vieta

Viète, often written with an accent as Viète, is a surname most closely associated with François Viète (1540–1603), a French mathematician who helped establish algebra as a symbolic and systematic discipline. He introduced the widespread use of letters to denote unknown quantities and coefficients, a convention that underpins modern algebraic notation. Viète also contributed to the solving of polynomial equations and influenced later mathematicians such as Descartes and Euler.

One of Viète’s lasting contributions is the set of relationships between the roots of a polynomial and its coefficients, now known as Viète’s formulas. For a monic polynomial P(x) = x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 with roots r_1, ..., r_n, the elementary symmetric sums of the roots relate to the coefficients: the sum of the roots is -a_{n-1}, the sum of all pairwise products is a_{n-2}, and in general the sum of products of k distinct roots equals (-1)^k a_{n-k}. In particular, for a quadratic x^2 + b x + c = 0, the roots satisfy r_1 + r_2 = -b and r_1 r_2 = c.

Legacy and influence: Viète’s algebraic notation and his root–coefficient relations became foundational tools in algebra and number theory, continuing to appear in teaching and research as Viète’s formulas and the symbolic approach he championed.