polynomial

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given that the expression ax^3-11x^2+bx+2 is exactly divisible by 2x^2-3x-2, find a and b then factorise completly (do not use long division).

  • polynomial -

    (cx-d)(2x^2 - 3x - 2) = ax^3 - 11x^2 + bx + 2
    2cx^3 - (3c+2d)x^2 - (2c-3d)x + 2d = ax^3 - 11x^2 + bx + 2

    2c = a
    3c+2d = 11
    2c-3d = -b
    2d = 2

    so,

    d=1
    3c+2=11
    c=3
    -3 = b
    a = 6

    check:

    (3x-1)(2x^2 - 3x - 2) =
    6x^3 - 11x^2 - 3x + 2

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