The Fundamental Theorem of Algebra states that a polynomial of degree \( n \) has exactly \( n \) roots in the complex number system (counting multiplicities).
In the given polynomial:
\[ x^3 + x^2 - 16x - 16 = 0 \]
the degree of the polynomial is 3 (since the highest power of \( x \) is 3).
According to the Fundamental Theorem of Algebra, this polynomial will have exactly 3 roots in the complex number system. These roots may be real or complex, and they can include repeated roots, but there will be a total of 3 roots when counted with their multiplicities.
So, the number of roots for the polynomial \( x^3 + x^2 - 16x - 16 = 0 \) is:
\[ \boxed{3} \]
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