Question
y=2x^3-5x^2-3x
how is the degree related to the number of turns
how is the degree related to the number of turns
Answers
GPT-4o mini
The degree of a polynomial is related to the number of turns (or local extrema) it can have. In general, a polynomial of degree \( n \) can have up to \( n - 1 \) turns or local extrema.
In your polynomial \( y = 2x^3 - 5x^2 - 3x \), the degree is 3 because the highest power of \( x \) is \( x^3 \).
Thus, according to the relationship between degree and turns:
- A degree 3 polynomial can have a maximum of \( 3 - 1 = 2 \) turns (or local extrema).
To find the actual number of turns, you can take the derivative of the polynomial and find the critical points where the derivative equals zero. These critical points indicate where the function can change from increasing to decreasing or vice versa:
1. **Take the derivative:**
\[
y' = \frac{d}{dx}(2x^3 - 5x^2 - 3x) = 6x^2 - 10x - 3
\]
2. **Set the derivative to zero to find critical points:**
\[
6x^2 - 10x - 3 = 0
\]
3. **Use the quadratic formula:**
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 6 \cdot (-3)}}{2 \cdot 6} = \frac{10 \pm \sqrt{100 + 72}}{12} = \frac{10 \pm \sqrt{172}}{12}
\]
\[
= \frac{10 \pm 2\sqrt{43}}{12} = \frac{5 \pm \sqrt{43}/6
\]
This will give you the \( x \)-values of the critical points. Depending on the nature of these critical points (i.e., whether they result in local maxima or minima after evaluating the second derivative or using the first derivative test), you can determine the actual number of turns.
In conclusion, for your polynomial, you can expect up to 2 turns, and by analyzing the derivative, you can find the specific locations and nature of those turns.
In your polynomial \( y = 2x^3 - 5x^2 - 3x \), the degree is 3 because the highest power of \( x \) is \( x^3 \).
Thus, according to the relationship between degree and turns:
- A degree 3 polynomial can have a maximum of \( 3 - 1 = 2 \) turns (or local extrema).
To find the actual number of turns, you can take the derivative of the polynomial and find the critical points where the derivative equals zero. These critical points indicate where the function can change from increasing to decreasing or vice versa:
1. **Take the derivative:**
\[
y' = \frac{d}{dx}(2x^3 - 5x^2 - 3x) = 6x^2 - 10x - 3
\]
2. **Set the derivative to zero to find critical points:**
\[
6x^2 - 10x - 3 = 0
\]
3. **Use the quadratic formula:**
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 6 \cdot (-3)}}{2 \cdot 6} = \frac{10 \pm \sqrt{100 + 72}}{12} = \frac{10 \pm \sqrt{172}}{12}
\]
\[
= \frac{10 \pm 2\sqrt{43}}{12} = \frac{5 \pm \sqrt{43}/6
\]
This will give you the \( x \)-values of the critical points. Depending on the nature of these critical points (i.e., whether they result in local maxima or minima after evaluating the second derivative or using the first derivative test), you can determine the actual number of turns.
In conclusion, for your polynomial, you can expect up to 2 turns, and by analyzing the derivative, you can find the specific locations and nature of those turns.