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Learn how to count the roots of a polynomial by examining sign changes in its coefficients. See the proof, examples, generalizations and applications of this mathematical theorem by René Descartes.
A polynomial is a mathematical expression involving addition, subtraction, multiplication and exponentiation of variables and coefficients. Learn how to write, classify and manipulate polynomials, and see examples and applications in various fields of mathematics and science.
Learn about the equalities that involve trigonometric functions and are true for every value of the variables. Find the Pythagorean, reciprocal, reflection, shift, periodicity, angle sum and difference, and Ptolemy identities with examples and proofs.
Learn how to solve quadratic equations using the quadratic formula, a closed-form expression that describes the roots of a quadratic function. Explore different methods of deriving the formula, such as completing the square, substitution, and Lagrange resolvents.
The binomial theorem describes the expansion of powers of a binomial, such as (x + y)n, into a sum of terms involving binomial coefficients. Learn the history, statement, examples, and applications of this algebraic formula and its geometric interpretation.
An inequation is a statement that an inequality holds between two values, usually written with a relational sign. Learn how to solve inequations, use chains and computer support, and distinguish from equations and inequalities.
The Collatz conjecture is an unsolved problem in mathematics that asks whether repeating 3x + 1 will eventually transform every positive integer into 1. Learn about the history, the statement, the hardness, and the popular culture of this conjecture.
An algebraic equation is a polynomial equation with coefficients in some field, often the rationals. Learn about the history, theory, and applications of algebraic equations, and how they relate to Galois theory, field theory, and transcendental numbers.