Answer
$\{1,1\}$
Work Step by Step
Step 1. Given $f(x)=x^4-2x^3+10x^2-18x+9$, list possible rational zeros as $\frac{p}{q}=\pm1,\pm3,\pm9$
Step 2. Use synthetic division as shown in the figure to find zeros $x=1,1$.
Step 3. Use the quotient and solve $x^2+9=0$ and there are no real zeros.
Step 4. Thus the real zeros are $\{1,1\}$