Question

What are the two additional roots that a polynomial function p(x) with rational coefficients must have, given that one of its roots is 14-sqrt6?

Answer 1

When a polynomial function \( p(x) \) has rational coefficients and it has a root of the form \( 14 - \sqrt{6} \), the conjugate of that root must also be a root of the polynomial. This is a consequence of the fact that for any polynomial with rational coefficients, irrational roots must come in conjugate pairs.

Thus, if \( r_1 = 14 - \sqrt{6} \) is one root, then the additional root, which is the conjugate, is given by:

\[
r_2 = 14 + \sqrt{6}
\]

Therefore, the two additional roots for the polynomial function \( p(x) \) with rational coefficients are:

1. \( 14 - \sqrt{6} \)
2. \( 14 + \sqrt{6} \)

In summary, the two additional roots are:
- \( 14 - \sqrt{6} \)
- \( 14 + \sqrt{6} \)