Question
y=2x^3-5x^2-3x
if the degree is 3 what does that tell us about the zeroes
if the degree is 3 what does that tell us about the zeroes
Answers
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GPT-4o mini
If a polynomial has a degree of 3, it indicates that the polynomial is a cubic polynomial. A cubic polynomial can have up to three real zeros (roots). The number of real zeros can vary depending on the nature of the roots:
- **Three distinct real zeros:** This occurs when the cubic intersects the x-axis at three different points.
- **One real zero and two complex conjugate zeros:** This happens when the cubic intersects the x-axis at one point (i.e., at one real root) and the other two roots are complex.
- **Multiple roots:** It's also possible for a cubic polynomial to have a double root or a triple root, which means that the same root is counted more than once, but still, the total count of roots (considering multiplicity) must equal three.
Additionally, the behavior of the graph of the cubic polynomial provides information about its zeros:
1. The graph of a cubic polynomial will approach \( +\infty \) or \( -\infty \) as \( x \) approaches \( +\infty \) or \( -\infty \), respectively.
2. It can change direction at most twice, which accounts for the local maxima and minima, affecting the number of times the graph intersects the x-axis.
Thus, while the degree tells us the maximum number of zeros, it does not specify how many will be real versus complex or if any will be repeated. To find the actual zeros, one would typically use methods like factoring, synthetic division, or numerical methods such as the Newton-Raphson method, or graphing techniques to identify them.
- **Three distinct real zeros:** This occurs when the cubic intersects the x-axis at three different points.
- **One real zero and two complex conjugate zeros:** This happens when the cubic intersects the x-axis at one point (i.e., at one real root) and the other two roots are complex.
- **Multiple roots:** It's also possible for a cubic polynomial to have a double root or a triple root, which means that the same root is counted more than once, but still, the total count of roots (considering multiplicity) must equal three.
Additionally, the behavior of the graph of the cubic polynomial provides information about its zeros:
1. The graph of a cubic polynomial will approach \( +\infty \) or \( -\infty \) as \( x \) approaches \( +\infty \) or \( -\infty \), respectively.
2. It can change direction at most twice, which accounts for the local maxima and minima, affecting the number of times the graph intersects the x-axis.
Thus, while the degree tells us the maximum number of zeros, it does not specify how many will be real versus complex or if any will be repeated. To find the actual zeros, one would typically use methods like factoring, synthetic division, or numerical methods such as the Newton-Raphson method, or graphing techniques to identify them.