Final answer:
The cubic polynomial with rational coefficients and roots 6√6 and 2/3 will also have the root -6√6 due to the Conjugate Root Theorem which states that non-real irrational roots come in conjugate pairs in polynomials with rational coefficients.
Step-by-step explanation:
The subject in question pertains to the roots of a cubic polynomial. Given that two of the roots are 6√6 and 2/3, and since the polynomial has rational coefficients, we can invoke the Conjugate Root Theorem. According to this theorem, if a polynomial has rational coefficients and a real irrational root, then the conjugate of this root must also be a root of the polynomial.
Since 6√6 is irrational and non-real, its conjugate is -6√6. Therefore, the cubic polynomial will have one positive irrational root, one negative irrational root which is its conjugate, and one rational root. Thus, the additional root of the cubic polynomial with rational coefficients, besides 6√6 and 2/3, is -6√6.