Which of the following properly describes the relationship between the degree of a polynomial and the number of relative extrema it has?
Responses
The number of relative extrema of a polynomial is always one less than the degree.
The number of relative extrema of a polynomial is always one less than the degree. - no response given
The number of relative extrema of a polynomial is equal to the degree.
The number of relative extrema of a polynomial is equal to the degree. - incorrect
The number of relative extrema of a polynomial is, at most, one less than the degree.
The number of relative extrema of a polynomial is, at most, one less than the degree. - not selected,
this is the correct answer The number of relative extrema of a polynomial is, at least, one less than the degree.
explain how you get that answer
To determine the relationship between the degree of a polynomial and the number of relative extrema it has, we can consider the behavior of the polynomial's graph.
The degree of a polynomial refers to the highest exponent of the variable in the polynomial expression. For example, a polynomial of degree 2 would have a term with the variable raised to the power of 2.
The number of relative extrema of a polynomial represents the number of points where the graph of the polynomial changes direction. These points can be either local maximums or local minimums.
The correct answer is "The number of relative extrema of a polynomial is, at most, one less than the degree."
This means that the maximum number of relative extrema a polynomial can have is one less than its degree. It is possible for a polynomial of degree n to have fewer than n-1 relative extrema or even no relative extrema at all.
This relationship can be observed by considering the highest degree term in the polynomial expression. For example, a polynomial of degree 2 (quadratic) can have at most one relative extremum, while a polynomial of degree 3 (cubic) can have at most two relative extrema.
Overall, the relationship between the degree of a polynomial and the number of relative extrema it has is that the number of relative extrema is, at most, one less than the degree.