Maximum and Minimum Roots

Algebra Level 5

Consider all polynomials \( P_c (x) = x^3 - 2x^2 - 4x + c \) which have 3 real roots. The difference between the maximum possible root of such a polynomial, and the minimum possible roots of such a polynomial, can be expressed as \( \frac{a}{b} \) for positive coprime integers. Find \( a + b \).

Note: The value of \(c\) need not be identical in the maximum and minimum cases.

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