 | Joseph Ray - Algebra - 1852 - 408 pages
...changed equation, and the variations in the former by permanences in the latter ; and since the changed equation cannot have a greater number of positive roots than there are variations of sign, the proposed equation cannot have a greater number of negative roots than there are permanences... | |
 | Benjamin Peirce - Algebra - 1855 - 296 pages
...multiplied by every different power of x fromthe highest to unity, and also a constant term, such as I. 296. Descartes' Theorem. A complete equation cannot...If the equation is that of art. 295, the values of *, U, U', &c. in art. 294, are u = xn-\-axn-1 -j-&.C. -f A z2 -f & z -j- Z U == n zn-i -f (n — 1)... | |
 | Benjamin Peirce - Algebra - 1855 - 308 pages
...multiplied by every different power of x from the highest to unity, and also a constant term, such as I. 296. Descartes' Theorem. A complete equation cannot...If the equation is that of art. 295, the values of «, U, U', &c. in art. 294, are u = Zn -fa Zn-1 -f &c. -f A z2 + kx -f I U = raa;n &c. The row of signs... | |
 | Joseph Ray - Algebra - 1857 - 408 pages
...changed equation, and the variations in the former by permanences in the latter ; and since the changed equation cannot have a greater number of positive roots than there are variations of sign, the proposed equation cannot have a greater number of negative roots than there are permanences... | |
 | Charles Davies - Algebra - 1860 - 412 pages
...converse of this proposition is evidently true. Descartes' Rule. 293. An equation of any degree whatever, cannot have a greater number of positive roots than there are variations in the signs of its terms, nor a greater number of negative roots than there are permanences of these signs.... | |
 | Charles Hutton - Mathematics - 1860 - 1020 pages
...negative, has at least two real roots; the one positive, and the other negative. PROPOSITION VII. An equation cannot have a greater number of positive roots than there are variations of signs in the successive terms from + to — , or from — to +, nor can it have a greater number... | |
 | Benjamin Peirce - Algebra - 1864 - 314 pages
...as the number of roots comprised between p and q. Descartes' Theorem. 295. Definition. An equation is said to be complete in its form, when it contains...If the equation is that of art. 295, the values of M, U, U', &c. in art. 294, are M = xn + ax"'1 +&c. + Az2+ kx-\-l U = nx"-i (n — laz" &c. The row... | |
 | Horatio Nelson Robinson - Algebra - 1864 - 444 pages
...root, a •^Vi — 6, it has also the root, a — \ — b. RULE OF DES CARTES. 440. An equation can not have a greater number of positive roots than there are variations in the signs of its terms, nor a greater »umber of negative roots than there are permanences of signs. NOTK.—... | |
 | Benjamin Greenleaf - Algebra - 1864 - 420 pages
...-f- 8 = 0 is 2 — 2 \/2 ; what are the others ? Ans. 2 + 2\/2, —1, and 2. DESCARTES' RULE. 452. A complete equation cannot have a greater number of positive roots than it has variations of sign ; nor a greater number of negative roots than it has permanences of sign.... | |
 | Encyclopedias and dictionaries - 1865 - 854 pages
...of such factors, the coeflicients of the resulting polynomial are formed, we arrive at this : that a complete equation cannot have a greater number of positive roots than these chanyea of sign from + to - and from — to + in the scries of terms forming its first member... | |
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Rule. 324. An equation of any degree whatever cannot have a greater number of positive roots than there are variations in the signs of Us terms, nor a greater number of negative roots than there are permanences of these signs. 
