Answer
$\{-4,2\pm\sqrt 2\}$
Work Step by Step
Step 1. Given $f(x)=x^3-14x+8$, we can list possible rational zeros as $\pm1,\pm2,\pm4,\pm8$
Step 2. There are 2 sign changes in $f(x)$ indicating that there could be 2 or 0 positive real zeros.
Step 3. $f(-x)=-x^3+14x+8$, there is 1 sign change in $f(-x)$ indicating that there must be 1 negative real zero.
Step 4. Use synthetic division to find one real zero as shown in the figure.
Step 5. The resulting quotient gives $x^2-4x+2=0$ which gives $x=\frac{4\pm\sqrt {4^2-4(2)}}{2}=2\pm\sqrt 2$
Step 6. The zeros are $\{-4,2\pm\sqrt 2\}$
