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MATHS

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two zeros of cubic polynomial ax^3+3x^2-bx-6 are -1 and -2. find the third zero and values of a and b

  • MATHS -

    if x=-1, then -a + 3 + b - 6 = 0
    a - b = -3
    if x=-2, then -8a + 12 + 2b - 6 = 0
    8a - 2b = 6
    4a - b = 3
    subtract them:
    3a = 6
    a = 2
    in a-b=-3
    2-b = -3
    b = 5

    so expression is
    2x^3 + 3x^2 - 5x - 6

    which is (x+1)(x+2)( ?x + ?)
    by common sense, the last bracket must be
    (2x -3)

    so the third root is x=3/2

    check:
    (x+1)(x+2)(2x-3)
    = (x^2 + 3x + 2)(2x-3)
    = 2x^3 - 3x^2 + 6x^2 - 9x + 4x - 6
    = 2x^3 + 3x^2 - 5x - 6 , all is good

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