To determine the roots of the polynomial function and write its equation when one of the roots is irrational, we need to remember that for polynomials with real coefficients, if we have an irrational root, its conjugate must also be a root to ensure the coefficients remain real.
Given roots of the polynomial are [tex]\(2\)[/tex], [tex]\(\sqrt{3}\)[/tex], and [tex]\(5\)[/tex]. Since [tex]\(\sqrt{3}\)[/tex] is an irrational number, its conjugate [tex]\(-\sqrt{3}\)[/tex] must also be a root of the polynomial. Therefore, the roots are [tex]\(2\)[/tex], [tex]\(\sqrt{3}\)[/tex], [tex]\(-\sqrt{3}\)[/tex], and [tex]\(5\)[/tex].
We can now write the equation of the polynomial in its factored form:
[tex]\[ P(x) = (x - 2)(x - \sqrt{3})(x + \sqrt{3})(x - 5) \][/tex]
To summarize, the roots of the polynomial function include:
- [tex]\(2\)[/tex],
- [tex]\(\sqrt{3}\)[/tex],
- [tex]\(5\)[/tex],
- [tex]\(-\sqrt{3}\)[/tex].
From the list of choices given, the root that must also be a root of the function is:
[tex]\[ -\sqrt{3} \][/tex]
Thus, the correct choice is:
[tex]\[ -\sqrt{3} \][/tex]
