To find all real and complex zeros of the function \( f(x) = x^3 - 3x^2 + 2x - 6 \), we need to solve the equation \( f(x) = 0 \). Here are the steps to find the zeros:
1. Set the function equal to zero:
[tex]\[
x^3 - 3x^2 + 2x - 6 = 0
\][/tex]
2. Factor the polynomial, if possible:
- In this case, factoring directly might be complicated.
- We should consider the possible rational roots using the Rational Root Theorem. However, since the polynomial has complex zeros, it implies some roots may not be rational.
3. Solve the polynomial equation:
- Solve for the roots of the cubic polynomial either by factoring or by using algebraic methods such as synthetic division or by more advanced techniques like solving systems of equations.
4. Identify real and complex roots:
- Suppose we find roots directly, we would be able to find:
[tex]\[
\{3, -\sqrt{2}i, \sqrt{2}i\}
\][/tex]
Here, we have:
- One real root: \( 3 \)
- Two complex roots: \( -\sqrt{2}i \) and \( \sqrt{2}i \)
Thus, the zeros of the function \( f(x) = x^3 - 3x^2 + 2x - 6 \) are:
[tex]\[
3, -\sqrt{2}i, \sqrt{2}i
\][/tex]
