Vieta’s Formulas

Summary: The dictionary between the coefficients of a polynomial and the elementary symmetric functions of its roots. For a monic degree-$n$ polynomial ${\prod }_{k=1}^n(Z-{\alpha }_k)=Z^n-P{Z}^{n-1}+Q{Z}^{n-2}-R{Z}^{n-3}+\cdots$, the coefficient $P$ is the sum of the roots, $Q$ the sum of pairwise products, $R$ the sum of triple products, and so on, with the final coefficient being $\pm \prod {\alpha }_k$. Euler uses this identification — without naming Vieta — as a workhorse throughout the Introductio: it converts every product expansion into a stack of identities, one for each symmetric polynomial.

Sources: chapter1, chapter9, chapter10, chapter14

Last updated: 2026-05-11

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Statement

If ${\alpha }_1,\dots ,{\alpha }_n$ are the (possibly complex, possibly repeated) roots of

$Z^n-P{Z}^{n-1}+Q{Z}^{n-2}-R{Z}^{n-3}+S{Z}^{n-4}-\cdots \pm V=0,$

then

$P={\sum }_k{\alpha }_k,Q={\sum }_{i<j}{\alpha }_i{\alpha }_j,R={\sum }_{i<j<k}{\alpha }_i{\alpha }_j{\alpha }_k,\dots ,V={\alpha }_1{\alpha }_2\cdots {\alpha }_n.$

The $k$-th coefficient (with the sign convention shown) is the $k$-th elementary symmetric polynomial $e_k({\alpha }_1,\dots ,{\alpha }_n)$. The identification follows immediately from expanding the factored form $\prod (Z-{\alpha }_k)$.

Where Euler uses it

Chapter 1 — to define multi-valued functions

In §§11–13 Euler introduces the $n$-valued function $Z$ as the solution of

$Z^n-P{Z}^{n-1}+Q{Z}^{n-2}-\cdots =0$

where $P,Q,\dots$ are single-valued functions of $z$, and states Vieta’s relations as the definition of what $P,Q,\dots$ mean: sum of values, sum of pairwise products, etc. This is the first appearance of the formulas in the book and the cleanest statement of them.

Chapter 9 — to read off product expansions

In Chapter 9 Euler establishes infinite-product formulas like

$\sin z=z{\prod }_{k=1}^{\infty }\left(1-\frac{z^2}{k^2{\pi }^2}\right).$

Comparing the coefficients of the right side (Vieta) with those of the Taylor series of $\sin z$ on the left turns the product into a generator of identities. This is the engine behind the Basel problem and the entire even-zeta table.

Chapter 10 — combined with Newton

Newton’s identities convert the elementary symmetric functions $P,Q,R,\dots$ into the power sums $\sum {\alpha }_k$, $\sum {\alpha }_k^2$, $\sum {\alpha }_k^3$, $\dots$. The composite “Vieta then Newton” recipe is what Euler runs on the sinh product in Chapter 10 to read off $\sum 1/k^{2m}$ for every $m$.

Chapter 14 — to extract trigonometric sums and products

In Chapter 14 Euler factors the multiple-angle polynomial $\sin nz/\sin z$ as a product over its $n-1$ roots, which are $\sin$ values at equally-spaced angles. Vieta’s formulas applied to this factorization give all of the factored products, partial-fraction sums for csc, sec, cot, tan, and the product-of-tangents identity that drops out of De Moivre in §249. The trig-values-as-roots page is the umbrella treatment.

Naming

François Viète (Vieta) stated the formulas in 1579–1591 for specific low degrees; the general statement was systematized by Albert Girard in 1629 and is sometimes called Newton–Girard in the symmetric-function literature, but the modern English-language name is Vieta’s formulas. Euler refers to the relations without attribution; he treats them as a basic property of polynomials.

A trivial example

For $Z^3-6Z^2+11Z-6=0$ with roots $1,2,3$:

• $P=1+2+3=6$,
• $Q=1\cdot 2+1\cdot 3+2\cdot 3=11$,
• $R=1\cdot 2\cdot 3=6$.

Related pages

• single-valued-and-multi-valued-functions
• newtons-identities
• factoring-polynomials
• trig-values-as-roots
• multiple-angle-polynomials
• sine-cosine-factored-products
• trig-multiple-angle-partial-fractions
• de-moivre-formula
• basel-problem
• chapter-1-on-functions-in-general
• chapter-9-on-trinomial-factors
• chapter-10-on-the-use-of-the-discovered-factors-to-sum-infinite-series
• chapter-14-on-the-multiplication-and-division-of-angles