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forever119
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What are the requirements for the exercise on multiplicity and set of zeros?
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In summary, multiplicity refers to the number of times a particular value appears as a zero of a polynomial function. It is related to the behavior of the graph of the polynomial at that point, with even multiplicities resulting in the graph touching but not crossing the x-axis, and odd multiplicities resulting in the graph crossing the x-axis. A zero must have a positive multiplicity, as a negative multiplicity would mean the factor has been divided out. The multiplicity can be determined from the highest power of the corresponding factor in the factored form of the polynomial. A polynomial can have multiple zeros with the same value but different multiplicities.
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Opalg
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Hi forever119, and welcome to MHB.
For that exercise, there seems to be no need for $f(a)$ and $f(b)$ to be nonzero. It looks as though that requirement is only needed for the following exercise, which refers to the signs of $f(a)$ and $f(b)$. These need to be strictly positive or strictly negative for that exercise to make sense.
For that exercise, there seems to be no need for $f(a)$ and $f(b)$ to be nonzero. It looks as though that requirement is only needed for the following exercise, which refers to the signs of $f(a)$ and $f(b)$. These need to be strictly positive or strictly negative for that exercise to make sense.
FAQ: What are the requirements for the exercise on multiplicity and set of zeros?
What is multiplicity in relation to zeros of a function?
Multiplicity refers to the number of times a particular value appears as a zero of a function. It is determined by the exponent of the corresponding factor in the factored form of the function.
How does the multiplicity of a zero affect the graph of a function?
The multiplicity of a zero affects the behavior of the graph near that point. If the multiplicity is even, the graph will touch or cross the x-axis at that point. If the multiplicity is odd, the graph will cross the x-axis at that point.
Can a function have multiple zeros with the same value but different multiplicities?
Yes, a function can have multiple zeros with the same value but different multiplicities. This means that the graph of the function will touch or cross the x-axis at that point multiple times, depending on the multiplicities.
What is the relationship between the number of zeros and the degree of a polynomial function?
The number of zeros of a polynomial function is equal to its degree, or the highest exponent in the polynomial. For example, a polynomial of degree 3 can have up to 3 zeros.
How can we determine the multiplicity of a zero from the factored form of a function?
The multiplicity of a zero can be determined by the exponent of the corresponding factor in the factored form of the function. For example, if the factor (x+2) appears twice in the factored form, the multiplicity of the zero x=-2 is 2.