Answer
$\{-\frac{1}{2},1,\pm 2i\}$
Work Step by Step
Step 1. Given $f(x)=2x^4-x^3+7x^2-4x-4$, we can list possible rational zeros as $\pm1,\pm2,\pm4,\pm\frac{1}{2}$
Step 2. There are 3 sign changes in $f(x)$ indicating that there could be 3, or 1 positive real zeros.
Step 3. $f(-x)=2x^4+x^3+7x^2+4x-4$, there is 1 sign change in $f(-x)$ indicating that there will be 1 negative real zero.
Step 4. Use synthetic division to find two real zeros as shown in the figure.
Step 5. The resulting quotient gives $2x^2+8=0$ which gives $x=\pm 2i$
Step 6. The zeros are $\{-\frac{1}{2},1,\pm 2i\}$
