In mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. They are named after François Viète (1540-1603), more commonly referred to by the Latinised form of his name, "Franciscus Vieta."
Basic formulas Any general polynomial of degree n
P ( x ) = a 0 x n + a 1 x n − 1 + ⋯ + a n − 1 x + a n {\displaystyle P(x)=a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n-1}x+a_{n}}
(with the coefficients being real or complex numbers and a0 ≠ 0) has n (not necessarily distinct) complex roots r1, r2, ..., rn by the fundamental theorem of algebra. Vieta's formulas relate the polynomial coefficients to signed sums of products of the roots r1, r2, ..., rn as follows:
where E k ( n ) ( r 1 , … , r n ) {\displaystyle E_{k}^{(n)}(r_{1},\ldots ,r_{n})} is the k {\displaystyle k} th elementary symmetric polynomial in n {\displaystyle n} indeterminates; that is,
E k ( n ) ( r 1 , … , r n ) = ∑ 1 ≤ i 1 < i 2 < ⋯ < i k ≤ n ( ∏ j = 1 k r i j ) {\displaystyle E_{k}^{(n)}(r_{1},\ldots ,r_{n})=\sum _{1\leq i_{1}<i_{2}<\cdots <i_{k}\leq n}\left(\prod _{j=1}^{k}r_{i_{j}}\right)}
is the sum of all products of k {\displaystyle k} r i {\displaystyle r_{i}} with different indices. In particular,
E 1 ( n ) ( r 1 , … , r n ) = r 1 + r 2 ⋯ + r n , E n ( n ) ( r 1 , … , r n ) = r 1 r 2 ⋯ r n , {\displaystyle {\begin{aligned}E_{1}^{(n)}(r_{1},\ldots ,r_{n})&=r_{1}+r_{2}\cdots +r_{n},\\E_{n}^{(n)}(r_{1},\ldots ,r_{n})&=r_{1}r_{2}\cdots r_{n},\end{aligned}}}
and the above expression of degree 2 in the r i {\displaystyle r_{i}} equals E 2 ( n ) ( r 1 , … , r n ) {\displaystyle E_{2}^{(n)}(r_{1},\ldots ,r_{n})} .
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