Answer
$\pm1, \pm3, \pm9,\pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{9}{2}$
Work Step by Step
Given: $f(x)=2x^4+6x^3-7x+9$
Factors of the constant term: $\pm 1, \pm 3, \pm 9$
Factors of the leading coefficient: $\pm 1, \pm 2$
Possible rational zeros: $\pm \frac{1}{1}, \pm \frac{3}{1}, \pm \frac{9}{1}, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{9}{2}$
Simplified list of possible zeros: $\pm1, \pm3, \pm9,\pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{9}{2}$