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Math Help - Cubic Equation

  1. #1
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    Post Cubic Equation

    Solve this cubic equation for x:x^3-4x^2+x+6=0
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  2. #2
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    Quote Originally Posted by Sawas View Post
    Solve this cubic equation for x:x^3-4x^2+x+6=0
    numerically or analytically?
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  3. #3
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    From the rational root test, the potential rational roots are: \pm 1, \ \pm 2, \ \pm 3, \ \pm 6

    By trial and error we find that x=2 is a root: (2)^3-4(2)^2+(2)+6 = 0, i.e. (x-2) is one of the factors.

    Long divide to get the blue factor: x^3 - 4x^2 + x + 6 = (x-2){\color{blue}(x^2 - 2x - 3)}

    Now factor again and you should be on your way.
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  4. #4
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    Solution

    x^3-4x^2+x+6=0
    Using synthetic division we get:

    +1 -4 +1 +6
    -1 0 -1 +5 -6
    +1 -5 +6 0
    so, -1 is a root of the given equation,so the given equ becomes:
    (x+1)(x^2-5x+6)=0
    which implies (x+1)(x^2-2x-3x+6)=0
    which implies (x+1)(x(x-2)-3(x-2))=0
    which implies (x+1)(x-2)(x-3)=0
    which implies x+1=0 ,x-2=0 ,x-3=0
    which implies x=-1, x=2,x=3
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