Multiplicity in a polynomial refers to the number of times a particular root or zero appears in the polynomial. When a polynomial is factored, you can identify its roots, and the multiplicity indicates how many times each root is repeated.
For instance, consider the polynomial expression f(x) = (x – 2)²(x + 3). In this case, the root x = 2 has a multiplicity of 2 because it appears as a factor squared, while the root x = -3 has a multiplicity of 1 since it appears only once.
The concept of multiplicity helps us understand how the graph of the polynomial behaves at the roots. If a root has an odd multiplicity, the graph will cross the x-axis at that point, whereas if a root has an even multiplicity, the graph will touch the x-axis and turn around at that root. This behavior provides insight into the shape and features of the polynomial’s graph.