An introduction to the modern theory of equations, by Florian Cajori.

CHAPTER III LOCATION OF THE ROOTS OF AN EQUATION 37. In this chapter we shall deduce theorems giving limits between which all the real roots of an equation with real coefficients lie. We shall also derive theorems which enable us to separate from each other all the distinct real roots, and to ascertain the exact number and location of the real roots. 38. An Upper Limit. If in the equation f(x) = 0 the coefficient of xa is unity, then the numerically greatest negative coefficient, increased by one, is an upper limit of the positive roots of the equation. Any positive value of x makes f(x) > 0, if it makes xn _ p(x'-1 + X-2 +...2 + 1) > 0, n,n _ 1 x -1 or, xZ-p _. >0, where p is the numerical value of the greatest negative coefficient. All the more is f(x) > 0, if a positive value of x makes n _ 1 (xn 1) -)- >O x-1 or, ) ( z) (ox — 1)1- x_ > 0. But this last expression is always > 0, or positive, if p < x- 1; that is, if x >p 1. Since any real value of x, greater than p+l, makes f(x)>0, every real value of x which makes f(x) equal to zero must be equal to or less than p + 1. Hence p + 1 is an upper limit of the real positive roots of f(x) = 0. 43

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Title
An introduction to the modern theory of equations, by Florian Cajori.
Author
Cajori, Florian, 1859-1930.
Canvas
Page 30
Publication
New York,: The Macmillan company,
1904.
Subject terms
Equations, Theory of
Group theory.

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"An introduction to the modern theory of equations, by Florian Cajori." In the digital collection University of Michigan Historical Math Collection. https://name.umdl.umich.edu/abv2146.0001.001. University of Michigan Library Digital Collections. Accessed May 28, 2025.
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