When dealing with polynomial functions that have real coefficients, a key property to remember is how complex and irrational roots behave. Given that one of the roots of the polynomial is an irrational number, in this case, √7, we need to consider whether it has a conjugate that is also a root.
For any polynomial function with real coefficients, if it has a root that includes an irrational number involved in a square root (like √7), then the negative of that root must also be included in the set of roots. This is because irrational numbers often appear in conjugate pairs when related to polynomial equations. Therefore, alongside √7, we also consider its negative counterpart.
Thus, if the polynomial function f(x) has roots 3 and √7, it logically follows that both:
- 3 (which is a rational root),
- √7 (which is the original irrational root),
- -√7 (the negative of the irrational root),
must also be roots of the polynomial f(x). Therefore, the additional root that must also be part of the function is -√7.