First Course in the Theory of EquationsJohn Wiley & Sons, Incorporated, 1922 - 168 Seiten |
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Häufige Begriffe und Wortgruppen
a+bi a₁ abscissa algebraic angle b₁ b₂ bend point C₁ circle column complex numbers compute 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 imaginary roots integral root interchanges intersection k₁ Lemma modulus multiple root Newton's method nth roots number of real number of variations obtain polynomial positive number positive root primitive nth root Prove quadratic equation quartic equation quotient r-rowed minor rational function rational root real coefficients real numbers regular polygon replace resulting root of f(x)=0 root of multiplicity roots of unity rows ruler and compasses single real root Solution Solve Sturm's theorem synthetic division Theory of Equations unknowns upper limit variations of sign WARING'S FORMULA x-axis y₁ zero
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Seite 103 - ... diagonal running from the upper left-hand corner to the lower right-hand corner of the symbol (11) for the determinant.
Seite 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-
Seite 31 - ... possible by ruler and compasses if, and only if, the numbers which define analytically the desired geometric elements can be derived from those defining the given elements by a finite number of rational operations and extractions of real square roots. 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...
Seite 47 - 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...
Seite 4 - The modulus of the product of two complex numbers is equal to the product of their moduli.
Seite 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"
Seite 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. A root of multiplicity m is here counted as m roots.
Seite 8 - 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.
Seite 83 - To find an upper limit to the number of real roots of f(x) = 0 between a and b, set a + by 1+j/ multiply by (1 + ?/)", and apply Descartes' Rule to the resulting equation in y.
Seite 67 - If the coefficients of the polynomial f(x) are real and if a and b are real numbers such that /(a) and f(b) have opposite signs, the equation f(x) = 0 has at least one real root between a and b; in fact, an odd number of such roots, if an m-fold root is counted as m roots. The only argument* given here is one based upon geometrical intuition. We are stating that, if the points...