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a+bi abscissa algebra angle bend point circle common divisor complex numbers constant term cube roots cubic equation decimal places denote derivative Descartes determinant of order discriminant double root elements equal example EXERCISES factor given graph of y=f(x Hence identical imaginary roots inflexion tangent integral coefficients integral root interchanges left member Lemma linear equations modulus multiple root Newton's method number of real number of variations obtain odd number opposite signs polynomial positive number positive root primitive nth root proof Prove quadratic equation quartic equation quotient r-rowed minor rational function rational root real coefficients real numbers regular polygon replace resulting root of multiplicity roots of unity rows rule of signs ruler and compasses single real root Solution Solve Sturm's functions Sturm's theorem symmetric functions synthetic division Taylor's theorem Theory of Equations unknowns upper limit variations of sign Waring's formula x-axis zero
Page 153 - The limit of the product of two functions is equal to the product of their limits, ie, lim [V (x) (ф x)] = [lim V (x)] [lim ф (х)] x-»a x-»a х-
Page 31 - Suppose, first, that the construction is possible. The straight lines and circles drawn in making the construction are located by means of points either initially given or obtained as the intersections of two straight lines, a straight line and a circle, or two circles.
Page 47 - Article ; for the product of the squares 01 the differences of all the roots is made up of the product of the squares of the differences of the roots of...
Page 4 - The modulus of the product of two complex numbers is equal to the product of their moduli.
Page 22 - 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 8 - THEOREM 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 72 - Descartes' rule states that the number of positive real roots of an equation with real coefficients is either equal to the number of its variations of sign or is less than that number by a positive even integer.
Page 83 - Vb is either the number of real roots of f(x) = 0 between a and b or exceeds the number of those roots by an even integer. A root of multiplicity m is here counted as m roots.