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Math Help - help with polynomial roots question

  1. #1
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    help with polynomial roots question

    Use synthetic division to show that all of the real zeros of f(x)=x^3-3x^2-4x+5 lie in the interval [-2,4]
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  2. #2
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    Hello, cityismine!

    Use synthetic division to show that all of the real zeros of f(x) \:=\: x^3-3x^2-4x+5
    lie in the interval [-2,4]
    I'm not sure why Synthetic Divison is used here . . .

    We find that: . f(-2) \:=\:-8-12+8+5 \:=\:-7
    . . The graph is below the x-axis.

    We find that: . f(0) \:=\: 0 - 0 - 9 + 5 \:=\:5
    . . The graph is above the x-axis.

    We find that: . f(2) \:=\:8 - 12 - 8 + 5 \:=\:-7
    . . The graph is below the x-axis.

    We find that:. f(4) \:=\:64 - 48 - 16 + 5 \:=\:5
    . . The graph is above the x-axis.

    So far, the graph looks like this:
    Code:
                |
                |         (4,5)
           (0,5)*           *
                |           :
         -2     |     2     :
        --+-----+-----+-----+----
          :     |     :     4
          :     |     :
          :     |     :
          *     |     *
       (-2,-7)  |   (2,-7}
                |

    Since a polynomial function is continuous, the graph must cross the x-axis
    . . on the intervals (-2,0),\:(0,2).\:(2,4)
    And those are the three (real) zeros of the cubic function.
    Code:
                |
                *           *
              * | *         :
         -2     |     2     :
        --+--o--+--o--+---o-+----
          :     |     :     4
          :     |     :  *
          : *   |   * : *
          *     |     *
                |
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