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

ELIMINATION 93 coefficients of the equations, the expression R is called the eliminant or resultant of the given equations. In the above example the elimination was performed with the aid of symmetric functions. This method generalized is as follows: 73. Elimination by Symmetric Functions. To find the conditions that the two equations f(x) _ax + alx-' + a2'-2 + *.. + can = 0, F(x) E-XM Co Cm + cl C2- + C- + c = 0, shall have a common root. For this purpose it is necessary and sufficient that some one of the roots f/31, 'n, /.m of F(x) = 0 shall satisfy f(x) = 0, in which case the product f(/1). f(#2) e ' f(PM) must vanish. We have f(f31) - ca0ft + allib-1 +... + an, f (/2) - aof"2 + alf2t-1 + ~... + a, f (Pm) E=- alom" + CT/- + + aI. Multiplying these together, we obtain, after substituting for the symmetric functions of f1, /2, * -, f3, which occur in the product their values in terms of co,c, *, c,, and after clearing of fractions, = c. (). ) -R = Co',.;(31) -.f()...f(,.). Here R is the eliminant and is a rational integral function of the coefficients of f(x) and F(x). Its vanishing is the condition that the two givein equations hace a root in, common. The degree of the resultant in the coefficients of the given equations is in general m + n. It is easy to see that we obtain the same eliminant by substituting the roots a1,a2, *, an of f(x) = O, in succession, for x in the polynomial F(x).

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Title
An introduction to the modern theory of equations, by Florian Cajori.
Author
Cajori, Florian, 1859-1930.
Canvas
Page 90
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 31, 2025.
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