The Analysis and Solution of Cubic and Biquadratic Equations: Forming a Sequal to the Elements of Algebra, and an Introduction to the Theory and Solution of Equations of the Higher Orders

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Souter and Law, 1842 - 247 pages

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Page 202 - Divide the given number into periods of three figures eacn, as in the common method, and find the nearest cube to the first period, subtract it therefrom, and put the root in the quotient ; then thrice the square of this root will be the trial divisor for finding the next figure.
Page 53 - If in any equation each negative coefficient be taken positively, and divided by the sum of all the positive coefficients which precede it, the greatest quotient thus formed increased by unity is a superior limit of the positive roots. Let the equation be Ootf...
Page 13 - ... and — . The preceding equation is only of the fourth power or degree ; but it is manifest that the above remark applies to equations of higher or lower dimensions : viz, that in general . an equation of any degree whatever has as many roots as there are units in the exponent of the highest power of the unknown quantity, and that each root has the property of rendering, by its substitution in place of the unknown quantity, the aggregate' of all the terms of the equation equal to nothing.
Page 33 - 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 to , the number of permanences.
Page 25 - Since the coefficient of the second term is the sum of all the roots with their signs changed...
Page 55 - If each negative coefficient be taken positively and divided by the sum of all the positive coefficients which precede it, the greatest of all the fractions thus formed increased by unity, is a superior limit of lhe positive roots. Let the equation be f(x) = 0, where f(x) denotes p,l'•K" + plx"~l+p,xn~13-pllx"~a + p,x"~í + ... -px"
Page 11 - In general, any equation has as many roots as there are units in the exponent of the unknown quantity. EXAMPLE. — Solve the equation ~ — * ~ = ^. SOLUTION. — Clearing of fractions by multiplying each term by 80, 5x> - 16(.r
Page 28 - An equation cannot have a greater number of positive roots than there are variations of...

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