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abscissas Algebra approximate bend point column common divisor complex number constant corresponding cube roots decimal places denote derivative determinant of order discriminant double root elementary symmetric functions equals equation in Ex equation of degree example EXERCISES expression factor factor x fi(x given graph greatest common divisor Hence imaginary roots inflexion integral root interchanges intersection left member method multiple roots Newton's Newton's identities Newton's Method nth root number of real number of variations pairs positive number positive roots proof prove quadratic equation quartic equation quotient rational root real coefficients real numbers reduced cubic equation replace root of f(x roots of unity rows ruler and compasses set of solutions single real root Sturm's functions Sturm's theorem symmetric polynomial tangent theorem tion vanish variables variations of signs vectors x-axis zero
Page 33 - Article ; for the product of the squares ot the differences of all the roots is made up of the product of the squares of the differences of the roots of...
Page 171 - These criteria were derived by a new transformation, namely the one which yields an equation whose roots are the squares of the differences of the roots of the given...
Page 140 - ... column is the sum of the products of the elements of the Tth row of the first determinant by the corresponding elements of the cth column of the second determinant.
Page 25 - Hence the modulus of the product of two complex numbers equals the product of their moduli, and the amplitude of the product equals the sum of their amplitudes.
Page 91 - Hence the coordinates of the various points located by the construction, and therefore also the length ± xi of the segment joining two of them, are found by a finite number of rational operations and extractions of real square roots, performed upon rational numbers and numbers obtained by earlier ones of these operations.
Page 27 - THEOREM s. 377. If the number of sides of a regular inscribed polygon is indefinitely increased, the apothem of the polygon approaches the radius of the circle as its limit. Given a regular polygon of n sides inscribed in the circle of radius OA, s being one side and a the apothem. To prove that a approaches r as a limit, if n is increased indefinitely. Proof. We know that a < r.
Page 102 - И 0. For, if it were zero, x — p would by (1) be a common factor of / and /i. We may now proceed as in the second case in § 4. The third case requires a modified proof only when r is a multiple root. Let r be a root of multiplicity т, т Ш 2.
Page 90 - Another famous problem of antiquity was the construction of a cube whose volume shall be double that of a given cube.