Chapter 4 - Polynomial and Rational Functions - 4.5 The Real Zeros of a Polynomial Function - 4.5 Assess Your Understanding - Page 233: 57

$\{-1,2 \}$

Step 1. Given $f(x)=x^4-x^3+2x^2-4x-8$, list possible rational zeros as $\frac{p}{q}=\pm1,\pm2,\pm4,\pm8$ Step 2. Use synthetic division as shown in the figure to find two zero $x=-1,2$. Step 3. Use the quotient to solve $x^2+4=0$ or $x^2=-4$, thus $x=\pm 2 i$ Step 4. Thus the real zeros are $\{-1,2 \}$