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

CHAPTER VIII ELIMINATION 72. Resultants or Eliminants. Let us determine the condition that the two equations f(x) aox2+ a x + a2 = 0, F(x) = CoX2 + C1 x + C2 = 0, shall have a root in common. Designate the roots of the second equation by /,, /2. The necessary and sufficient condition that /p or /2 shall satisfy the equation f(x)= 0 is that f(B1) or f(/2) shall vanish; in other words, that the product f(/') f(/ () shall be zero. Multiplying together f(l,) 0 0312 + C 1p, + a2, f(92) - o022 + (Cl 18 + a2 we get ao21,p22 + a,0a(/3,/22 + 12/3,) + ao02 (/132 + /22) + a2fil,832 + t,,2(/1 + 92) + aC2. Multiplying by c02 and substituting for the symmetric functions of /, and f, their values in terms of the coefficients of F(x) = 0, we have ao2C2 - aaC, a - 2 aoaaCo,c + a,2coc2 - alcCoC + a22cO'. This expression is called the eliminant or resultant. Its vanishing is the condition that the given equations shall have a root in common. If from n equations involving n - 1 variables we eliminate the variables and obtain an equation R = 0 involving only the 92

/ 251
Pages

Actions

file_download Download Options Download this page PDF - Pages 90-109 Image - Page 90 Plain Text - Page 90

About this Item

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.

Technical Details

Link to this Item
https://name.umdl.umich.edu/abv2146.0001.001
Link to this scan
https://quod.lib.umich.edu/u/umhistmath/abv2146.0001.001/103

Rights and Permissions

The University of Michigan Library provides access to these materials for educational and research purposes. These materials are in the public domain in the United States. If you have questions about the collection, please contact Historical Mathematics Digital Collection Help at [email protected]. If you have concerns about the inclusion of an item in this collection, please contact Library Information Technology at [email protected].

DPLA Rights Statement: No Copyright - United States

Manifest
https://quod.lib.umich.edu/cgi/t/text/api/manifest/umhistmath:abv2146.0001.001

Cite this Item

Full citation
"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 29, 2025.
Do you have questions about this content? Need to report a problem? Please contact us.