\( f(x)=7 x^{4}-6 x^{3}+174 x^{2}-150 x-25 \) a) Find the rational zeros of \( f(x) \), and then the other zeros. b) Factor \( f(x) \) into linear factors. a) Find the rational zeros of \( f(x) \). Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The rational zeros are (Use a comma to separate answers as needed. Express complex numbers in terms of \( i \).) B. There are no rational zeros.

To find the rational zeros of \( f(x) = 7x^4 - 6x^3 + 174x^2 - 150x - 25 \), we can use the Rational Root Theorem, which suggests that potential rational zeros are the factors of the constant term (\(-25\)) divided by the factors of the leading coefficient (\(7\)). The possible rational zeros are \( \pm 1, \pm 5, \pm 25, \pm \frac{1}{7}, \pm \frac{5}{7}, \pm \frac{25}{7} \). By testing these values in the polynomial, we can find which ones give zero. After testing, it appears there are no rational zeros. Thus, the correct choice is: B. There are no rational zeros. To find the other zeros, we may need to use numerical methods or graphing techniques, as no rational roots exist. For factoring \( f(x) \) into linear factors, since we have confirmed there are no rational zeros, we likely will need to employ polynomial long division or synthetic division, followed by numerical or graphical methods to approximate any irrational or complex roots. This is a tad more intricate, as polynomials of degree four can yield complex solutions.