Chapter 4 - Polynomial and Rational Functions - 4.5 The Real Zeros of a Polynomial Function - 4.5 Assess Your Understanding - Page 234: 63

$\{-3,-2\}$

Step 1. Given $f(x)=x^4+4x^3+2x^2-x+6$, list possible rational zeros as $\frac{p}{q}=\pm1,\pm2,\pm3,\pm6$ Step 2. Use synthetic division as shown in the figure to find a zero $x=-2,-3$. Step 3. Use the quotient and solve $x^2-x+1=0$ and there are no real zeros. Step 4. Thus the real zeros are $\{-3,-2\}$