# Vieta's formulas
**Vieta's formulas** are relations between the coefficients of a polynomial and sums and products of its roots: for a degree-n polynomial, each elementary symmetric polynomial in the roots equals a signed ratio of two coefficients.[1](https://mathworld.wolfram.com/VietasFormulas.html) For the quadratic \( ax^2 + bx + c = 0 \) with roots \( r_1, r_2 \), they read \( r_1 + r_2 = -b/a \) and \( r_1 r_2 = c/a \).[2](https://www.parabola.unsw.edu.au/sites/default/files/2024-05/vol60_no1_8.pdf) The formulas are named after François Viète, who proved a restricted version in the sixteenth century; Albert Girard established the general form.[3](https://encyclopediaofmath.org/index.php?title=Vi%C3%A8te_theorem)
| Key fact | Detail |
|---|---|
| Quadratic | For \( ax^2 + bx + c = 0 \): \( r_1 + r_2 = -b/a \), \( r_1 r_2 = c/a \)[2](https://www.parabola.unsw.edu.au/sites/default/files/2024-05/vol60_no1_8.pdf) |
| Cubic | For \( ax^3 + bx^2 + cx + d = 0 \): \( x_1+x_2+x_3 = -b/a \), \( x_1x_2+x_1x_3+x_2x_3 = c/a \), \( x_1x_2x_3 = -d/a \)[4](https://www.mathwords.com/v/vietas_formulas.htm) |
| General degree n | \( e_k(r_1,\ldots,r_n) = (-1)^k a_{n-k}/a_n \) for \( 0 \le k \le n \), roots counted with multiplicity[5](https://mathweb.ucsd.edu/~trshin/handouts/vieta_in_action.pdf) |
| Monic form | \( c_k = (-1)^{n-k} e_{n-k}(x_1,\ldots,x_n) \), formally verified in the Isabelle Archive of Formal Proofs[6](https://isa-afp.org/browser_info/current/AFP/Symmetric_Polynomials/outline.pdf) |
| Scope of validity | Holds for real, irrational, complex, and repeated roots, provided all complex roots are counted with multiplicity[7](http://www.amsi.org.au/ESA_Senior_Years/SeniorTopic2/2e/2e_3links_2.html) • [8](https://www.varsitytutors.com/practice/subjects/ib-mathematics-analysis-and-approaches/lessons/vietas-formulas) |
| History | Viète proved the relations for positive roots only; Girard established the general form[3](https://encyclopediaofmath.org/index.php?title=Vi%C3%A8te_theorem) |
| Notable application | Vieta jumping, the key technique for 1988 IMO Problem 6[2](https://www.parabola.unsw.edu.au/sites/default/files/2024-05/vol60_no1_8.pdf) |
## General statement for degree n
Let \( a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0 \) be a polynomial with roots \( r_1, \ldots, r_n \) counted with multiplicity. Then for \( 0 \le k \le n \),
\[ e_k(r_1, \ldots, r_n) = (-1)^k \frac{a_{n-k}}{a_n}, \]
where \( e_k \) is the k-th elementary symmetric polynomial: \( e_1 = r_1 + \cdots + r_n \), \( e_2 = \sum_{i[5](https://mathweb.ucsd.edu/~trshin/handouts/vieta_in_action.pdf) In particular \( e_1 = -a_{n-1}/a_n \), \( e_2 = a_{n-2}/a_n \), and \( e_n = (-1)^n a_0/a_n \).[9](https://chapelhillmathcircle.org/wp-content/uploads/2025/03/2025-03-15_-_vietas_formulas_and_symmetric_polynomials_students_copy.pdf) The \( e_k \) are symmetric polynomials, invariant under every permutation of the variables.[9](https://chapelhillmathcircle.org/wp-content/uploads/2025/03/2025-03-15_-_vietas_formulas_and_symmetric_polynomials_students_copy.pdf)
For a monic polynomial \( f(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_0 \) with roots \( \alpha_1, \ldots, \alpha_n \), the theorem gives \( a_{n-1} = -(\alpha_1 + \cdots + \alpha_n) \) and \( a_0 = (-1)^n \alpha_1 \cdots \alpha_n \), with intermediate coefficients the signed elementary symmetric sums.[3](https://encyclopediaofmath.org/index.php?title=Vi%C3%A8te_theorem) Equivalently, the k-th coefficient of a univariate monic polynomial is \( c_k = (-1)^{n-k} e_{n-k}(x_1, \ldots, x_n) \); this form has been machine-verified in the Isabelle proof assistant, paired with the Fundamental Theorem of Symmetric Polynomials.[6](https://isa-afp.org/browser_info/current/AFP/Symmetric_Polynomials/outline.pdf) The formalized statement covers a finite set \( A \) of roots: the coefficient of \( \prod_{x \in A} (X - f(x)) \) at index k equals \( (-1)^{\mathrm{card}\,A - k} \) times the sum of products over subsets of \( A \) of the matching size.[10](https://isa-afp.org/browser_info/current/AFP/Factor_Algebraic_Polynomial/Symmetric_Polynomials.Vieta.html)
When the leading coefficient \( a_n \) is not 1, each symmetric sum is divided by \( a_n \), which is the only change.[11](https://github.com/antoniolupetti/algebrica/blob/main/polynomials/vieta-formulas.md)
## Proof and why it works
The proof is a coefficient comparison. For the quadratic, factor and expand:
\[ ax^2 + bx + c = a(x - r)(x - s) = ax^2 + (-a r - a s)x + (a r s). \]
Equating coefficients gives \( r + s = -b/a \) and \( rs = c/a \).[2](https://www.parabola.unsw.edu.au/sites/default/files/2024-05/vol60_no1_8.pdf) MathWorld notes the same mechanism for a second-degree polynomial: the formulas can be seen by multiplying out the factored form.[1](https://mathworld.wolfram.com/VietasFormulas.html)
The argument extends to degree n because expanding \( a_n \prod_{i=1}^{n} (x - r_i) \) produces, at each power of x, exactly one elementary symmetric sum of the roots, with sign \( (-1)^k \). The fundamental theorem of algebra guarantees that a nonconstant complex polynomial of degree n has precisely n complex roots including multiplicity, so the factored form always exists over the complex numbers.[9](https://chapelhillmathcircle.org/wp-content/uploads/2025/03/2025-03-15_-_vietas_formulas_and_symmetric_polynomials_students_copy.pdf)
## By the numbers
**Worked quadratic.** For \( p(x) = ax^2 + bx + c \), the roots are \( r_i = (-b \pm \sqrt{b^2 - 4ac})/2 \). When \( b^2 - 4ac = 0 \) the two roots coincide; when it is positive the roots are real and distinct; when it is negative they are non-real.[9](https://chapelhillmathcircle.org/wp-content/uploads/2025/03/2025-03-15_-_vietas_formulas_and_symmetric_polynomials_students_copy.pdf) The discriminant can be written in root form as \( \Delta = a^2(\alpha - \beta)^2 = a^2\left[(\alpha + \beta)^2 - 4\alpha\beta\right] \), an expression that uses both Vieta relations, so the discriminant is itself a consequence of the formulas.[8](https://www.varsitytutors.com/practice/subjects/ib-mathematics-analysis-and-approaches/lessons/vietas-formulas) For a monic polynomial of degree \( n \ge 1 \), the discriminant generalizes as \( \mathrm{Disc}_p = \prod_{1 \le i < j \le n} (r_i - r_j)^2 \).[9](https://chapelhillmathcircle.org/wp-content/uploads/2025/03/2025-03-15_-_vietas_formulas_and_symmetric_polynomials_students_copy.pdf)
**Cubic and quartic.** For \( f(x) = a_3 x^3 + a_2 x^2 + a_1 x + a_0 \) with roots \( \alpha, \beta, \gamma \): \( \alpha + \beta + \gamma = -a_2/a_3 \), \( \alpha\beta + \beta\gamma + \gamma\alpha = a_1/a_3 \), and \( \alpha\beta\gamma = -a_0/a_3 \).[7](http://www.amsi.org.au/ESA_Senior_Years/SeniorTopic2/2e/2e_3links_2.html) The formulas apply not just to cubics but to polynomials of any degree, including quartics.[7](http://www.amsi.org.au/ESA_Senior_Years/SeniorTopic2/2e/2e_3links_2.html)
**Matrices.** The formulas connect the trace and determinant of a matrix to its eigenvalues, since the characteristic polynomial is monic with the eigenvalues as roots.[4](https://www.mathwords.com/v/vietas_formulas.htm) For \( z^n - 1 \), all non-leading coefficients vanish except the constant term, so the sum of the n-th roots of unity is 0 for \( n \ge 2 \) and their product equals \( (-1)^{n+1} \).[11](https://github.com/antoniolupetti/algebrica/blob/main/polynomials/vieta-formulas.md)
## Comparison with Newton's identities and related results
Vieta's formulas and [Newton's identities](https://www.edgepedia.org/newtons-identities) answer different questions about the same data. Vieta's formulas express the coefficients as elementary symmetric polynomials of the roots; Newton's identities relate the elementary symmetric polynomials \( e_k \) to the power sums \( p_k = \sum_i r_i^k \).[5](https://mathweb.ucsd.edu/~trshin/handouts/vieta_in_action.pdf) Together they form a reversible pipeline from coefficients to power sums: Vieta's formulas convert coefficients to \( e_k \), and Newton's identities convert \( e_k \) to \( p_k \) and back.[5](https://mathweb.ucsd.edu/~trshin/handouts/vieta_in_action.pdf) [Isaac Newton](https://www.edgepedia.org/isaac-newton) extended Vieta's formulas to his identities in 1666, expressing power sums recursively in terms of the elementary symmetric polynomials.[12](https://leangenius.org/proof/vietas-formulas)
The surrounding structure is the fundamental theorem of symmetric polynomials: every symmetric polynomial in n variables with integer coefficients can be written uniquely as a polynomial in \( e_1, \ldots, e_n \).[5](https://mathweb.ucsd.edu/~trshin/handouts/vieta_in_action.pdf) Because the coefficients are symmetric functions of the roots, any [Galois group](https://www.edgepedia.org/galois-group) must permute the roots while preserving the coefficients, which makes Vieta's formulas the conceptual foundation of [Galois theory](https://www.edgepedia.org/galois-theory) and, ultimately, of the [Abel–Ruffini theorem](https://www.edgepedia.org/abel-ruffini-theorem) that no general radical solution exists for degree \( \ge 5 \).[12](https://leangenius.org/proof/vietas-formulas)
## Applications
**Vieta jumping.** [Vieta jumping](https://www.edgepedia.org/vieta-jumping), derived from Vieta's formulas for quadratics, is an important Olympiad tool for difficult number-theoretic problems, and is the key technique for 1988 IMO Problem 6, which asks for a proof that \( (a^2 + b^2)/(ab + 1) \) is a square of an integer whenever a and b are positive integers.[2](https://www.parabola.unsw.edu.au/sites/default/files/2024-05/vol60_no1_8.pdf) The method exploits the fact that if a and b satisfy a quadratic relation, the formulas produce a second root: if \( a \ge b \) and k are positive integers satisfying \( (ab+1)k = a^2 + b^2 \), then a smaller non-negative integer \( a' < a \) exists that also satisfies the equation with b and k, giving a descent.[2](https://www.parabola.unsw.edu.au/sites/default/files/2024-05/vol60_no1_8.pdf) The same technique analyzes the Markov equation \( x^2 + y^2 + z^2 = 3xyz \), showing that its solutions form Markov's tree.[2](https://www.parabola.unsw.edu.au/sites/default/files/2024-05/vol60_no1_8.pdf) For the rational [Diophantine equation](https://www.edgepedia.org/diophantine-equation) \( a + 1/b + b + 1/a = k \) with positive integers a, b, the Vieta jumps of a solution pair \( (a, b) \) are \( (kb - 1 - a, b) \) and \( (a, ka - 1 - b) \), both again solutions, and k must be 3 or 4.[13](https://web.williams.edu/Mathematics/sjmiller/public_html/math/papers/FibNumbVietaJumpRatDiophEq_MNS_v10.pdf)
**Matrix polynomials.** Vieta-type spectral identities extend the scalar relations to complex matrix polynomials. For a monic matrix polynomial, the eigenvalues taken together have a total equal to the combined sum of the roots of the diagonal scalar polynomials when these are multiplied together; the same total also equals a constant times the sum of the roots of the polynomial formed by adding the diagonal entries, while the eigenvalues' product is governed by the determinant of the constant coefficient matrix.[14](https://arxiv.gg/abs/2607.06606) These monic-case identities do not extend directly to non-monic matrix polynomials with nonsingular leading coefficients.[14](https://arxiv.gg/abs/2607.06606)
**Partial root information.** A 2026 paper in Results in [Mathematics](https://www.edgepedia.org/mathematics) generalizes the classical formulas to the case where only some of a polynomial's roots are known, using complete homogeneous symmetric polynomials; the generalized coefficient condition reads \( a_{m-s} = -\sum_j a_j \mathcal{H}_{m-s-j}(x_0, \ldots, x_s) \).[15](https://link.springer.com/article/10.1007/s00025-026-02632-5) The same paper provides new identities for complete homogeneous symmetric polynomials when the variables form a geometric progression and new identities involving roots of unity.[15](https://link.springer.com/article/10.1007/s00025-026-02632-5)
## History
François Viète (1540–1603), a lawyer at the court of King Henry II in Tours and Paris, wrote the treatises known collectively as *The Analytic Art*, originally published in 1591.[16](https://books.google.com/books/about/The_Analytic_Art.html?id=omHqrYBzMpAC) In *In artem analyticam isagoge*, published at Tours in 1591, he introduced the first systematic algebraic notation.[17](https://mathshistory.st-andrews.ac.uk/Biographies/Viete/) His convention used vowels for unknowns and consonants for known quantities, streamlining the solution of equations.[16](https://books.google.com/books/about/The_Analytic_Art.html?id=omHqrYBzMpAC) He was the first to use letters for the parameters of an equation as well, so that while Cardano solved particular cubic equations stated in words, such as "a cube and 6 things are equal to 20" (\( x^3 + 6x = 20 \)), Viète could treat the general cubic.[18](https://math.berkeley.edu/~robin/Viete/work.html) • [19](https://mathematics.edwardworthlibrary.ie/notation/viete/) He insisted that equations be homogeneous, refusing to add terms of different dimensions, and framed algebra as the restored classical Greek method of analysis.[19](https://mathematics.edwardworthlibrary.ie/notation/viete/)
His coefficient–root relations carried a restriction: he did not consider negative numbers and had no symbol for equality.[20](https://arxiv.org/html/2210.12545v1) He proved the relation for all n but for positive roots only, and the general form was established by Albert Girard.[3](https://encyclopediaofmath.org/index.php?title=Vi%C3%A8te_theorem) Girard extended the results to polynomials of any degree, introducing the concept of elementary symmetric polynomials and acknowledging complex roots.[8](https://www.varsitytutors.com/practice/subjects/ib-mathematics-analysis-and-approaches/lessons/vietas-formulas)
The publication record is stated differently by different references. MathWorld dates Viète's proof to 1579, for positive roots only.[1](https://mathworld.wolfram.com/VietasFormulas.html) The Encyclopedia of Mathematics says the result was published in *Opera mathematica* (Leiden, 1646, edited by F. van Schouten), while a Springer paper likewise gives 1646, the publication date of *Opera mathematica*, for real-coefficient polynomials with only positive roots.[3](https://encyclopediaofmath.org/index.php?title=Vi%C3%A8te_theorem) • [15](https://link.springer.com/article/10.1007/s00025-026-02632-5) A formalized-mathematics reference instead states that the formulas were published posthumously in *De aequationum recognitione et emendatione tractatus duo* (1615).[12](https://leangenius.org/proof/vietas-formulas) What the sources agree on is the substance: Viète knew the connection between the positive roots of an equation and its coefficients, presented in his *De aequationum recognitione et emendatione* (1615) methods for solving equations of second, third, and fourth degree.[21](https://mathshistory.st-andrews.ac.uk/SH/viete_sh.pdf)
Viète also gave an original geometric recipe for solving any cubic with three real roots, the casus irreducibilis, the case in which Cardano's formulas require cube roots of imaginary quantities.[20](https://arxiv.org/html/2210.12545v1)
## Generalizations and edge cases
**Repeated and complex roots.** The formulas remain valid for multiple roots provided each root is listed in the elementary symmetric polynomials according to its multiplicity.[11](https://github.com/antoniolupetti/algebrica/blob/main/polynomials/vieta-formulas.md) In the quadratic case they hold for every value of the discriminant, including pairs of complex conjugate roots.[11](https://github.com/antoniolupetti/algebrica/blob/main/polynomials/vieta-formulas.md) The essential restriction is different: the roots referred to are all the complex roots of the polynomial. If attention is restricted to real roots only, the result is not true.[7](http://www.amsi.org.au/ESA_Senior_Years/SeniorTopic2/2e/2e_3links_2.html)
**Noncommutative coefficients.** Over a skew-field, where roots and coefficients need not commute, a noncommutative Vieta theorem expresses the coefficients \( a_k \) via Vandermonde quasideterminants, giving \( a_k = (-1)^k \sum y_{i_k} \cdots y_{i_1} \) for \( k = 1, \ldots, n \), without requiring trace or determinant.[22](https://ar5iv.labs.arxiv.org/html/q-alg/9507010) Earlier formulas in this line of work expressed the coefficients only as rational functions of the roots, and a prior result covered the case of complex matrices using the Amitsur theorem.[22](https://ar5iv.labs.arxiv.org/html/q-alg/9507010) The expressions \( \Lambda_k = (-1)^k a_k \) are symmetric functions of the roots, proved inside free skew-fields over a commutative field of characteristic zero.[22](https://ar5iv.labs.arxiv.org/html/q-alg/9507010)
**Common sign error.** A frequent mistake is forgetting the alternating signs and writing the quadratic sum of roots as \( +b/a \) instead of \( -b/a \); the correct sum for \( ax^2 + bx + c = 0 \) carries the factor \( (-1)^1 \).[4](https://www.mathwords.com/v/vietas_formulas.htm)
## References
1. [Vieta's Formulas, Wolfram MathWorld](https://mathworld.wolfram.com/VietasFormulas.html)
2. [Vieta jumping, Parabola (UNSW) Vol. 60 No. 1 (2024)](https://www.parabola.unsw.edu.au/sites/default/files/2024-05/vol60_no1_8.pdf)
3. [Viète theorem, Encyclopedia of Mathematics](https://encyclopediaofmath.org/index.php?title=Vi%C3%A8te_theorem)
4. [Vieta's Formulas, Mathwords](https://www.mathwords.com/v/vietas_formulas.htm)
5. [Vieta's formulae, UCSD course handout](https://mathweb.ucsd.edu/~trshin/handouts/vieta_in_action.pdf)
6. [Symmetric Polynomials, Archive of Formal Proofs (Isabelle)](https://isa-afp.org/browser_info/current/AFP/Symmetric_Polynomials/outline.pdf)
7. [Sums and products of roots, Australian Mathematical Sciences Institute](http://www.amsi.org.au/ESA_Senior_Years/SeniorTopic2/2e/2e_3links_2.html)
8. [Vieta's Formulas, IB HL lesson, Varsity Tutors](https://www.varsitytutors.com/practice/subjects/ib-mathematics-analysis-and-approaches/lessons/vietas-formulas)
9. [Vieta's Formulas and Symmetric Polynomials, Chapel Hill Math Circle (2025)](https://chapelhillmathcircle.org/wp-content/uploads/2025/03/2025-03-15_-_vietas_formulas_and_symmetric_polynomials_students_copy.pdf)
10. [Theory Symmetric_Polynomials.Vieta, Isabelle AFP](https://isa-afp.org/browser_info/current/AFP/Factor_Algebraic_Polynomial/Symmetric_Polynomials.Vieta.html)
11. [Vieta's formulas, Algebrica](https://github.com/antoniolupetti/algebrica/blob/main/polynomials/vieta-formulas.md)
12. [Vieta's formulas, LeanGenius annotated proofs](https://leangenius.org/proof/vietas-formulas)
13. [Fibonacci Numbers and Vieta Jumping for a Rational Diophantine Equation, Williams College](https://web.williams.edu/Mathematics/sjmiller/public_html/math/papers/FibNumbVietaJumpRatDiophEq_MNS_v10.pdf)
14. [Vieta-Type Formulas for Matrix Polynomials, arXiv](https://arxiv.gg/abs/2607.06606)
15. [Complete Homogeneous Symmetric Polynomials and Generalized Vieta's Formulas, Results in Mathematics](https://link.springer.com/article/10.1007/s00025-026-02632-5)
16. [The Analytic Art, François Viète, trans. T. Richard Witmer](https://books.google.com/books/about/The_Analytic_Art.html?id=omHqrYBzMpAC)
17. [François Viète (1540–1603), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Viete/)
18. [François Viète, UC Berkeley archive](https://math.berkeley.edu/~robin/Viete/work.html)
19. [Viète, Edward Worth Library](https://mathematics.edwardworthlibrary.ie/notation/viete/)
20. [François Viète and his contribution to mathematics, arXiv](https://arxiv.org/html/2210.12545v1)
21. [Francois Viète and the Foundations of Algebra, MacTutor](https://mathshistory.st-andrews.ac.uk/SH/viete_sh.pdf)
22. [Noncommutative Vieta Theorem and Symmetric Functions, arXiv](https://ar5iv.labs.arxiv.org/html/q-alg/9507010)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Algebraic structures › Abstract algebra — overview*
*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*
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