Answer
$\{\frac{1}{3},\pm\sqrt 5\}$
Work Step by Step
Step 1. Given $f(x)=3x^3-x^2-15x+5$, list possible rational zeros as $\frac{p}{q}=\pm1,\pm5,\pm\frac{1}{3},\pm\frac{5}{3}$
Step 2. Use synthetic division as shown in the figure to find a zero $x=\frac{1}{3}$.
Step 3. Use the quotient and solve $3x^2-15=0$ or $x^2=5$, thus $x=\pm\sqrt 5$.
Step 4. Thus the real zeros are $\{\frac{1}{3},\pm\sqrt 5\}$