Final answer:
To solve for the rational roots of the cubic equation 4x^3 + 12x^2 - 25x - 75, one would use the Rational Root Theorem and then factor the equation or apply polynomial division, reducing it to a quadratic form, if possible, to solve for the roots.
Step-by-step explanation:
The question asks us to find the rational roots of the cubic equation 4x^3 + 12x^2 - 25x - 75. To solve for the rational roots of a polynomial, we can use the Rational Root Theorem which suggests that any rational root, expressed in its simplest form p/q, is such that p is a factor of the constant term and q is a factor of the leading coefficient.
However, since the equation provided is not a quadratic but a cubic, the quadratic formula that is generally used for ax² + bx + c = 0 cannot be applied directly. Cubic equations can sometimes be solved by factoring by grouping, synthetic division, or using the rational root theorem followed by polynomial division. If a cubic equation can be factored into a product of a linear term and a quadratic term, the roots can be found by setting each factor equal to zero.
If a rational root is found, it can be used to simplify the cubic to a quadratic through polynomial long division or synthetic division, after which the quadratic formula can be used to find the remaining roots.