Multiplicity 1
In mathematics, multiplicity refers to the number of times a given value or root appears in a polynomial function
In mathematics, multiplicity refers to the number of times a given value or root appears in a polynomial function. When a root has a multiplicity of 1, it means that it appears once as a solution for the polynomial equation.
To understand this concept better, let’s consider a simple example. Suppose we have a polynomial function f(x) = (x – 2)(x – 3)(x – 3)(x – 4). In this case, the roots of the function are 2, 3, and 4. However, the root 2 has a multiplicity of 1 since it appears only once as a factor in the expression (x – 2).
Similarly, the root 4 also has a multiplicity of 1 since it appears only once as a factor in the expression (x – 4). However, the root 3 has a multiplicity of 2 because it appears twice as a factor in the expression (x – 3)(x – 3).
The multiplicity of a root is important because it helps determine the behavior of the graph of the polynomial near that root. For a root with multiplicity 1, the graph will cross the x-axis at that point. On the other hand, for a root with multiplicity greater than 1, the graph will touch or bounce off the x-axis at that point, depending on the specific multiplicity.
Understanding the multiplicity of roots can be particularly useful when analyzing the behavior of polynomial functions, finding their x-intercepts, or determining their end behaviors.
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