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" VARIATIONS of signs, nor the number of negative roots greater than the number of PERMANENCES. Consequence. 328. When the roots of an equation are all real, the number of positive roots is equal to the number of variations, and the number of negative roots... "
Elements of Algebra: Including Sturm's Theorem - Page 348
1847 - 368 pages
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Elements of Algebra: Tr. from the French of M. Bourdon, for the ..., Volume 1

Bourdon (M., Louis Pierre Marie) - Algebra - 1831 - 446 pages
...of sign, nor the number of negative roots greater than the number of PERMANENCES. 331 . Consequence. When the roots of an equation are all real, the number...number of variations, and the number of negative roots is equal to the number of permanences. For, let in denote the degree of the equation, n the number...
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Elements of Algebra: Tr. from the French of M. Bourdon. Revised and Adapted ...

Charles Davies - Algebra - 1835 - 378 pages
...of sign, nor the number of negative roots greater than the number of PERMANENCES. 303. Consequence. When the roots of an equation are all real, the number of positive roots is equal to Hie number of variations, and the number of negative roots is equal to the number of permanences. For,...
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On the theory and solution of algebraical equations

John Radford Young - 1835 - 302 pages
...necessarily, p=p' and r = t^ ; consequently, when the roots are all real, the number of positive roots will be equal to the number of variations, and the number of negative roots equal to the number of permanencies.' CHAPTER. II. ON THE TRANSFORMATION OF EQUATIONS. (19.) Algebraical...
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Elements of Algebra

Algebra - 1838 - 372 pages
...of sign, nor the number of negative roots greater than the number of PERMANENCES. 325. Consequence. When the roots of an equation are all real, the number...number of variations, and the number of negative roots is equal to the number of permanences. For, let m denote the degree of the equation, n the number of...
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The elements of algebra

Andrew Bell (writer on mathematics.) - 1839 - 500 pages
...lie discovered the important theorem, called " the rule of signs," that in an equation whose roots are all real, the number of positive roots is equal to the number of variations of the signs of its terms tяken in succession, and the number of the negative roots to that of the...
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Elements of Algebra

Charles Davies - Algebra - 1842 - 368 pages
...of sign, nor the number of negative roots greater than the number of PERMANENCES. 325. Consequence. When the roots of an equation are all real, the number...number of variations, and the number of negative roots is equal to the number of permanences. For, let m denote the degree of the equation, n the number of...
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The Analysis and Solution of Cubic and Biquadratic Equations: Forming a ...

John Radford Young - Equations - 1842 - 276 pages
...and » = »' Consequently, when the roots are all real, the number of positive roots will be exactly equal to the number of variations, and the number of negative roots to the number of permanencies. It must be borne in mind, however, that whether the roots are all real or not, the equation...
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A Treatise on Algebra: For the Use of Schools and Colleges

Stephen Chase - Algebra - 1849 - 348 pages
...362). Therefore, Cor. i. If the roots of an equation be all real, the number of positive roots must be equal to the number of variations ; and the number of negative roots, to the number of permanences. See § 218. 1, 2, 3. § 362. J.) If any term of the equation be wanting, a cypher may be put in its...
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Ray's Algebra Part Second: An Analytical Treatise, Designed for High Schools ...

Joseph Ray - Algebra - 1852 - 408 pages
...ciphers). Therefore, if the roots of an equation be alt real, the number of positive roots must be equal to the number of variations, and the number of negative roots to the number of permanences. (See examples, pages 343, 345.) For example, the equation 0:2+16=0, may be written a5±0a;+10=0. Now,...
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Treatise on Algebra, for the Use of Schools and Colleges

William Smyth - Algebra - 1855 - 370 pages
...degree of the equation. If all the roots, therefore, are real, the number of positive roots will be equal to the number of variations, and the number of negative roots will be equal to the number of permanences. 234. In what precedes, the equation is supposed to be complete....
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