Answer
$\{2,2,2, \pm i\sqrt 2\}$
Work Step by Step
Step 1. Given $f(x)=x^5-6x^4+14x^3-20x^2+24x-16$, we can list possible rational zeros as $\pm1,\pm2,\pm4,\pm8,\pm16$
Step 2. There are 5 sign changes in $f(x)$ indicating that there could be 5,3, or 1 positive real zeros.
Step 3. $f(-x)=-x^5-6x^4-14x^3-20x^2-24x-16$, there is 0 sign change in $f(-x)$ indicating that there will be no negative real zero.
Step 4. Use synthetic division to find three real zeros as shown in the figure.
Step 5. The resulting quotient gives $x^2+2=0$ which gives $x=\pm i\sqrt 2$
Step 6. The zeros are $\{2,2,2, \pm i\sqrt 2\}$
