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Learn how to count the roots of a polynomial by examining sign changes in its coefficients, as described by René Descartes in his La Géométrie. See examples, proofs, generalizations and related topics.
Learn about the history, principles and methods of finding roots of polynomials, from linear to high degrees. Compare different algorithms, such as Newton's method, Francis QR algorithm, Aberth method and others.
An nth root of a number x is a number r that, when raised to the power of n, yields x. The number x is called the radicand and the index or degree of the root is n. Learn more about the history, notation, properties and operations of nth roots.
If a < b are two real numbers, then W(a) – W(b) is the number of roots of P in the interval (,] such that Q(a) > 0 minus the number of roots in the same interval such that Q(a) < 0. Combined with the total number of roots of P in the same interval given by Sturm's theorem, this gives the number of roots of P such that Q ( a ) > 0 and the ...
Learn about the origins and solutions of cubic equations, which are equations of the form ax3 + bx2 + cx + d = 0. Find out how ancient and medieval mathematicians from different cultures approached and solved cubic equations using algebra, geometry, and numerical approximations.
Newton's method, also known as the Newton–Raphson method, is a numerical technique to approximate the roots of a function. It uses the derivative of the function to construct a tangent line and find the x-intercept as a better approximation of the root.
Learn about different algorithms for approximating the non-negative square root of a positive real number, such as Heron's method, Newton's method, and continued fractions. Compare the accuracy, complexity, and history of various methods, and how to choose a suitable initial estimate.
A complex number is an expression of the form a + bi, where a and b are real numbers, and i is an imaginary unit. The absolute value (or modulus) of a complex number z is the non-negative square root of the sum of its squares, and the argument of z is the angle from the positive real axis to z.
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