List all possible rational zeros, find all rational zeros, and factor \( f(x) \). \( f(x)=x^{3}-11 x^{2}+31 x-21 \) Choose the possible rational zeros for \( f(x)=x^{3}-11 x^{2}+31 x-21 \) \( \begin{array}{ll}\text { A. } 1,3,7,21 & \text { B. } 1,3,7 \\ \text { C. } \pm 1, \pm 3, \pm 7, \pm 21 & \text { D. } \pm 1, \pm 3, \pm 7\end{array} \)

To find the possible rational zeros of the polynomial \( f(x) = x^3 - 11x^2 + 31x - 21 \), we can use the Rational Root Theorem. According to this theorem, the possible rational roots are given by the factors of the constant term divided by the factors of the leading coefficient. Here, the constant term is -21 and the leading coefficient is 1. Factors of -21 are \( \pm 1, \pm 3, \pm 7, \pm 21 \). Therefore, the possible rational zeros are \( \pm 1, \pm 3, \pm 7, \pm 21 \). The answer is C. Feeling confused about rational zeros? You're not alone! Even seasoned mathematicians can get tangled in the myriad of potential roots. A good tip is to always start by applying the Rational Root Theorem, which provides you with a systematic approach—making the task of finding zeros less daunting and more structured! Jumping into real-world application, knowing about rational roots can come in handy in various engineering and physics problems where polynomial equations frequently arise. For instance, in designing structures or electronic circuits, determining critical points, such as stress or voltage, often leads to solving polynomials. Mastering the concept of rational zeros can empower you in these practical scenarios!