Answer:
The multiplicity of a zero describes how many times a zero of a function is a factor.
Explanation:
The multiplicity of a zero is a concept in algebra that determines how many times a particular value, such as zero, appears as a factor of a polynomial function.
For example, let's consider the function f(x) = (x - 2)^3 * (x + 1)^2 * (x - 3). In this function, we have three distinct factors: (x - 2), (x + 1), and (x - 3). Each factor represents a zero of the function, as setting them equal to zero will give us the corresponding values of x where the function equals zero.
The multiplicity of a zero refers to the exponent or power to which a factor appears in the function. In our example, the zero x = 2 has a multiplicity of 3 because the factor (x - 2) appears with an exponent of 3. Similarly, the zero x = -1 has a multiplicity of 2 because the factor (x + 1) appears with an exponent of 2.
The multiplicity of a zero provides information about the behavior of the graph of the function near that zero. A zero with an odd multiplicity (such as 1, 3, 5, etc.) results in the graph crossing the x-axis at that zero. On the other hand, a zero with an even multiplicity (such as 2, 4, 6, etc.) results in the graph touching or bouncing off the x-axis at that zero.
The concept of multiplicity is important in understanding the behavior and features of polynomial functions.