Since x^3-5x^2+2x+8 = (x + 1)(x^2 - 6x + 8) we need to find the solution of the equation:

x^2 - 6x + 8 = 0

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click here to see the equation solved for x

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x1 = 4

x2 = 2

The zeros of the equations  x^3-5x^2+2x+8 = 0 are -1, 2 and 4.