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Multiplicity of Zeros

Concept

Multiplicity of Zeros

When solving an equation with the Zero Product Property, the first step is factoring the polynomial. Then, each factor is set equal to 00 and solved as its own equation. x35x2+7x3=0(x3)(x1)(x1)=0x3=0x1=0x1=0\begin{gathered} x^3-5x^2+7x-3=0\\ \Downarrow \\ (x-3)(x-1)(x-1)=0\\ \Downarrow \\ x-3=0\\ x-1=0\\ x-1=0 \end{gathered} In this case, even though there are three factors, solving them will result in only two unique zeros: x=3x=3 and x=1.x=1. This occurrence is described as the multiplicity of zeros; in the example equation above, the zero x=3x=3 has multiplicity 11 because it only occurs once. On the other hand, x=1x=1 has multiplicity 22 because there are two (x1)(x-1) factors.

Note: The multiplicity of a zero being greater than 11 implies that, algebraically, the zero occurs multiple times. The graph, however, does not intercept the x-x\text{-}axis at this point multiple times. In this example, the zero x=1x=1 has a multiplicity of 22 but the graph intercepts the x-x\text{-}axis at x=1x=1 only once.