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Can the upper bound of a polynomial be equal to the greatest real zero of a polynomial or does it have to greater than it?
TL;DR Title
My friend and I got worked up on this and unfortunately no teacher was present during the time to quench our curiosity, searching through the internet proved unfruitful.
Basically he asked me to give him a polynomial equation for an exercise and I gave him a 5th degree with mostly arbitrary coefficients.
By the Bounds of zeros theorem, finding the upper bound requires obtaining all positive coefficients when doing synthetic division, but how do we do that when the first step of synthetic division is bringing down the coefficient of the first number, which is already negative?
EDIT: Okay sorry for any confusion I may cause but yahoo decided to keep my details for a different question.
2 Answers
- TomVLv 73 years ago
The upper bound of a polynomial of even degree can be equal to the greatest real zero of the polynomial. As an example, look at any downward opening parabola tangent to the x-axis.
The equation of a parabola is a polynomial
The upper bound of the parabola is the point of tangency with the x-axis
The greatest (only) real zero is the point of tangency with the x-axis
Hence it is true that the upper bound of a polynomial can be equal to the greatest real zero of the polynomial.
The upper bound of a polynomial of odd degree will be infinite so you can say that the upper bound of a polynomial of odd degree will be greater than the greatest real zero, but the upper bound of some, but not all, polynomials of even degree can be equal to the greatest real zero.
