To determine the number of real and complex roots of the polynomial [tex]\( F(x) = x^3 + 2x^2 + 4x + 8 \)[/tex], we start by considering its degree. This polynomial is of degree 3, which means it has three roots in total (counting multiplicity), according to the Fundamental Theorem of Algebra.
We need to examine the nature of these roots more closely.
1. Finding the roots:
To solve for the roots of the polynomial [tex]\( F(x) = x^3 + 2x^2 + 4x + 8 \)[/tex], complex and real roots are identified through either algebraic methods or by using root-finding techniques for polynomials. However, we already know that the calculation has provided us with the roots, confirming that we have a detailed understanding of their nature.
2. Analyzing the root types:
From this process, it is confirmed there is a mixture of real and complex roots. Specifically:
- The polynomial [tex]\( F(x) = x^3 + 2x^2 + 4x + 8 \)[/tex] has 1 real root.
- The polynomial also has 2 complex roots.
Given the nature of these roots, we can conclude our answer.
So, the polynomial [tex]\( F(x) = x^3 + 2x^2 + 4x + 8 \)[/tex] has:
- 1 real root
- 2 complex roots
Therefore, the correct answer is:
B. 1 real root and 2 complex roots
