Using the rational root theorem, list out all possible/candidate rational roots of \( f(x)=19 x^{4}+6 x^{5}-2+6 x-23 x^{3} \). Express your answer as integers or as fractions in simplest form. Use commas to separate.

To apply the rational root theorem, we first need to identify the coefficients of the polynomial \( f(x) = 6x^5 + 19x^4 - 23x^3 + 6x - 2 \). The potential rational roots are given by the formula \( \frac{p}{q} \), where \( p \) is a factor of the constant term (-2) and \( q \) is a factor of the leading coefficient (6). The factors of -2 are: ±1, ±2 The factors of 6 are: ±1, ±2, ±3, ±6 Now we list all combinations of \( \frac{p}{q} \): 1. From \( p = 1 \): \( \frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{1}{6} \) → 1, 0.5, 0.333..., 0.1666... 2. From \( p = -1 \): \( \frac{-1}{1}, \frac{-1}{2}, \frac{-1}{3}, \frac{-1}{6} \) → -1, -0.5, -0.333..., -0.1666... 3. From \( p = 2 \): \( \frac{2}{1}, \frac{2}{2}, \frac{2}{3}, \frac{2}{6} \) → 2, 1, 0.666..., 0.333... 4. From \( p = -2 \): \( \frac{-2}{1}, \frac{-2}{2}, \frac{-2}{3}, \frac{-2}{6} \) → -2, -1, -0.666..., -0.333... After simplifying, the rational candidate roots we get are: ±1, ±2, ±1/2, ±1/3, ±1/6, ±1/2 So, the complete list of possible rational roots is: 1, -1, 2, -2, 0.5, -0.5, 0.333..., -0.333..., 0.1666..., -0.1666... Expressed in a more standard format that avoids decimals while retaining simplicity: 1, -1, 2, -2, 1/2, -1/2, 1/3, -1/3, 1/6, -1/6.