{"id":4377,"date":"2024-09-30T16:55:07","date_gmt":"2024-09-30T16:55:07","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=4377"},"modified":"2024-11-21T18:08:55","modified_gmt":"2024-11-21T18:08:55","slug":"power-and-polynomial-functions-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/power-and-polynomial-functions-get-stronger\/","title":{"raw":"Power and Polynomial Functions: Get Stronger","rendered":"Power and Polynomial Functions: Get Stronger"},"content":{"raw":"

Introduction to Power and Polynomial Functions<\/span><\/h2>\r\n

For the following exercises, identify the function as a power function, a polynomial function, or neither.<\/p>\r\n\r\n

    \r\n \t
  1. [latex]f(x)=(x^2)^3[\/latex]<\/li>\r\n \t
  2. [latex]f(x)=\\frac{x^2}{x^2\u22121}[\/latex]<\/li>\r\n \t
  3. [latex]f(x)=3^{x+1}[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, find the degree and leading coefficient for the given polynomial.\r\n
      \r\n \t
    1. [latex]7\u22122x^2[\/latex]<\/li>\r\n \t
    2. [latex]x(4\u2212x^2)(2x+1)[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, determine the end behavior of the functions.\r\n
        \r\n \t
      1. [latex]f(x)=x^4[\/latex]<\/li>\r\n \t
      2. [latex]f(x)=\u2212x^4[\/latex]<\/li>\r\n \t
      3. [latex]f(x)=\u22122x^4\u22123x^2+x\u22121[\/latex]<\/li>\r\n \t
      4. [latex]f(x)=x^2(2x^3-x+1)[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, find the intercepts of the functions.\r\n
          \r\n \t
        1. [latex]f(t)=2(t\u22121)(t+2)(t\u22123)[\/latex]<\/li>\r\n \t
        2. [latex]f(x)=x^4\u221216[\/latex]<\/li>\r\n \t
        3. [latex]f(x)=x(x^2\u22122x\u22128)[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, determine the least possible degree of the polynomial function shown.\r\n
            \r\n \t
          1. \"Graph<\/li>\r\n \t
          2. \"Graph<\/li>\r\n \t
          3. \"Graph<\/li>\r\n \t
          4. \"Graph<\/li>\r\n<\/ol>\r\nFor the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.\r\n
              \r\n \t
            1. \"Graph<\/li>\r\n \t
            2. \"Graph<\/li>\r\n \t
            3. \"Graph<\/li>\r\n \t
            4. \"Graph<\/li>\r\n \t
            5. [latex]V(m)=(3+2m)^3[\/latex], where [latex]m[\/latex] is the number of minutes elapsed.<\/li>\r\n \t
            6. [latex]V(x)=x(8-2x)^2[\/latex], where [latex]x[\/latex] is the length of the square's side in inches.<\/li>\r\n<\/ol>\r\n

              Graphs of Polynomial Functions<\/span><\/h2>\r\nFor the following exercises, find the [latex]x[\/latex]- or [latex]t[\/latex]-intercepts of the polynomial functions.\r\n
                \r\n \t
              1. [latex]C(t)=3(t+2)(t\u22123)(t+5)[\/latex]<\/li>\r\n \t
              2. [latex]C(t)=2t(t\u22123)(t+1)^2[\/latex]<\/li>\r\n \t
              3. [latex]C(t)=4t^4+12t^3\u221240t^2[\/latex]<\/li>\r\n \t
              4. [latex]f(x)=x^3+x^2\u221220x[\/latex]<\/li>\r\n \t
              5. [latex]f(x)=x^3+x^2\u22124x-4[\/latex]<\/li>\r\n \t
              6. [latex]f(x)=2x^3-x^2\u22128x+4[\/latex]<\/li>\r\n \t
              7. [latex]f(x)=2x^4+6x^2-8[\/latex]<\/li>\r\n \t
              8. [latex]f(x)=x^6-2x^4-3x^2[\/latex]<\/li>\r\n \t
              9. [latex]f(x)=x^5-5x^3+4x[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval.\r\n
                  \r\n \t
                1. [latex]f(x)=x^3\u22129x[\/latex], between [latex]x=2[\/latex] and [latex]x=4[\/latex].<\/li>\r\n \t
                2. [latex]f(x)=\u2212x^4+4[\/latex], between [latex]x=1[\/latex] and [latex]x=3[\/latex].<\/li>\r\n \t
                3. [latex]f(x)=x^3\u2212100x+2[\/latex], between [latex]x=0.01[\/latex] and [latex]x=0.1[\/latex].<\/li>\r\n<\/ol>\r\nFor the following exercises, find the zeros and give the multiplicity of each.\r\n
                    \r\n \t
                  1. [latex]f(x)=x^2(2x+3)^5(x\u22124)^2[\/latex]<\/li>\r\n \t
                  2. [latex]f(x)=x^2(x^2+4x+4)[\/latex]<\/li>\r\n \t
                  3. [latex]f(x)=(3x+2)^5(x^2\u221210x+25)[\/latex]<\/li>\r\n \t
                  4. [latex]f(x)=x^6-x^5-2x^4[\/latex]<\/li>\r\n \t
                  5. [latex]f(x)=4x^5-12x^4+9x^3[\/latex]<\/li>\r\n \t
                  6. [latex]f(x)=4x^4(9x^4-12x^3+4x^2)[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, graph the polynomial functions. Note [latex]x[\/latex]- and [latex]y[\/latex]- intercepts, multiplicity, and end behavior.\r\n
                      \r\n \t
                    1. [latex]g(x)=(x+4)(x\u22121)^2[\/latex]<\/li>\r\n \t
                    2. [latex]k(x)=(x\u22123)^3(x\u22122)^2[\/latex]<\/li>\r\n \t
                    3. [latex]n(x)=\u22123x(x+2)(x\u22124)[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, use the graphs to write the formula for a polynomial function of least degree.\r\n
                        \r\n \t
                      1. \"Graph<\/li>\r\n \t
                      2. \"Graph<\/li>\r\n<\/ol>\r\nFor the following exercises, use the graph to identify zeros and multiplicity.\r\n
                          \r\n \t
                        1. \"Graph<\/li>\r\n \t
                        2. \"Graph<\/li>\r\n<\/ol>\r\nFor the following exercises, use the given information about the polynomial graph to write the equation.\r\n
                            \r\n \t
                          1. Degree [latex]3[\/latex]. Zeros at [latex]x=\u20132, x=1[\/latex], and [latex]x=3[\/latex]. [latex]y[\/latex]-intercept at [latex](0,\u20134)[\/latex].<\/li>\r\n \t
                          2. Degree [latex]5[\/latex]. Roots of multiplicity [latex]2[\/latex] at [latex]x=3[\/latex] and [latex]x=1[\/latex], and a root of multiplicity [latex]1[\/latex] at [latex]x=\u20133[\/latex]. [latex]y[\/latex]-intercept at [latex](0,9)[\/latex].<\/li>\r\n \t
                          3. Degree [latex]5[\/latex]. Double zero at [latex]x=1[\/latex], and triple zero at [latex]x=3[\/latex]. Passes through the point [latex](2,15)[\/latex].<\/li>\r\n \t
                          4. Degree 3. Zeros at [latex]x=-3[\/latex], [latex]x=-2[\/latex] and [latex]x=1[\/latex]. y-intercept at [latex](0,12)[\/latex].<\/li>\r\n \t
                          5. Degree 4. Roots of multiplicity 2 at [latex]x=\\dfrac{1}{2}[\/latex] and roots of multiplicity 1 at [latex]x=6[\/latex] and [latex]x=-2[\/latex]. y-intercept at [latex](0,18)[\/latex].<\/li>\r\n<\/ol>\r\n

                            Dividing Polynomials<\/span><\/h2>\r\nFor the following exercises, use long division to divide. Specify the quotient and the remainder.\r\n
                              \r\n \t
                            1. [latex](x^2+5x\u22121)\u00f7(x\u22121)[\/latex]<\/li>\r\n \t
                            2. [latex](3x^2+23x+14)\u00f7(x+7)[\/latex]<\/li>\r\n \t
                            3. [latex](6x^2\u221225x\u221225)\u00f7(6x+5)[\/latex]<\/li>\r\n \t
                            4. [latex](2x^2\u22123x+2)\u00f7(x+2)[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.)\r\n
                                \r\n \t
                              1. [latex](2x^3\u22126x^2\u22127x+6)\u00f7(x\u22124)[\/latex]<\/li>\r\n \t
                              2. [latex](4x^3\u221212x^2\u22125x\u22121)\u00f7(2x+1)[\/latex]<\/li>\r\n \t
                              3. [latex](3x^3\u22122x^2+x\u22124)\u00f7(x+3)[\/latex]<\/li>\r\n \t
                              4. [latex](2x^3+7x^2\u221213x\u22123)\u00f7(2x\u22123)[\/latex]<\/li>\r\n \t
                              5. [latex](4x^3-5x^2+13)\u00f7(x+4)[\/latex]<\/li>\r\n \t
                              6. [latex](x^3-21x^2+147x\u2212343)\u00f7(x-7)[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.\r\n
                                  \r\n \t
                                1. [latex]x\u22122[\/latex], [latex]3x^4\u22126x^3\u22125x+10[\/latex]<\/li>\r\n \t
                                2. [latex]x\u22122[\/latex], [latex]4x^4\u221215x^2\u22124[\/latex]<\/li>\r\n \t
                                3. [latex]x+\\frac{1}{3}[\/latex], [latex]3x^4+x^3\u22123x+1[\/latex]<\/li>\r\n<\/ol>\r\n

                                  Zeros of Polynomial Functions<\/span><\/h2>\r\nFor the following exercises, use the Remainder Theorem to find the remainder.\r\n
                                    \r\n \t
                                  1. [latex](3x^3\u22122x^2+x\u22124)\u00f7(x+3)[\/latex]<\/li>\r\n \t
                                  2. [latex](\u22123x^2+6x+24)\u00f7(x\u22124)[\/latex]<\/li>\r\n \t
                                  3. [latex](x^4\u22121)\u00f7(x\u22124)[\/latex]<\/li>\r\n \t
                                  4. [latex](4x^3+5x^2\u22122x+7)\u00f7(x+2)[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, use the given factor and the Factor Theorem to find all real zeros for the given polynomial function.\r\n
                                      \r\n \t
                                    1. [latex]f(x)=2x^3+x^2\u22125x+2; x+2[\/latex]<\/li>\r\n \t
                                    2. [latex]f(x)=2x^3+3x^2+x+6; x+2[\/latex]<\/li>\r\n \t
                                    3. [latex]x^3+3x^2+4x+12; x+3[\/latex]<\/li>\r\n \t
                                    4. [latex]2x^3+5x^2\u221212x\u221230; 2x+5[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation.\r\n
                                        \r\n \t
                                      1. [latex]2x^3+7x^2\u221210x\u221224=0[\/latex]<\/li>\r\n \t
                                      2. [latex]x^3+5x^2\u221216x\u221280=0[\/latex]<\/li>\r\n \t
                                      3. [latex]2x^3\u22123x^2\u221232x\u221215=0[\/latex]<\/li>\r\n \t
                                      4. [latex]2x^3\u22123x^2\u2212x+1=0[\/latex]<\/li>\r\n \t
                                      5. [latex]2x^3\u22125x^2+9x-9=0[\/latex]<\/li>\r\n<\/ol>\r\n

                                        For the following exercises, find all complex solutions (real and non-real).<\/p>\r\n\r\n

                                          \r\n \t
                                        1. [latex]x^3-8x^2+25x-26=0[\/latex] Solutions: [latex]x=2[\/latex], [latex]x=3\\pm i[\/latex]<\/li>\r\n \t
                                        2. [latex]3x^3-4x^2+11x+10=0[\/latex] Solutions: [latex]x=-1[\/latex], [latex]x=\\dfrac{2\\pm i\\sqrt{11}}{3}[\/latex]<\/li>\r\n \t
                                        3. [latex]2x^3-3x^2+32x+17=0[\/latex] Solutions: [latex]x=-1[\/latex], [latex]x=2\\pm2i[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, use Descartes\u2019 Rule to determine the possible number of positive and negative solutions. Confirm with the given graph.\r\n
                                            \r\n \t
                                          1. [latex]f(x)=x^4-x^2-1[\/latex]<\/li>\r\n \t
                                          2. [latex]f(x)=x^3-2x^2+x-1[\/latex]<\/li>\r\n \t
                                          3. [latex]f(x)=2x^3+37x^2+200x+300[\/latex]<\/li>\r\n \t
                                          4. [latex]f(x)=2x^4-5x^3-5x^2+5x+3[\/latex]<\/li>\r\n \t
                                          5. [latex]f(x)=10x^4-21x^2+11[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, list all possible rational zeros for the functions.\r\n
                                              \r\n \t
                                            1. [latex]f(x)=2x^3+3x^2\u22128x+5[\/latex]<\/li>\r\n \t
                                            2. [latex]f(x)=6x^4\u221210x^2+13x+1[\/latex]<\/li>\r\n<\/ol>\r\n

                                              For the following exercises, find the dimensions of the box described.<\/p>\r\n\r\n

                                                \r\n \t
                                              1. The length is twice as long as the width. The height is [latex]2[\/latex] inches greater than the width. The volume is [latex]192[\/latex] cubic inches.<\/li>\r\n \t
                                              2. The length is one inch more than the width, which is one inch more than the height. The volume is [latex]86.625[\/latex] cubic inches.<\/li>\r\n \t
                                              3. The length is [latex]3[\/latex] inches more than the width. The width is [latex]2[\/latex] inches more than the height. The volume is [latex]120[\/latex] cubic inches.<\/li>\r\n<\/ol>\r\n

                                                For the following exercises, find the dimensions of the right circular cylinder described.<\/p>\r\n\r\n

                                                  \r\n \t
                                                1. The height is one less than one half the radius. The volume is [latex]72\\pi[\/latex] cubic meters.<\/li>\r\n \t
                                                2. The radius and height differ by two meters. The height is greater and the volume is [latex]28.125\\pi[\/latex] cubic meters.<\/li>\r\n<\/ol>","rendered":"

                                                  Introduction to Power and Polynomial Functions<\/span><\/h2>\n

                                                  For the following exercises, identify the function as a power function, a polynomial function, or neither.<\/p>\n

                                                    \n
                                                  1. [latex]f(x)=(x^2)^3[\/latex]<\/li>\n
                                                  2. [latex]f(x)=\\frac{x^2}{x^2\u22121}[\/latex]<\/li>\n
                                                  3. [latex]f(x)=3^{x+1}[\/latex]<\/li>\n<\/ol>\n

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

                                                      \n
                                                    1. [latex]7\u22122x^2[\/latex]<\/li>\n
                                                    2. [latex]x(4\u2212x^2)(2x+1)[\/latex]<\/li>\n<\/ol>\n

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

                                                        \n
                                                      1. [latex]f(x)=x^4[\/latex]<\/li>\n
                                                      2. [latex]f(x)=\u2212x^4[\/latex]<\/li>\n
                                                      3. [latex]f(x)=\u22122x^4\u22123x^2+x\u22121[\/latex]<\/li>\n
                                                      4. [latex]f(x)=x^2(2x^3-x+1)[\/latex]<\/li>\n<\/ol>\n

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

                                                          \n
                                                        1. [latex]f(t)=2(t\u22121)(t+2)(t\u22123)[\/latex]<\/li>\n
                                                        2. [latex]f(x)=x^4\u221216[\/latex]<\/li>\n
                                                        3. [latex]f(x)=x(x^2\u22122x\u22128)[\/latex]<\/li>\n<\/ol>\n

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

                                                            \n
                                                          1. \"Graph<\/li>\n
                                                          2. \"Graph<\/li>\n
                                                          3. \"Graph<\/li>\n
                                                          4. \"Graph<\/li>\n<\/ol>\n

                                                            For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.<\/p>\n

                                                              \n
                                                            1. \"Graph<\/li>\n
                                                            2. \"Graph<\/li>\n
                                                            3. \"Graph<\/li>\n
                                                            4. \"Graph<\/li>\n
                                                            5. [latex]V(m)=(3+2m)^3[\/latex], where [latex]m[\/latex] is the number of minutes elapsed.<\/li>\n
                                                            6. [latex]V(x)=x(8-2x)^2[\/latex], where [latex]x[\/latex] is the length of the square’s side in inches.<\/li>\n<\/ol>\n

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

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

                                                                \n
                                                              1. [latex]C(t)=3(t+2)(t\u22123)(t+5)[\/latex]<\/li>\n
                                                              2. [latex]C(t)=2t(t\u22123)(t+1)^2[\/latex]<\/li>\n
                                                              3. [latex]C(t)=4t^4+12t^3\u221240t^2[\/latex]<\/li>\n
                                                              4. [latex]f(x)=x^3+x^2\u221220x[\/latex]<\/li>\n
                                                              5. [latex]f(x)=x^3+x^2\u22124x-4[\/latex]<\/li>\n
                                                              6. [latex]f(x)=2x^3-x^2\u22128x+4[\/latex]<\/li>\n
                                                              7. [latex]f(x)=2x^4+6x^2-8[\/latex]<\/li>\n
                                                              8. [latex]f(x)=x^6-2x^4-3x^2[\/latex]<\/li>\n
                                                              9. [latex]f(x)=x^5-5x^3+4x[\/latex]<\/li>\n<\/ol>\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

                                                                  \n
                                                                1. [latex]f(x)=x^3\u22129x[\/latex], between [latex]x=2[\/latex] and [latex]x=4[\/latex].<\/li>\n
                                                                2. [latex]f(x)=\u2212x^4+4[\/latex], between [latex]x=1[\/latex] and [latex]x=3[\/latex].<\/li>\n
                                                                3. [latex]f(x)=x^3\u2212100x+2[\/latex], between [latex]x=0.01[\/latex] and [latex]x=0.1[\/latex].<\/li>\n<\/ol>\n

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

                                                                    \n
                                                                  1. [latex]f(x)=x^2(2x+3)^5(x\u22124)^2[\/latex]<\/li>\n
                                                                  2. [latex]f(x)=x^2(x^2+4x+4)[\/latex]<\/li>\n
                                                                  3. [latex]f(x)=(3x+2)^5(x^2\u221210x+25)[\/latex]<\/li>\n
                                                                  4. [latex]f(x)=x^6-x^5-2x^4[\/latex]<\/li>\n
                                                                  5. [latex]f(x)=4x^5-12x^4+9x^3[\/latex]<\/li>\n
                                                                  6. [latex]f(x)=4x^4(9x^4-12x^3+4x^2)[\/latex]<\/li>\n<\/ol>\n

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

                                                                      \n
                                                                    1. [latex]g(x)=(x+4)(x\u22121)^2[\/latex]<\/li>\n
                                                                    2. [latex]k(x)=(x\u22123)^3(x\u22122)^2[\/latex]<\/li>\n
                                                                    3. [latex]n(x)=\u22123x(x+2)(x\u22124)[\/latex]<\/li>\n<\/ol>\n

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

                                                                        \n
                                                                      1. \"Graph<\/li>\n
                                                                      2. \"Graph<\/li>\n<\/ol>\n

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

                                                                          \n
                                                                        1. \"Graph<\/li>\n
                                                                        2. \"Graph<\/li>\n<\/ol>\n

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

                                                                            \n
                                                                          1. Degree [latex]3[\/latex]. Zeros at [latex]x=\u20132, x=1[\/latex], and [latex]x=3[\/latex]. [latex]y[\/latex]-intercept at [latex](0,\u20134)[\/latex].<\/li>\n
                                                                          2. Degree [latex]5[\/latex]. Roots of multiplicity [latex]2[\/latex] at [latex]x=3[\/latex] and [latex]x=1[\/latex], and a root of multiplicity [latex]1[\/latex] at [latex]x=\u20133[\/latex]. [latex]y[\/latex]-intercept at [latex](0,9)[\/latex].<\/li>\n
                                                                          3. Degree [latex]5[\/latex]. Double zero at [latex]x=1[\/latex], and triple zero at [latex]x=3[\/latex]. Passes through the point [latex](2,15)[\/latex].<\/li>\n
                                                                          4. Degree 3. Zeros at [latex]x=-3[\/latex], [latex]x=-2[\/latex] and [latex]x=1[\/latex]. y-intercept at [latex](0,12)[\/latex].<\/li>\n
                                                                          5. Degree 4. Roots of multiplicity 2 at [latex]x=\\dfrac{1}{2}[\/latex] and roots of multiplicity 1 at [latex]x=6[\/latex] and [latex]x=-2[\/latex]. y-intercept at [latex](0,18)[\/latex].<\/li>\n<\/ol>\n

                                                                            Dividing Polynomials<\/span><\/h2>\n

                                                                            For the following exercises, use long division to divide. Specify the quotient and the remainder.<\/p>\n

                                                                              \n
                                                                            1. [latex](x^2+5x\u22121)\u00f7(x\u22121)[\/latex]<\/li>\n
                                                                            2. [latex](3x^2+23x+14)\u00f7(x+7)[\/latex]<\/li>\n
                                                                            3. [latex](6x^2\u221225x\u221225)\u00f7(6x+5)[\/latex]<\/li>\n
                                                                            4. [latex](2x^2\u22123x+2)\u00f7(x+2)[\/latex]<\/li>\n<\/ol>\n

                                                                              For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.)<\/p>\n

                                                                                \n
                                                                              1. [latex](2x^3\u22126x^2\u22127x+6)\u00f7(x\u22124)[\/latex]<\/li>\n
                                                                              2. [latex](4x^3\u221212x^2\u22125x\u22121)\u00f7(2x+1)[\/latex]<\/li>\n
                                                                              3. [latex](3x^3\u22122x^2+x\u22124)\u00f7(x+3)[\/latex]<\/li>\n
                                                                              4. [latex](2x^3+7x^2\u221213x\u22123)\u00f7(2x\u22123)[\/latex]<\/li>\n
                                                                              5. [latex](4x^3-5x^2+13)\u00f7(x+4)[\/latex]<\/li>\n
                                                                              6. [latex](x^3-21x^2+147x\u2212343)\u00f7(x-7)[\/latex]<\/li>\n<\/ol>\n

                                                                                For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.<\/p>\n

                                                                                  \n
                                                                                1. [latex]x\u22122[\/latex], [latex]3x^4\u22126x^3\u22125x+10[\/latex]<\/li>\n
                                                                                2. [latex]x\u22122[\/latex], [latex]4x^4\u221215x^2\u22124[\/latex]<\/li>\n
                                                                                3. [latex]x+\\frac{1}{3}[\/latex], [latex]3x^4+x^3\u22123x+1[\/latex]<\/li>\n<\/ol>\n

                                                                                  Zeros of Polynomial Functions<\/span><\/h2>\n

                                                                                  For the following exercises, use the Remainder Theorem to find the remainder.<\/p>\n

                                                                                    \n
                                                                                  1. [latex](3x^3\u22122x^2+x\u22124)\u00f7(x+3)[\/latex]<\/li>\n
                                                                                  2. [latex](\u22123x^2+6x+24)\u00f7(x\u22124)[\/latex]<\/li>\n
                                                                                  3. [latex](x^4\u22121)\u00f7(x\u22124)[\/latex]<\/li>\n
                                                                                  4. [latex](4x^3+5x^2\u22122x+7)\u00f7(x+2)[\/latex]<\/li>\n<\/ol>\n

                                                                                    For the following exercises, use the given factor and the Factor Theorem to find all real zeros for the given polynomial function.<\/p>\n

                                                                                      \n
                                                                                    1. [latex]f(x)=2x^3+x^2\u22125x+2; x+2[\/latex]<\/li>\n
                                                                                    2. [latex]f(x)=2x^3+3x^2+x+6; x+2[\/latex]<\/li>\n
                                                                                    3. [latex]x^3+3x^2+4x+12; x+3[\/latex]<\/li>\n
                                                                                    4. [latex]2x^3+5x^2\u221212x\u221230; 2x+5[\/latex]<\/li>\n<\/ol>\n

                                                                                      For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation.<\/p>\n

                                                                                        \n
                                                                                      1. [latex]2x^3+7x^2\u221210x\u221224=0[\/latex]<\/li>\n
                                                                                      2. [latex]x^3+5x^2\u221216x\u221280=0[\/latex]<\/li>\n
                                                                                      3. [latex]2x^3\u22123x^2\u221232x\u221215=0[\/latex]<\/li>\n
                                                                                      4. [latex]2x^3\u22123x^2\u2212x+1=0[\/latex]<\/li>\n
                                                                                      5. [latex]2x^3\u22125x^2+9x-9=0[\/latex]<\/li>\n<\/ol>\n

                                                                                        For the following exercises, find all complex solutions (real and non-real).<\/p>\n

                                                                                          \n
                                                                                        1. [latex]x^3-8x^2+25x-26=0[\/latex] Solutions: [latex]x=2[\/latex], [latex]x=3\\pm i[\/latex]<\/li>\n
                                                                                        2. [latex]3x^3-4x^2+11x+10=0[\/latex] Solutions: [latex]x=-1[\/latex], [latex]x=\\dfrac{2\\pm i\\sqrt{11}}{3}[\/latex]<\/li>\n
                                                                                        3. [latex]2x^3-3x^2+32x+17=0[\/latex] Solutions: [latex]x=-1[\/latex], [latex]x=2\\pm2i[\/latex]<\/li>\n<\/ol>\n

                                                                                          For the following exercises, use Descartes\u2019 Rule to determine the possible number of positive and negative solutions. Confirm with the given graph.<\/p>\n

                                                                                            \n
                                                                                          1. [latex]f(x)=x^4-x^2-1[\/latex]<\/li>\n
                                                                                          2. [latex]f(x)=x^3-2x^2+x-1[\/latex]<\/li>\n
                                                                                          3. [latex]f(x)=2x^3+37x^2+200x+300[\/latex]<\/li>\n
                                                                                          4. [latex]f(x)=2x^4-5x^3-5x^2+5x+3[\/latex]<\/li>\n
                                                                                          5. [latex]f(x)=10x^4-21x^2+11[\/latex]<\/li>\n<\/ol>\n

                                                                                            For the following exercises, list all possible rational zeros for the functions.<\/p>\n

                                                                                              \n
                                                                                            1. [latex]f(x)=2x^3+3x^2\u22128x+5[\/latex]<\/li>\n
                                                                                            2. [latex]f(x)=6x^4\u221210x^2+13x+1[\/latex]<\/li>\n<\/ol>\n

                                                                                              For the following exercises, find the dimensions of the box described.<\/p>\n

                                                                                                \n
                                                                                              1. The length is twice as long as the width. The height is [latex]2[\/latex] inches greater than the width. The volume is [latex]192[\/latex] cubic inches.<\/li>\n
                                                                                              2. The length is one inch more than the width, which is one inch more than the height. The volume is [latex]86.625[\/latex] cubic inches.<\/li>\n
                                                                                              3. The length is [latex]3[\/latex] inches more than the width. The width is [latex]2[\/latex] inches more than the height. The volume is [latex]120[\/latex] cubic inches.<\/li>\n<\/ol>\n

                                                                                                For the following exercises, find the dimensions of the right circular cylinder described.<\/p>\n

                                                                                                  \n
                                                                                                1. The height is one less than one half the radius. The volume is [latex]72\\pi[\/latex] cubic meters.<\/li>\n
                                                                                                2. The radius and height differ by two meters. The height is greater and the volume is [latex]28.125\\pi[\/latex] cubic meters.<\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":35,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":206,"module-header":"practice","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4377"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4377\/revisions"}],"predecessor-version":[{"id":5975,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4377\/revisions\/5975"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/206"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4377\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=4377"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=4377"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=4377"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=4377"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}