Vieta's formulas

Vieta's formulas express the coefficients of a polynomial in terms of sums and products of its roots. They apply whether the roots are real or complex, counted with multiplicity.

Elementary symmetric polynomial: A polynomial formed by summing products of distinct variables, such as the sum of all pairwise products. Each coefficient of a monic polynomial is, up to sign, one of these expressions in its roots.
Quadratic equation: An equation in which the highest power of the unknown is two. Its roots sum to the negative linear coefficient divided by the leading coefficient, and multiply to the constant term divided by it.
François Viète: A French mathematician whose work helped develop symbolic algebra in the late sixteenth century. His writings established the use of systematic coefficient-root relations in algebra.
Quadratic formula: A formula giving the solutions of a quadratic equation from its coefficients. It computes the roots explicitly, while Vieta's formulas relate their combinations to coefficients.
Fundamental theorem of algebra: Every nonconstant polynomial with complex coefficients has a complex root and factors into linear factors over the complex numbers. Factoring into linear terms provides the roots from which the formulas recover coefficients.
Rational root theorem: A theorem restricting rational roots of an integer-coefficient polynomial to fractions built from its constant and leading coefficients. Vieta's relations can narrow candidate rational roots by their required sums and products.
Algebra: A branch of mathematics that studies symbols, equations, and the structures governing their operations. Vieta's formulas belong to algebra's study of equations and their solutions.
Cardano's formula: A formula expressing solutions of a general cubic equation in terms of its coefficients and radicals. It seeks explicit cubic roots rather than only their symmetric sums and products.
Polynomial: An expression built from variables and coefficients using nonnegative integer powers and addition or multiplication. Vieta's formulas connect a polynomial's coefficients to the roots of its linear factors.
Root bounds: Estimates that constrain where the roots of an equation or polynomial can lie. Known coefficient-root sums and products can constrain possible root values.

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