The zeros are -2, 0, and 4, each with a multiplicity of 2, 1 and 2 respectively.
A zero of a function is an input value where the function's output is zero. In other words, the graph of the function intersects the x-axis at that point.
Looking at the graph, we see that it crosses the x-axis at three points:
- x = -2: The graph touches the x-axis at this point and bounces off, indicating a zero with even multiplicity (in this case, multiplicity 2).
- x = 0: The graph crosses the x-axis at this point, changes direction, and continues smoothly, indicating a zero with odd multiplicity (multiplicity 1).
- x = 4: Similar to x = -2, the graph touches the x-axis at this point and bounces off, indicating a zero with even multiplicity (multiplicity 2).
Therefore, the zeros of the function are:
- x = -2 with multiplicity 2
- x = 0 with multiplicity 1
- x = 4 with multiplicity 2
Calculating Multiplicity:
The multiplicity of a zero indicates how many times the corresponding factor appears in the function's factored form. We can estimate the multiplicity by looking at the behavior of the graph near the zero:
- Even multiplicity (2, 4, etc.): If the graph touches the x-axis at the zero and bounces off, it indicates that the corresponding factor is squared in the function's factored form. For example, if the function is in the form f(x) = (x - a)^2, then the graph will touch the x-axis at x = a with multiplicity 2.
- Odd multiplicity (1, 3, etc.): If the graph crosses the x-axis at the zero and changes direction smoothly, it indicates that the corresponding factor is simply x - a in the function's factored form. For example, if the function is in the form f(x) = (x - a), then the graph will cross the x-axis at x = a with multiplicity 1.