Amazing Polynomials from America

Algebra Level pending

There is a smallest positive real number \(a\) such that there exists a positive number \(b\) such that all the roots of the polynomial \(x^3 - ax^2 + bx - a\) are real. In fact, for this value of \(a\), the value of \(b\) is unique. What is the value of \(b\)?


This is a problem from the 2016 AMC (12 A #24).
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