Answer
$\{-\frac{1}{3}\}$
Work Step by Step
Step 1. Given $f(x)=x^3-\frac{2}{3}x^2+\frac{8}{3}x+1$, list possible rational zeros as $\frac{p}{q}=\pm1$
Step 2. Use synthetic division as shown in the figure to find zero(s) $x=-\frac{1}{3}$.
Step 3. Use the quotient and solve $x^2-x+3=0$ and there are no real zeros.
Step 4. Thus the real zeros are $\{-\frac{1}{3}\}$