Answer the following questions about the equation below. \( x^{3}-8 x+8=0 \) (a) List all rational roots that are possible according to the Rational Zero Theorem. Choose the correct answer below. A. \( 1,2,4,8 \) B. \( \pm 8 \) C. \( \pm 1, \pm 2, \pm 4, \pm 8 \) D. \( \pm 1 \)

The Rational Root Theorem states that any rational solution of the polynomial equation, in the form of \( \frac{p}{q} \), must have \( p \) as a factor of the constant term and \( q \) as a factor of the leading coefficient. In this equation \( x^3 - 8x + 8 = 0 \), the constant term is 8 (factors: \( \pm 1, \pm 2, \pm 4, \pm 8 \)) and the leading coefficient is 1 (factors: \( \pm 1 \)). This leads us to possible rational roots being \( \pm 1, \pm 2, \pm 4, \pm 8 \). So, the correct choice is C: \( \pm 1, \pm 2, \pm 4, \pm 8 \). Now, let's dive into some interesting details! In the early history of algebra, mathematicians like Descartes and Viete laid the groundwork for the Rational Root Theorem. They explored relationships between coefficients and roots in polynomials. These breakthroughs were crucial for understanding polynomial equations and paved the way for future mathematicians! When tackling polynomial equations, a common mistake is to forget to check all possible rational roots listed by the Rational Root Theorem! Be sure to simplify your process: start testing the simplest candidates like \( \pm 1 \) before moving to larger numbers. This not only saves time but also builds confidence in your problem-solving approach!