Algebra-2

The Conjugate Root Theorem states that if a polynomial with rational coefficients has a complex root (where and are real numbers and ), then its complex conjugate is also a root of the polynomial.

In this problem, we are given that the quartic polynomial has rational coefficients. Since is a root, by the Conjugate Root Theorem, its conjugate must also be a root. This is because complex roots of polynomials with rational coefficients always occur in conjugate pairs.

Now, for the root . If a polynomial with rational coefficients has a root of the form , then is also a root. Here and . So if is a root, then

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