Question
what is the smallest degree a polynomial with 5 turning points could have.
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GPT-4o mini
A polynomial's turning points are locations on its graph where it changes direction, meaning it goes from increasing to decreasing or vice versa. The relationship between the degree of a polynomial and its number of turning points is essential here.
The key takeaway is:
1. A polynomial of degree \( n \) can have at most \( n - 1 \) turning points. This is because the maximum number of times a polynomial can change direction is limited by one less than its degree.
Given this relationship, we want to find the minimum degree \( n \) for a polynomial that can have 5 turning points. If a polynomial can have at most \( n - 1 \) turning points, we set up the inequality:
\[
n - 1 \geq 5
\]
This simplifies to:
\[
n \geq 6
\]
Thus, the smallest degree a polynomial with 5 turning points could have is \( n = 6 \).
To summarize, a polynomial of degree 6 can have up to 5 turning points, fulfilling the condition required for a polynomial with 5 turning points.
The key takeaway is:
1. A polynomial of degree \( n \) can have at most \( n - 1 \) turning points. This is because the maximum number of times a polynomial can change direction is limited by one less than its degree.
Given this relationship, we want to find the minimum degree \( n \) for a polynomial that can have 5 turning points. If a polynomial can have at most \( n - 1 \) turning points, we set up the inequality:
\[
n - 1 \geq 5
\]
This simplifies to:
\[
n \geq 6
\]
Thus, the smallest degree a polynomial with 5 turning points could have is \( n = 6 \).
To summarize, a polynomial of degree 6 can have up to 5 turning points, fulfilling the condition required for a polynomial with 5 turning points.
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