use the rational zero's theorem to list the potential zero's of the polynomial function. Do not attempt to find the zero's f(x)=-169x^7-x^6+x+13 and then the second part is find the potential rational zero's.

any rational zeros have numerators which divide 13 and denominators which divide 169. That means

±(1,13)/±(1,13,169)
or,
±1 ±13 ±1/13 ±1/169

A little synthetic division shows that there are no rational roots.

To use the Rational Zero's Theorem and list the potential zeros of a polynomial function, we need to consider the factors of the constant term and the factors of the leading coefficient.

In the given polynomial function f(x) = -169x^7 - x^6 + x + 13, the constant term is 13, and the leading coefficient is -169.

The factors of the constant term 13 are ±1 and ±13.
The factors of the leading coefficient -169 are ±1 and ±169.

Therefore, the potential rational zeros (or rational roots) of the polynomial function f(x) = -169x^7 - x^6 + x + 13 can be obtained by considering all possible combinations of these factors.

The potential rational zeros are calculated by taking each factor of the constant term divided by each factor of the leading coefficient. Thus, the potential rational zeros are:
±1/169, ±1/13, ±13, ±1.

Note that these values are only potential rational zeros, and they need to be further evaluated or tested to determine if they are actual zeros of the function.