>\n

7.\u00a0[latex]14+7i[\/latex]<\/p>\n

9.\u00a0[latex]-\\frac{23}{29}+\\frac{15}{29}i[\/latex]<\/p>\n

11.\u00a02 real and 0 nonreal<\/p>\n

13.
\n\"Graph<\/p>\n

15.
\n\"Graph<\/p>\n

17.\u00a0[latex]8-i[\/latex]<\/p>\n

19.\u00a0[latex]-11+4i[\/latex]<\/p>\n

21.\u00a0[latex]2 - 5i[\/latex]<\/p>\n

23.\u00a0[latex]6+15i[\/latex]<\/p>\n

25.\u00a0[latex]-16+32i[\/latex]<\/p>\n

27.\u00a0[latex]-4 - 7i[\/latex]<\/p>\n

29.\u00a025<\/p>\n

31.\u00a0[latex]2-\\frac{2}{3}i[\/latex]<\/p>\n

33.\u00a0[latex]4 - 6i[\/latex]<\/p>\n

35.\u00a0[latex]\\frac{2}{5}+\\frac{11}{5}i[\/latex]<\/p>\n

37. 15i<\/em><\/p>\n

39.\u00a0[latex]1+i\\sqrt{3}[\/latex]<\/p>\n

41. 1<\/p>\n

43. \u20131<\/p>\n

45.\u00a0128i<\/em><\/p>\n

47.\u00a0[latex]{\\left(\\frac{\\sqrt{3}}{2}+\\frac{1}{2}i\\right)}^{6}=-1[\/latex]<\/p>\n

49. 3i<\/em><\/p>\n

51.\u00a00<\/p>\n

53.\u00a05 \u2013 5i<\/em><\/p>\n

55. \u20132i<\/em><\/p>\n

57.\u00a0[latex]\\frac{9}{2}-\\frac{9}{2}i[\/latex]<\/p>\n

Section 2.3 Solutions<\/h2>\n

1.\u00a0When written in that form, the vertex can be easily identified.<\/p>\n

3.\u00a0If [latex]a=0[\/latex] then the function becomes a linear function.<\/p>\n

5.\u00a0If possible, we can use factoring. Otherwise, we can use the quadratic formula.<\/p>\n

7.\u00a0[latex]f\\left(x\\right)={\\left(x+1\\right)}^{2}-2[\/latex], Vertex [latex]\\left(-1,-4\\right)[\/latex]<\/p>\n

9.\u00a0[latex]f\\left(x\\right)={\\left(x+\\frac{5}{2}\\right)}^{2}-\\frac{33}{4}[\/latex], Vertex [latex]\\left(-\\frac{5}{2},-\\frac{33}{4}\\right)[\/latex]<\/p>\n

11.\u00a0[latex]f\\left(x\\right)=3{\\left(x - 1\\right)}^{2}-12[\/latex], Vertex [latex]\\left(1,-12\\right)[\/latex]<\/p>\n

13.\u00a0[latex]f\\left(x\\right)=3{\\left(x-\\frac{5}{6}\\right)}^{2}-\\frac{37}{12}[\/latex], Vertex [latex]\\left(\\frac{5}{6},-\\frac{37}{12}\\right)[\/latex]<\/p>\n

15.\u00a0Minimum 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.\u00a0Minimum 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

19.\u00a0Minimum is [latex]-\\frac{7}{2}[\/latex] and occurs at \u20133. Axis of symmetry is [latex]x=-3[\/latex].<\/p>\n

21.\u00a0Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]. Range is [latex]\\left[2,\\infty \\right)[\/latex].<\/p>\n

23.\u00a0Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]. Range is [latex]\\left[-5,\\infty \\right)[\/latex].<\/p>\n

25.\u00a0Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]. Range is [latex]\\left[-12,\\infty \\right)[\/latex].<\/p>\n

27.\u00a0[latex]\\left\\{2i\\sqrt{2},-2i\\sqrt{2}\\right\\}[\/latex]<\/p>\n

29.\u00a0[latex]\\left\\{3i\\sqrt{3},-3i\\sqrt{3}\\right\\}[\/latex]<\/p>\n

31.\u00a0[latex]\\left\\{2+i,2-i\\right\\}[\/latex]<\/p>\n

33.\u00a0[latex]\\left\\{2+3i,2 - 3i\\right\\}[\/latex]<\/p>\n

35.\u00a0[latex]\\left\\{5+i,5-i\\right\\}[\/latex]<\/p>\n

37.\u00a0[latex]\\left\\{2+2\\sqrt{6}, 2 - 2\\sqrt{6}\\right\\}[\/latex]<\/p>\n

39.\u00a0[latex]\\left\\{-\\frac{1}{2}+\\frac{3}{2}i, -\\frac{1}{2}-\\frac{3}{2}i\\right\\}[\/latex]<\/p>\n

41.\u00a0[latex]\\left\\{-\\frac{3}{5}+\\frac{1}{5}i, -\\frac{3}{5}-\\frac{1}{5}i\\right\\}[\/latex]<\/p>\n

43.\u00a0[latex]\\left\\{-\\frac{1}{2}+\\frac{1}{2}i\\sqrt{7}, -\\frac{1}{2}-\\frac{1}{2}i\\sqrt{7}\\right\\}[\/latex]<\/p>\n

45.\u00a0[latex]f\\left(x\\right)={x}^{2}-4x+4[\/latex]<\/p>\n

47.\u00a0[latex]f\\left(x\\right)={x}^{2}+1[\/latex]<\/p>\n

49.\u00a0[latex]f\\left(x\\right)=\\frac{6}{49}{x}^{2}+\\frac{60}{49}x+\\frac{297}{49}[\/latex]<\/p>\n

51.\u00a0[latex]f\\left(x\\right)=-{x}^{2}+1[\/latex]<\/p>\n

53.\u00a0Vertex [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.\u00a0Vertex [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.\u00a0Vertex [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.\u00a0[latex]f\\left(x\\right)={x}^{2}-4x+1[\/latex]<\/p>\n

61.\u00a0[latex]f\\left(x\\right)=-2{x}^{2}+8x - 1[\/latex]<\/p>\n

63.\u00a0[latex]f\\left(x\\right)=\\frac{1}{2}{x}^{2}-3x+\\frac{7}{2}[\/latex]<\/p>\n

65.\u00a0[latex]f\\left(x\\right)={x}^{2}+1[\/latex]<\/p>\n

67.\u00a0[latex]f\\left(x\\right)=2-{x}^{2}[\/latex]<\/p>\n

69.\u00a0[latex]f\\left(x\\right)=2{x}^{2}[\/latex]<\/p>\n

71.\u00a0The graph is shifted up or down (a vertical shift).<\/p>\n

73.\u00a050 feet<\/p>\n

75.\u00a0Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]. Range is [latex]\\left[-2,\\infty \\right)[\/latex].<\/p>\n

77.\u00a0Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex] Range is [latex]\\left(-\\infty ,11\\right][\/latex].<\/p>\n

79.\u00a0[latex]f\\left(x\\right)=2{x}^{2}-1[\/latex]<\/p>\n

81.\u00a0[latex]f\\left(x\\right)=3{x}^{2}-9[\/latex]<\/p>\n

83.\u00a0[latex]f\\left(x\\right)=5{x}^{2}-77[\/latex]<\/p>\n

85.\u00a050 feet by 50 feet. Maximize [latex]f\\left(x\\right)=-{x}^{2}+100x[\/latex].<\/p>\n

87.\u00a0125 feet by 62.5 feet. Maximize [latex]f\\left(x\\right)=-2{x}^{2}+250x[\/latex].<\/p>\n

89. 6 and \u20136; product is \u201336; maximize [latex]f\\left(x\\right)={x}^{2}+12x[\/latex].<\/p>\n

91.\u00a02909.56 meters<\/p>\n

93.\u00a0$10.70<\/p>\n

Section 2.4 Solutions<\/h2>\n

1.\u00a0The 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

3. The maximum number of turning points is always 1 less than the degree.<\/p>\n

5. The factored form is used to find the zeros.\u00a0 The behavior at each zero can be found from the factored form, and this aids in graphing.<\/p>\n

7. Degree 5 polynomial; [latex]a_{n}=6[\/latex]<\/p>\n

9. Not a polynomial<\/p>\n

11. Degree 4 polynomial; [latex]a_{n}=2[\/latex]<\/p>\n

13. Not a polynomial<\/p>\n

15. 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

17.\u00a0y<\/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

19.\u00a0y<\/em>-intercept is [latex]\\left(0,0\\right)[\/latex]. x-intercepts are [latex]\\left(0,0\\right),\\left(4,0\\right)[\/latex], and [latex]\\left(-2, 0\\right)[\/latex].<\/p>\n

21.\u00a0 2<\/p>\n

23.\u00a0 1<\/p>\n

25.\u00a0 4<\/p>\n

27.\u00a0Yes. Number of turning points is 2. Least possible degree is 3.<\/p>\n

29.\u00a0Yes. Number of turning points is 1. Least possible degree is 2.<\/p>\n

31.\u00a0Yes. Number of turning points is 0. Least possible degree is 1.<\/p>\n

33.\u00a0Yes. Number of turning points is 0. Least possible degree is 1.<\/p>\n

33.\u00a0 0 with multiplicity 2 and behavior: [latex]4x^{2}[\/latex]
\n[latex]\u20132[\/latex] with multiplicity 2 and behavior: [latex]4(x+2)^{2}[\/latex]<\/p>\n

35.[latex]\\text{ }-\\frac{2}{3}[\/latex] with multiplicity 5 and behavior: [latex]\\frac{289}{9}(3x+2)^3[\/latex]
\n5 with multiplicity 2 and behavior: [latex]4913(x-5)^{2}[\/latex]<\/p>\n

37.\u00a00 with multiplicity 4 and behavior: [latex]-2x^{4}[\/latex]
\n2 with multiplicity 1 and behavior: [latex]48(x-2)[\/latex]
\n-1 with multiplicity 1 and behavior: [latex]-(x+1)[\/latex]<\/p>\n

39.\u00a0[latex]\\text{ }\\frac{3}{2}[\/latex] with multiplicity 2 and behavior: [latex]\\frac{27}{8}(2x-3)^2[\/latex]
\n0 with multiplicity 3 and behavior: [latex]9x^{3}[\/latex]<\/p>\n

41.\u00a00 with multiplicity 6 and behavior: [latex]16x^{6}[\/latex]
\n[latex]\\frac{2}{3}[\/latex] with multiplicity 2 and behavior: [latex]\\frac{729}{16}(3x-2)^{2}[\/latex]<\/p>\n

43.\u00a0zeros [latex]\\left(1, 0\\right)[\/latex]\u00a0with multiplicity 2, [latex]\\left(-4, 0\\right)[\/latex] with multiplicity 1, y-<\/em>intercept [latex]\\left(0, 4\\right)[\/latex]. As\u00a0[latex]x\\to -\\infty[\/latex] ,\u00a0[latex]f\\left(x\\right)\\to -\\infty[\/latex] , as\u00a0[latex]x\\to \\infty[\/latex] ,\u00a0[latex]f\\left(x\\right)\\to \\infty[\/latex].
\nAt [latex]x=-4[\/latex] the behavior is: [latex]25(x+4)[\/latex]
\nAt [latex]x=1[\/latex] the behavior is: [latex]5(x-1)^{2}[\/latex].
\n\"Graph<\/p>\n

45.\u00a0zeros\u00a0[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\u00a0[latex]x\\to -\\infty[\/latex],\u00a0[latex]f\\left(x\\right)\\to -\\infty[\/latex] , as [latex]x\\to \\infty[\/latex] ,\u00a0[latex]f\\left(x\\right)\\to \\infty[\/latex].
\nAt [latex]x=2[\/latex] the behavior is: [latex]-(x-2)^{2}[\/latex].
\nAt [latex]x=3[\/latex] the behavior is: [latex](x-3)^{3}[\/latex].
\n\"Graph<\/p>\n

47.\u00a0x-intercepts [latex]\\left(0, 0\\right),\\left(-2, 0\\right),\\left(4, 0\\right)[\/latex]\u00a0with multiplicity 1, y<\/em>-intercept [latex]\\left(0, 0\\right)[\/latex]. As\u00a0[latex]x\\to -\\infty[\/latex] ,\u00a0[latex]f\\left(x\\right)\\to \\infty[\/latex] , as\u00a0[latex]x\\to \\infty[\/latex] ,\u00a0[latex]f\\left(x\\right)\\to -\\infty[\/latex].
\nAt [latex]x=0[\/latex] the behavior is: [latex]24x[\/latex]
\nAt [latex]x=-2[\/latex] the behavior is: [latex]-36(x+2)[\/latex]
\nAt [latex]x=4[\/latex] the behavior is: [latex]-72(x-4)[\/latex].
\n\"Graph<\/p>\n

49.\u00a0The <\/span>y<\/em>-intercept is [latex]\\left(0, 0\\right)[\/latex].\u00a0The\u00a0zero<\/span>s 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]<\/span>
\nAt [latex]x=0[\/latex] the behavior is: [latex]-2x^3[\/latex]
\nAt [latex]x=2[\/latex] the behavior is: [latex]8(x-2)[\/latex].
\n\"Graph<\/p>\n

51.\u00a0[latex]f\\left(x\\right)=x^{2}+x-6[\/latex]<\/p>\n

53.\u00a0[latex]f\\left(x\\right)=x(x-2)^{2}[\/latex] or\u00a0[latex]f\\left(x\\right)=x^{2}(x-2)[\/latex]<\/p>\n

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

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

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

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

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

65.\u00a0[latex]f\\left(2\\right)=-10[\/latex]\u00a0and [latex]f\\left(4\\right)=28[\/latex].\u00a0Sign change confirms.<\/p>\n

67.\u00a0[latex]f\\left(1\\right)=3[\/latex]\u00a0and [latex]f\\left(3\\right)=-77[\/latex].\u00a0Sign change confirms.<\/p>\n

69.\u00a0[latex]f\\left(0.01\\right)=1.000001[\/latex]\u00a0and [latex]f\\left(0.1\\right)=-7.999[\/latex].\u00a0Sign change confirms.<\/p>\n

Section 2.5 Solutions<\/h2>\n

1.\u00a0The binomial is a factor of the polynomial.<\/p>\n

3.\u00a0[latex]x+6+\\frac{5}{x - 1}\\text{,}\\text{quotient:}x+6\\text{,}\\text{remainder:}\\text{5}[\/latex]<\/p>\n

5.\u00a0[latex]3x+2\\text{,}\\text{ quotient: }3x+2\\text{, }\\text{remainder: 0}[\/latex]<\/p>\n

7.\u00a0[latex]x - 5\\text{,}\\text{quotient: }x - 5\\text{,}\\text{remainder: }\\text{0}[\/latex]<\/p>\n

9.\u00a0[latex]2x - 7+\\frac{16}{x+2}\\text{,}\\text{quotient: }\\text{ }2x - 7\\text{,}\\text{remainder: }\\text{16}[\/latex]<\/p>\n

11.\u00a0[latex]x - 2+\\frac{6}{3x+1}\\text{,}\\text{quotient: }x - 2\\text{,}\\text{remainder: }\\text{6}[\/latex]<\/p>\n

13.\u00a0[latex]2{x}^{2}-3x+5\\text{,}\\text{quotient:}2{x}^{2}-3x+5\\text{,}\\text{remainder: }\\text{0}[\/latex]<\/p>\n

15.\u00a0[latex]2{x}^{2}+2x+1+\\frac{10}{x - 4}[\/latex]<\/p>\n

17.\u00a0[latex]2{x}^{2}-7x+1-\\frac{2}{2x+1}[\/latex]<\/p>\n

19.\u00a0[latex]3{x}^{2}-11x+34-\\frac{106}{x+3}[\/latex]<\/p>\n

21.\u00a0[latex]{x}^{2}+5x+1[\/latex]<\/p>\n

23.\u00a0[latex]4{x}^{2}-21x+84-\\frac{323}{x+4}[\/latex]<\/p>\n

25.\u00a0[latex]{x}^{2}-14x+49[\/latex]<\/p>\n

27.\u00a0[latex]3{x}^{2}+x+\\frac{2}{3x - 1}[\/latex]<\/p>\n

29.\u00a0[latex]{x}^{3}-3x+1[\/latex]<\/p>\n

31.\u00a0[latex]{x}^{3}-{x}^{2}+2[\/latex]<\/p>\n

33.\u00a0[latex]{x}^{3}-6{x}^{2}+12x - 8[\/latex]<\/p>\n

35.\u00a0[latex]{x}^{3}-9{x}^{2}+27x - 27[\/latex]<\/p>\n

37.\u00a0[latex]2{x}^{3}-2x+2[\/latex]<\/p>\n

39.\u00a0[latex]\\left(x - 1\\right)\\left({x}^{2}+2x+4\\right)[\/latex]<\/p>\n

41.\u00a0[latex]\\left(x - 5\\right)\\left({x}^{2}+x+1\\right)[\/latex]<\/p>\n

43.\u00a0[latex]\\text{Quotient: }4{x}^{2}+8x+16\\text{,}\\text{remainder: }-1[\/latex]<\/p>\n

45.\u00a0[latex]\\text{Quotient: }3{x}^{2}+3x+5\\text{,}\\text{remainder: }0[\/latex]<\/p>\n

47.\u00a0[latex]\\text{Quotient: }{x}^{3}-2{x}^{2}+4x - 8\\text{,}\\text{remainder: }-6[\/latex]<\/p>\n

49.\u00a0[latex]{x}^{6}-{x}^{5}+{x}^{4}-{x}^{3}+{x}^{2}-x+1[\/latex]<\/p>\n

51.\u00a0[latex]{x}^{3}-{x}^{2}+x - 1+\\frac{1}{x+1}[\/latex]<\/p>\n

53.\u00a0[latex]1+\\frac{1+i}{x-i}[\/latex]<\/p>\n

55.\u00a0[latex]1+\\frac{1-i}{x+i}[\/latex]<\/p>\n

57.\u00a0[latex]{x}^{2}-ix - 1+\\frac{1-i}{x-i}[\/latex]<\/p>\n

59.\u00a0[latex]2{x}^{2}+3[\/latex]<\/p>\n

61.\u00a0[latex]2x+3[\/latex]<\/p>\n

63.\u00a0[latex]x+2[\/latex]<\/p>\n

65.\u00a0[latex]x - 3[\/latex]<\/p>\n

67.\u00a0[latex]3{x}^{2}-2[\/latex]<\/p>\n

Section 2.6 Solutions<\/h2>\n

1. The theorem can be used to evaluate a polynomial.<\/p>\n

3.\u00a0Rational zeros can be expressed as fractions whereas real zeros include irrational numbers.<\/p>\n

5.\u00a0Polynomial functions can have repeated zeros, so the fact that number is a zero doesn\u2019t preclude it being a zero again.<\/p>\n

7. \u2013106<\/p>\n

9.\u00a00<\/p>\n

11.\u00a0255<\/p>\n

13. \u20131<\/p>\n

15. \u20132, 1, [latex]\\frac{1}{2}[\/latex]<\/p>\n

17. \u20132<\/p>\n

19. \u20133<\/p>\n

21.\u00a0[latex]-\\frac{5}{2}, \\sqrt{6}, -\\sqrt{6}[\/latex]<\/p>\n

23.\u00a0[latex]2, -4, -\\frac{3}{2}[\/latex]<\/p>\n

25. 4, \u20134, \u20135<\/p>\n

27.\u00a0[latex]5, -3, -\\frac{1}{2}[\/latex]<\/p>\n

29.\u00a0[latex]\\frac{1}{2}, \\frac{1+\\sqrt{5}}{2}, \\frac{1-\\sqrt{5}}{2}[\/latex]<\/p>\n

31.\u00a0[latex]\\frac{3}{2}[\/latex]<\/p>\n

33. 2, 3, \u20131, \u20132<\/p>\n

35.\u00a0[latex]\\frac{1}{2}, -\\frac{1}{2}, 2, -3[\/latex]<\/p>\n

37.\u00a0[latex]-1, -1, \\sqrt{5}, -\\sqrt{5}[\/latex]<\/p>\n

39.\u00a0[latex]-\\frac{3}{4}, -\\frac{1}{2}[\/latex]<\/p>\n

41.\u00a0[latex]2, 3+2i, 3 - 2i[\/latex]<\/p>\n

43.\u00a0[latex]-\\frac{2}{3}, 1+2i, 1 - 2i[\/latex]<\/p>\n

45.\u00a0[latex]-\\frac{1}{2}, 1+4i, 1 - 4i[\/latex]<\/p>\n

47.\u00a01 positive, 1 negative
\n\"Graph<\/p>\n

49.\u00a03 or 1 positive, 0 negative
\n\"Graph<\/p>\n

51.\u00a00 positive, 3 or 1 negative
\n\"Graph<\/p>\n

53.\u00a02 or 0 positive, 2 or 0 negative
\n\"Graph<\/p>\n

55.\u00a02 or 0 positive, 2 or 0 negative
\n\"Graph<\/p>\n

57.\u00a0[latex]\\pm 5, \\pm 1, \\pm \\frac{5}{2}[\/latex]<\/p>\n

59.\u00a0[latex]\\pm 1, \\pm \\frac{1}{2}, \\pm \\frac{1}{3}, \\pm \\frac{1}{6}[\/latex]<\/p>\n

61.\u00a0[latex]1, \\frac{1}{2}, -\\frac{1}{3}[\/latex]<\/p>\n

63.\u00a0[latex]2, \\frac{1}{4}, -\\frac{3}{2}[\/latex]<\/p>\n

65.\u00a0[latex]\\frac{5}{4}[\/latex]<\/p>\n

67.\u00a0[latex]f\\left(x\\right)=\\frac{4}{9}\\left({x}^{3}+{x}^{2}-x - 1\\right)[\/latex]<\/p>\n

69.\u00a0[latex]f\\left(x\\right)=-\\frac{1}{5}\\left(4{x}^{3}-x\\right)[\/latex]<\/p>\n

71.\u00a08 by 4 by 6 inches<\/p>\n

73.\u00a05.5 by 4.5 by 3.5 inches<\/p>\n

75.\u00a08 by 5 by 3 inches<\/p>\n

77.\u00a0Radius = 6 meters, Height = 2 meters<\/p>\n

79.\u00a0Radius = 2.5 meters, Height = 4.5 meters<\/p>\n

Section 2.7 Solutions<\/h2>\n

1. Critical points are where the polynomial is equal to zero. It allows you to create the different intervals on the number line for testing.<\/p>\n

3. [latex]\\left(-3,4\\right)[\/latex]<\/p>\n

5. [latex]\\left(-\\infty,1\\right]\\cup\\left[\\frac{4}{3},\\infty\\right)[\/latex]<\/p>\n

7. [latex]\\left(-4,-1\\right)[\/latex]<\/p>\n

9.\u00a0[latex]\\left[-8,-2\\right]\\cup\\left[3,\\infty\\right)[\/latex]<\/p>\n

11.\u00a0[latex]\\left(-\\infty,-8\\right)\\cup\\left(-8,-7\\right)\\cup\\left(-5,\\infty\\right)[\/latex]<\/p>\n

13.\u00a0[latex]\\left(-\\infty,-4\\right]\\cup\\left[-3,\\frac{1}{2}\\right)[\/latex]<\/p>\n

15. [latex]\\left(-5,-1\\right)\\cup\\left(1,2\\right)[\/latex]<\/p>\n

17. [latex]\\left(1,7\\right)[\/latex]<\/p>\n

19. [latex]\\left[-\\infty,-4\\right]\\cup\\left[\\frac{3}{2},\\infty\\right)[\/latex]<\/p>\n

21.\u00a0[latex]\\left(-6,7\\right][\/latex]<\/p>\n

23.\u00a0[latex]\\left[-5,-3\\right]\\cup\\left(-2,\\infty\\right)[\/latex]<\/p>\n

25.\u00a0[latex]\\left(-\\infty,-\\frac{2}{3}\\right)\\cup\\left[2,7\\right][\/latex]<\/p>\n

27.\u00a0[latex]\\left(-\\infty,-6\\right]\\cup\\left(-3,8\\right)[\/latex]<\/p>\n

29.\u00a0[latex]\\left(-7,3\\right)[\/latex]<\/p>\n

31.\u00a0[latex]\\left(-3,-2\\right)\\cup\\left(\\frac{1}{2},\\infty\\right)[\/latex]<\/p>\n

Section 2.8 Solutions<\/h2>\n

1.\u00a0The rational function will be represented by a quotient of polynomial functions.<\/p>\n

3.\u00a0The numerator and denominator must have a common factor.<\/p>\n

5.\u00a0Yes. The numerator of the formula of the functions would have only complex roots and\/or factors common to both the numerator and denominator.<\/p>\n

7.\u00a0[latex]\\text{All reals }x\\ne -1, 1[\/latex]<\/p>\n

9.\u00a0[latex]\\text{All reals }x\\ne -1, -2, 1, 2[\/latex]<\/p>\n

11.\u00a0V.A. at [latex]x=-\\frac{2}{5}[\/latex]; H.A. at [latex]y=0[\/latex]; Domain is all reals [latex]x\\ne -\\frac{2}{5}[\/latex]<\/p>\n

13.\u00a0V.A. at [latex]x=4, -9[\/latex]; H.A. at [latex]y=0[\/latex]; Domain is all reals [latex]x\\ne 4, -9[\/latex]<\/p>\n

15.\u00a0V.A. at [latex]x=0, 4, -4[\/latex]; H.A. at [latex]y=0[\/latex]; Domain is all reals [latex]x\\ne 0,4, -4[\/latex]<\/p>\n

17.\u00a0V.A. at [latex]x=-5[\/latex]; H.A. at [latex]y=0[\/latex]; Domain is all reals [latex]x\\ne 5,-5[\/latex]<\/p>\n

19.\u00a0V.A. at [latex]x=\\frac{1}{3}[\/latex]; H.A. at [latex]y=-\\frac{2}{3}[\/latex]; Domain is all reals [latex]x\\ne \\frac{1}{3}[\/latex].<\/p>\n

21.\u00a0none<\/p>\n

23.\u00a0[latex]x\\text{-intercepts none, }y\\text{-intercept }\\left(0,\\frac{1}{4}\\right)[\/latex]<\/p>\n

25.\u00a0Local behavior: [latex]x\\to -{\\frac{1}{2}}^{+},f\\left(x\\right)\\to -\\infty ,x\\to -{\\frac{1}{2}}^{-},f\\left(x\\right)\\to \\infty[\/latex]<\/p>\n

End behavior: [latex]x\\to \\pm \\infty ,f\\left(x\\right)\\to \\frac{1}{2}[\/latex]<\/p>\n

27.\u00a0Local behavior: [latex]x\\to {6}^{+},f\\left(x\\right)\\to -\\infty ,x\\to {6}^{-},f\\left(x\\right)\\to \\infty[\/latex], End behavior: [latex]x\\to \\pm \\infty ,f\\left(x\\right)\\to -2[\/latex]<\/p>\n

29.\u00a0Local behavior: [latex]x\\to -{\\frac{1}{3}}^{+},f\\left(x\\right)\\to \\infty ,x\\to -{\\frac{1}{3}}^{-}[\/latex], [latex]f\\left(x\\right)\\to -\\infty ,x\\to {\\frac{5}{2}}^{-},f\\left(x\\right)\\to \\infty ,x\\to {\\frac{5}{2}}^{+}[\/latex] ,\u00a0[latex]f\\left(x\\right)\\to -\\infty[\/latex]<\/p>\n

End behavior: [latex]x\\to \\pm \\infty\\\\[\/latex], [latex]f\\left(x\\right)\\to \\frac{1}{3}[\/latex]<\/p>\n

31.\u00a0[latex]y=2x+4[\/latex]<\/p>\n

33.\u00a0[latex]y=2x[\/latex]<\/p>\n

35.\u00a0[latex]V.A.\\text{ }x=0,H.A.\\text{ }y=2[\/latex]
\n\"Graph<\/p>\n

37.\u00a0[latex]V.A.\\text{ }x=2,\\text{ }H.A.\\text{ }y=0[\/latex]
\n\"Graph<\/p>\n

39.\u00a0[latex]V.A.\\text{ }x=-4,\\text{ }H.A.\\text{ }y=2;\\left(\\frac{3}{2},0\\right);\\left(0,-\\frac{3}{4}\\right)[\/latex]
\n\"Graph<\/p>\n

41.\u00a0[latex]V.A.\\text{ }x=2,\\text{ }H.A.\\text{ }y=0,\\text{ }\\left(0,1\\right)[\/latex]
\n\"Graph<\/p>\n

43.\u00a0[latex]V.A.\\text{ }x=-4,\\text{ }x=\\frac{4}{3},\\text{ }H.A.\\text{ }y=1;\\left(5,0\\right);\\left(-\\frac{1}{3},0\\right);\\left(0,\\frac{5}{16}\\right)[\/latex]<\/p>\n

45.\u00a0[latex]V.A.\\text{ }x=-1,\\text{ }H.A.\\text{ }y=1;\\left(-3,0\\right);\\left(0,3\\right)[\/latex]
\n\"Graph<\/p>\n

47.\u00a0[latex]V.A.\\text{ }x=4,\\text{ }S.A.\\text{ }y=2x+9;\\left(-1,0\\right);\\left(\\frac{1}{2},0\\right);\\left(0,\\frac{1}{4}\\right)[\/latex]
\n\"Graph<\/p>\n

49.\u00a0[latex]V.A.\\text{ }x=-2,\\text{ }x=4,\\text{ }H.A.\\text{ }y=1,\\left(1,0\\right);\\left(5,0\\right);\\left(-3,0\\right);\\left(0,-\\frac{15}{16}\\right)[\/latex]
\n\"Graph<\/p>\n

51.\u00a0[latex]y=50\\frac{{x}^{2}-x - 2}{{x}^{2}-25}[\/latex]<\/p>\n

53.\u00a0[latex]y=7\\frac{{x}^{2}+2x - 24}{{x}^{2}+9x+20}[\/latex]<\/p>\n

55.\u00a0[latex]y=\\frac{1}{2}\\frac{{x}^{2}-4x+4}{x+1}[\/latex]<\/p>\n

57.\u00a0[latex]y=4\\frac{x - 3}{{x}^{2}-x - 12}[\/latex]<\/p>\n

59.\u00a0[latex]y=-9\\frac{x - 2}{{x}^{2}-9}[\/latex]<\/p>\n

61.\u00a0[latex]y=\\frac{1}{3}\\frac{{x}^{2}+x - 6}{x - 1}[\/latex]<\/p>\n

63.\u00a0[latex]y=-6\\frac{{\\left(x - 1\\right)}^{2}}{\\left(x+3\\right){\\left(x - 2\\right)}^{2}}[\/latex]<\/p>\n

65.<\/p>\n\n\n\n\n
x<\/em><\/td>\n2.01<\/td>\n2.001<\/td>\n2.0001<\/td>\n1.99<\/td>\n1.999<\/td>\n<\/tr>\n
y<\/em><\/td>\n100<\/td>\n1,000<\/td>\n10,000<\/td>\n\u2013100<\/td>\n\u20131,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n
x<\/em><\/td>\n10<\/td>\n100<\/td>\n1,000<\/td>\n10,000<\/td>\n100,000<\/td>\n<\/tr>\n<\/tbody>\n
y<\/em><\/td>\n.125<\/td>\n.0102<\/td>\n.001<\/td>\n.0001<\/td>\n.00001<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Vertical asymptote [latex]x=2[\/latex], Horizontal asymptote [latex]y=0[\/latex]<\/p>\n

67.<\/p>\n\n\n\n\n
x<\/em><\/td>\n\u20134.1<\/td>\n\u20134.01<\/td>\n\u20134.001<\/td>\n\u20133.99<\/td>\n\u20133.999<\/td>\n<\/tr>\n
y<\/em><\/td>\n82<\/td>\n802<\/td>\n8,002<\/td>\n\u2013798<\/td>\n\u20137998<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n
x<\/em><\/td>\n10<\/td>\n100<\/td>\n1,000<\/td>\n10,000<\/td>\n100,000<\/td>\n<\/tr>\n
y<\/em><\/td>\n1.4286<\/td>\n1.9331<\/td>\n1.992<\/td>\n1.9992<\/td>\n1.999992<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Vertical asymptote [latex]x=-4[\/latex], Horizontal asymptote [latex]y=2[\/latex]<\/p>\n

69.<\/p>\n\n\n\n\n
x<\/em><\/td>\n\u2013.9<\/td>\n\u2013.99<\/td>\n\u2013.999<\/td>\n\u20131.1<\/td>\n\u20131.01<\/td>\n<\/tr>\n
y<\/em><\/td>\n81<\/td>\n9,801<\/td>\n998,001<\/td>\n121<\/td>\n10,201<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n
x<\/em><\/td>\n10<\/td>\n100<\/td>\n1,000<\/td>\n10,000<\/td>\n100,000<\/td>\n<\/tr>\n
y<\/i><\/td>\n.82645<\/td>\n.9803<\/td>\n.998<\/td>\n.9998<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Vertical asymptote [latex]x=-1[\/latex], Horizontal asymptote [latex]y=1[\/latex]<\/p>\n

71.\u00a0[latex]\\left(\\frac{3}{2},\\infty \\right)[\/latex]
\n\"Graph<\/p>\n

73.\u00a0[latex]\\left(-2,1\\right)\\cup \\left(4,\\infty \\right)[\/latex]
\n\"Graph<\/p>\n

75.\u00a0[latex]\\left(2,4\\right)[\/latex]<\/p>\n

77.\u00a0[latex]\\left(2,5\\right)[\/latex]<\/p>\n

79.\u00a0[latex]\\left(-1,\\text{1}\\right)[\/latex]<\/p>\n

81.\u00a0[latex]C\\left(t\\right)=\\frac{8+2t}{300+20t}[\/latex]<\/p>\n

83.\u00a0After about 6.12 hours.<\/p>\n

85.\u00a0[latex]A\\left(x\\right)=50{x}^{2}+\\frac{800}{x}[\/latex]. 2 by 2 by 5 feet.<\/p>\n

87.\u00a0[latex]A\\left(x\\right)=\\pi {x}^{2}+\\frac{100}{x}[\/latex]. Radius = 2.52 meters.<\/p>\n

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