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Real Root of Cubic Equation

Source: 2013 AIME I Problem 5

March 15, 2013
algebrapolynomialgeometry3D geometrydifference of squaresspecial factorizationsfactorization

Problem Statement

The real root of the equation 8x3−3x2−3x−1=08x^3 - 3x^2 - 3x - 1 = 0 can be written in the form a3+b3+1c\frac{\sqrt[3]a + \sqrt[3]b + 1}{c}, where aa, bb, and cc are positive integers. Find a+b+ca+b+c.
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