Using the rational root theorem, list out all possible/candidate rational roots of \( f(x)=22 x^{2}+4 x-26 x^{3}-5 x^{5}+17 x^{4}-10 \). Express your answer as integers as fractions in simplest form. Use commas to separate.

To apply the rational root theorem, we first need to consider the leading coefficient and the constant term of the polynomial. The polynomial \( f(x) = -5x^5 + 17x^4 - 26x^3 + 22x^2 + 4x - 10 \) has a leading coefficient of \(-5\) and a constant term of \(-10\). The possible rational roots are given by the factors of the constant term divided by the factors of the leading coefficient. The factors of \(-10\) are: \(\pm 1, \pm 2, \pm 5, \pm 10\). The factors of \(-5\) are: \(\pm 1, \pm 5\). Now, we take the factors of \(-10\) and divide them by the factors of \(-5\): Possible rational roots: 1. \( \frac{1}{1}, \frac{1}{-5}, \frac{2}{1}, \frac{2}{-5}, \frac{5}{1}, \frac{5}{-5}, \frac{10}{1}, \frac{10}{-5} \) 2. \( -1, -2, -5, -10, \frac{1}{5}, \frac{2}{5}, \frac{5}{5}, \frac{10}{5} \) Simplifying the fractions, we have: Possible rational roots: \( \pm 1, \pm 2, \pm 5, \pm 10, \pm \frac{1}{5}, \pm \frac{2}{5} \). So the complete list of possible rational roots is: **Answer**: \(-10, -5, -2, -1, \frac{-2}{5}, \frac{-1}{5}, 1, 2, 5, 10, \frac{1}{5}, \frac{2}{5}\)