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Quotient polynomials with positive coefficients
Published online by Cambridge University Press: 23 January 2015
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In a recent paper [1] Michael Hirschhorn and the present author had to prove that the polynomial
is positive for all integers n ≥ 4. We did so by writing down the identity
Since all the coefficients of the quotient polynomial are positive as well as the remainder, this shows by inspection that f(n) > 0 for n ≥ 5 and computing f(4) = 12025 completes the proof. This example illustrates a useful and efficient method for proving that polynomials have strictly positive values for all real numbers exceeding a given one. We leamed this method from a paper by Chen [2], (see [3] and the references cited there), and we have subsequently used it in our own researches [1,4].
We also conjectured in [1] that the theoretical basis of the method is a true theorem and the present paper is dedicated to proving this conjecture.
Theorem 1: Let f(x) be a polynomial with real coefficients whose leading coefficient is positive and with at least one positive root x = a. Then there exists an x = b ≥ a such that
where f(b), and all the coefficients of the quotient polynomial g(x), are non-negative.
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References
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