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  2. Vieta's formulas - Wikipedia

    en.wikipedia.org/wiki/Vieta's_formulas

    Vieta's formulas can equivalently be written as. for k = 1, 2, ..., n (the indices ik are sorted in increasing order to ensure each product of k roots is used exactly once). The left-hand sides of Vieta's formulas are the elementary symmetric polynomials of the roots. Vieta's system (*) can be solved by Newton's method through an explicit ...

  3. De Moivre's formula - Wikipedia

    en.wikipedia.org/wiki/De_Moivre's_formula

    Roots of complex numbers. A modest extension of the version of de Moivre's formula given in this article can be used to find the n-th roots of a complex number for a non-zero integer n. (This is equivalent to raising to a power of 1 / n). If z is a complex number, written in polar form as

  4. Halley's method - Wikipedia

    en.wikipedia.org/wiki/Halley's_method

    Halley's method. In numerical analysis, Halley's method is a root-finding algorithm used for functions of one real variable with a continuous second derivative. Edmond Halley was an English mathematician and astronomer who introduced the method now called by his name. The algorithm is second in the class of Householder's methods, after Newton's ...

  5. Descartes' rule of signs - Wikipedia

    en.wikipedia.org/wiki/Descartes'_rule_of_signs

    In particular, if the number of sign changes is zero or one, the number of positive roots equals the number of sign changes. Negative roots. As a corollary of the rule, the number of negative roots is the number of sign changes after multiplying the coefficients of odd-power terms by −1, or fewer than it by an even number. This procedure is ...

  6. Square root - Wikipedia

    en.wikipedia.org/wiki/Square_root

    Square root. Notation for the (principal) square root of x. For example, √ 25 = 5, since 25 = 5 ⋅ 5, or 52 (5 squared). In mathematics, a square root of a number x is a number y such that ; in other words, a number y whose square (the result of multiplying the number by itself, or ) is x. [1] For example, 4 and −4 are square roots of 16 ...

  7. Doubling the cube - Wikipedia

    en.wikipedia.org/wiki/Doubling_the_cube

    In algebraic terms, doubling a unit cube requires the construction of a line segment of length x, where x 3 = 2; in other words, x = , the cube root of two. This is because a cube of side length 1 has a volume of 1 3 = 1, and a cube of twice that volume (a volume of 2) has a side length of the cube root of 2. The impossibility of doubling the ...

  8. Cubic field - Wikipedia

    en.wikipedia.org/wiki/Cubic_field

    Definition. If K is a field extension of the rational numbers Q of degree [ K: Q ] = 3, then K is called a cubic field. Any such field is isomorphic to a field of the form. where f is an irreducible cubic polynomial with coefficients in Q. If f has three real roots, then K is called a totally real cubic field and it is an example of a totally ...

  9. Mahāvīra (mathematician) - Wikipedia

    en.wikipedia.org/wiki/Mahāvīra_(mathematician)

    He also found out the formula for n C r as [n (n − 1) (n − 2) ... (n − r + 1)] / [r (r − 1) (r − 2) ... 2 * 1]. He devised a formula which approximated the area and perimeters of ellipses and found methods to calculate the square of a number and cube roots of a number. He asserted that the square root of a negative number does not exist.

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