To find the zeros of the polynomial function [tex]\( f(x) = x^3 + 6x^2 + 25x + 68 \)[/tex], we need to determine the values of [tex]\( x \)[/tex] that satisfy [tex]\( f(x) = 0 \)[/tex].
1. Setup the Polynomial Equation:
We start by setting the polynomial equal to zero:
[tex]\[
x^3 + 6x^2 + 25x + 68 = 0
\][/tex]
2. Identifying the Zeros:
Through the process of solving this cubic polynomial, we determine that the zeros of the polynomial are:
[tex]\[
-4, -1 - 4i, -1 + 4i
\][/tex]
3. Real and Complex Zeros:
- The first zero is [tex]\(-4\)[/tex], which is a real number.
- The next two zeros involve imaginary numbers. They are [tex]\(-1 - 4i\)[/tex] and [tex]\(-1 + 4i\)[/tex], which are complex conjugates of each other.
4. Reviewing the Zeros:
These zeros can be verified using polynomial division or synthetic division, and further supported by the fact that non-real zeros of polynomials with real coefficients appear in conjugate pairs.
Therefore, the zeros of the polynomial function [tex]\( f(x) = x^3 + 6x^2 + 25x + 68 \)[/tex] are:
[tex]\[
-4, \, -1 - 4i, \, -1 + 4i
\][/tex]
