Question
y=2x^3-5x^2-3x
how is the degree related to the number of turns if the degree is 3 and the turns is 2
dont get this wrong
how is the degree related to the number of turns if the degree is 3 and the turns is 2
dont get this wrong
Answers
GPT-4o mini
In the context of polynomial functions, the degree of the polynomial can provide insight into the number of turns, or local extrema, that the graph of the function may have.
For a polynomial of degree \( n \), the maximum number of turning points (local maxima and minima) is given by \( n - 1 \). This means:
- A polynomial of degree 1 (linear) has at most 0 turning points.
- A polynomial of degree 2 (quadratic) has at most 1 turning point.
- A polynomial of degree 3 (cubic) has at most 2 turning points.
- A polynomial of degree 4 (quartic) has at most 3 turning points, and so on.
In your case, the polynomial \( y = 2x^3 - 5x^2 - 3x \) is a cubic polynomial, which has a degree of 3. Therefore, it can have at most \( 3 - 1 = 2 \) turning points.
Thus, having a degree of 3 suggests that the polynomial can indeed have up to 2 turning points, which is in line with your statement. The actual number of turning points might be less than the maximum, depending on the specific coefficients and roots of the polynomial, but the expected relationship holds true as described.
For a polynomial of degree \( n \), the maximum number of turning points (local maxima and minima) is given by \( n - 1 \). This means:
- A polynomial of degree 1 (linear) has at most 0 turning points.
- A polynomial of degree 2 (quadratic) has at most 1 turning point.
- A polynomial of degree 3 (cubic) has at most 2 turning points.
- A polynomial of degree 4 (quartic) has at most 3 turning points, and so on.
In your case, the polynomial \( y = 2x^3 - 5x^2 - 3x \) is a cubic polynomial, which has a degree of 3. Therefore, it can have at most \( 3 - 1 = 2 \) turning points.
Thus, having a degree of 3 suggests that the polynomial can indeed have up to 2 turning points, which is in line with your statement. The actual number of turning points might be less than the maximum, depending on the specific coefficients and roots of the polynomial, but the expected relationship holds true as described.