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