How many turning points can a polynomial with a degree of 7 have?
A. 6 turning points
B. 7 turning points
C. 8 turning points
D. 9 turning points
Answer (1)
A polynomial of degree n can have at most n − 1 turning points.
The given polynomial has a degree of 7.
Therefore, the maximum number of turning points is 7 − 1 = 6 .
The polynomial with a degree of 7 can have a maximum of 6 turning points.
Explanation
Understanding Turning Points A turning point of a polynomial is a point where the polynomial's derivative is equal to zero. The number of turning points of a polynomial is at most one less than its degree.
Calculating the Maximum Number of Turning Points Since the polynomial has a degree of 7, the maximum number of turning points it can have is 7 - 1 = 6.
Conclusion Therefore, a polynomial with a degree of 7 can have at most 6 turning points.
Examples
Understanding turning points of polynomials is crucial in various fields, such as physics and engineering. For instance, when designing a roller coaster, engineers analyze polynomial functions to model the track's curves. The turning points help determine the maximum and minimum heights, ensuring a thrilling yet safe ride. Similarly, in economics, polynomial functions can model cost curves, where turning points indicate the levels of production that minimize costs or maximize profits. These applications highlight the practical significance of understanding polynomial behavior.
