Multiplicities of zeroes refer to the number of times a polynomial function intersects the x-axis at a specific value. These multiplicities can indicate the behavior of the graph near that point.
To find the multiplicity of a zero, you need to factor the polynomial and look at the exponent of the factor corresponding to the zero. The exponent represents the multiplicity of that zero.
The multiplicities of zeroes can tell us about the behavior of the graph near those points. For example, a zero with an odd multiplicity will cross the x-axis, while a zero with an even multiplicity will bounce off the x-axis.
Yes, a polynomial function can have multiple zeroes with the same value but different multiplicities. This means that the graph will intersect the x-axis at that value multiple times, with each intersection having a different behavior depending on the multiplicity of the zero.
Multiplicities of zeroes can help you determine the shape and behavior of the graph near each zero. You can use this information to plot points on the graph and make accurate sketches of the function.