- #1
forever119
- 1
- 0
What are the requirements for the exercise on multiplicity and set of zeros?
- MHB
- Thread starter forever119
- Start date
-
- Tags
- multiplicity Set
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.
- #2
Opalg
Gold Member
MHB
- 2,778
- 13
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.
Related to What are the requirements for the exercise on multiplicity and set of zeros?
1. 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.
2. 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.
3. 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.
4. 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.
5. 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.
Similar threads
-
Calculus
-
Calculus
-
Set Theory, Logic, Probability, Statistics
-
General Engineering
-
Linear and Abstract Algebra