Use the rational zeros theorem to list the potential rational zeros of the polynomial function. Do not attempt to find the zeros. \[ f(x)=-169 x^{7}-x^{6}+x+13 \] Find the potential rational zeros. Choose the correct answer below. O A. \( -1,1,-13,13,-\frac{1}{13}, \frac{1}{13},-\frac{1}{169}, \frac{1}{169} \) B. \( -1,1,-169,169,-\frac{1}{13}, \frac{1}{13},-\frac{1}{169}, \frac{1}{169} \) 45 C. \( -1,1,-13,13,-\frac{1}{13}, \frac{1}{13} \) OD. \( -1,1,-13,13,-169,169,-\frac{1}{13}, \frac{1}{13},-\frac{1}{169}, \frac{1}{169} \)

Recall that the Rational Zeros Theorem states that all possible rational zeros of a polynomial   f(x) = aₙxⁿ + ⋯ + a₀ are of the form   p/q where p is a factor of the constant term (a₀) and q is a factor of the leading coefficient (aₙ). For the polynomial   f(x) = -169x⁷ - x⁶ + x + 13 the constant term is 13 and the leading coefficient is -169. Step 1. List the factors of the constant term 13:  Possible p values: ±1, ±13 Step 2. List the factors of the leading coefficient -169.  Remember, 169 = 13², so the factors are:   Possible q values: ±1, ±13, ±169 Step 3. Form all fractions ±(factor of constant)/(factor of leading coefficient): Potential zeros = ±1/1, ±13/1, ±1/13, ±13/13, ±1/169, ±13/169 Simplify these: • 1/1 = 1 and 13/1 = 13 • 13/13 simplifies to 1 (so it’s not new) • 13/169 simplifies to 1/13 (so it’s already listed) Thus, the distinct potential rational zeros are:   ±1, ±13, ±1/13, and ±1/169 Looking at the provided options, Option A lists exactly these:   -1, 1, -13, 13, -1/13, 1/13, -1/169, 1/169 Therefore, the correct answer is Option A.