91Ó°ÊÓ

In complex analysis, the zeros of a function refer to the values of the variable that make the function equal to zero. For the function \( f(z) = z^4 + z^2 \), finding the zeros requires determining the values of \( z \) such that \( f(z) = 0 \).
This involves solving the equation \( f(z) = z^4 + z^2 = 0 \). By factoring the function to \( f(z) = z^2(z^2 + 1) \), we break it down into simpler components, making it easier to identify the zeros.
The zeros in this case are the solutions to the equations \( z^2 = 0 \) and \( z^2 + 1 = 0 \). Therefore, \( z = 0 \), \( z = i \), and \( z = -i \) are all zeros of the function. Understanding zeros is essential for analyzing the behavior of complex functions.

The multiplicity of a zero refers to the number of times a particular zero appears in the factorization of a function. It can indicate how the function behaves near that zero.
For our function \( f(z) = z^2(z^2 + 1) \), the zero \( z = 0 \) arises from the factor \( z^2 \), implying a multiplicity of 2. This means that \( z = 0 \) is not just a simple intersection but a point where the graph touches the axis before turning around.
On the other hand, the zeros \( z = i \) and \( z = -i \) both arise from \( z^2 + 1 = 0 \) and each has a multiplicity of 1. These are simple zeros where the function crosses the axis.
Understanding multiplicity is critical as it provides insight into the shape and movement of the graph around these points.

Complex numbers arise naturally when solving equations where real solutions do not exist. A complex number is of the form \( z = a + bi \) where \( a \) is the real part and \( b \) is the imaginary part. Here, \( i \) is the imaginary unit with the property \( i^2 = -1 \).
In our exercise, the solutions \( z = i \) and \( z = -i \) are examples of complex numbers. They solve the equation \( z^2 + 1 = 0 \), as squaring either \( i \) or \( -i \) results in \( -1 \), thereby fitting the complex number paradigm.
Complex numbers are crucial in many mathematical fields due to their ability to extend the scope of algebraic equations. They allow for a more comprehensive understanding of polynomial behaviors, even in cases where real solutions do not suffice. Integrating complex numbers expands the toolset for tackling diverse mathematical challenges.