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The multiplicity of a zero is important because it tells us how the graph of the polynomial will behave around the zero. For example, notice that the graph of f ( x ) = ( x − 1 ) ( x − 4 ) 2 behaves differently around the zero 1 than around the zero 4 , which is a double zero.
The number of times a given factor appears in the factored form of the equation of a polynomial is called the multiplicity. The zero associated with this factor, [latex]x=2[/latex], has multiplicity 2 because the factor [latex]\left(x - 2\right)[/latex] occurs twice.
Demonstrates how to recognize the multiplicity of a zero from the graph of its polynomial. Explains how graphs just "kiss" the x-axis where zeroes have even multiplicities.
Nov 1, 2021 · The higher the multiplicity, the flatter the curve is at the zero. The sum of the multiplicities is the degree of the polynomial function. For zeros with even multiplicities, the graphs touch or are tangent to the -axis. For zeros with odd multiplicities, the graphs cross or intersect the -axis.
The polynomial p(x)=(x-1)(x-3)² is a 3rd degree polynomial, but it has only 2 distinct zeros. This is because the zero x=3, which is related to the factor (x-3)², repeats twice. This is called multiplicity. It means that x=3 is a zero of multiplicity 2, and x=1 is a zero of multiplicity 1.
Jul 18, 2019 · The polynomial p (x)= (x-1) (x-3)² is a 3rd degree polynomial, but it has only 2 distinct zeros. This is because the zero x=3, which is related to the factor (x-3)², repeats twice. This is ...
The zeros of the function are 1 with odd multiplicity of 1 and \(−\frac{1}{2}\) with even multiplicity of 2. Analysis. Look at the graph of the function \(f\) shown on the right. Notice, at \(x =−0.5\), the graph bounces off the x-axis, indicating the even multiplicity \((2,\;4,\;6…)\) for the zero −0.5.
Mar 3, 2023 · The zeros of the function are 1 and \(−\frac{1}{2}\) with multiplicity 2. Analysis. Look at the graph of the function \(f\) in Figure \(\PageIndex{1}\). Notice, at \(x =−0.5\), the graph bounces off the x-axis, indicating the even multiplicity (2,4,6…) for the zero −0.5.
Given the graph of a polynomial and looking at its x-intercepts, we can determine the factors the polynomial must have. Additionally, we can determine whether those factors are raised to an odd power or to an even power (this is called the multiplicity of the factors).
Step 1: Find each zero by setting each factor equal to zero and solving the resulting equation. Step 2: Find the multiplicity of each factor by examining the exponent on the corresponding...
