Chapter 4 - Polynomial and Rational Functions - 4.6 Complex Zeros; Fundamental Theorem of Algebra - 4.6 Assess Your Understanding - Page 241: 42

$-25$

Step 1. Identify zeros for $f(x)$ as $x=3+i,3-i,2,-2$, thus $f(x)=(x-3+i)(x-3-i)(x-2)(x+2)=(x^2-6x+10)(x^2-4)$ Step 2. Identify zeros for $g(x)$ as $x=i,-i,2i,-2i$ and write $g(x)=a(x+i)(x-i)(x+2i)(x-2i)=a(x^2+1)(x^2+4)$ Step 3. As $g(0)=-4$, we have $g(0)=4a=-4$, thus $a=-1$ and $g(x)=-(x^2+1)(x^2+4)$ Step 4. $(f+g)(1)=f(1)+g(1)=(1^2-6+10)(1^2-4)-(1^2+1)(1^2+4)=-25$