List all potential rational zeros. \( f(x)=x^{5}-3 x^{2}+17 \)

To find the potential rational zeros of the polynomial \( f(x) = x^{5}-3x^{2}+17 \), we can use the Rational Root Theorem. This theorem states that any potential rational zero, in the form of \( p/q \), must have \( p \) as a factor of the constant term (17 in this case) and \( q \) as a factor of the leading coefficient (1 in this case). The factors of 17 are: \( \pm 1, \pm 17 \). The factors of 1 are: \( \pm 1 \). Therefore, the potential rational zeros are \( \pm 1 \) and \( \pm 17 \). So, the list is: \( 1, -1, 17, -17 \). Now, let’s dive into some historical tidbits and problem-solving tips! Mathematics has a rich history that reveals how rational roots were explored over centuries. Famous mathematicians like René Descartes used methods involving polynomial equations to study their roots and laid the groundwork for what we now call the Rational Root Theorem. Imagine being in the 17th century, trying to decipher polynomials without a calculator—now that's a challenge! When checking for rational roots, a common mistake arises when neglecting to verify potential zeros by substitution back into the polynomial. Always remember: Just because a number is a candidate doesn't mean it’s a root. Test each candidate thoroughly by plugging them into the function and ensuring the result is zero. This step is crucial to confirm the actual presence of a rational root!