List all potential rational zeros. \( f(x)=x^{5}-7 x^{2}+10 \)

To find the potential rational zeros of the polynomial \( f(x) = x^5 - 7x^2 + 10 \), we can use the Rational Root Theorem. This theorem states that any possible rational zero, 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. Here, the constant term is 10 and the leading coefficient is 1. The factors of 10 are \( \pm 1, \pm 2, \pm 5, \pm 10 \). Since the leading coefficient is 1, the potential rational zeros are just the factors of 10. Thus, the potential rational zeros of \( f(x) \) are \( \pm 1, \pm 2, \pm 5, \pm 10 \). Exploring the Rational Root Theorem not only gives us a structured approach to find potential zeros, but it also reminds us of the beauty of mathematics in problem-solving! Each candidate zero can be tested in the polynomial to see if it truly is a root, leading us down the path of further exploration of polynomial behavior. It’s also interesting to note that the actual process of testing these potential rational roots can sometimes lead you to solutions that reveal inherent patterns in the polynomial, adding an element of excitement to the challenge of finding roots! Have fun diving into the polynomial's values to see which of these candidates holds true!