Vieta's Formulas Calculator
Calculate relationships between polynomial coefficients and roots using Vieta's formulas
Category: Mathematics
Vieta's Formulas Calculator Inputs
Vieta's Formulas Calculator Formula
Equation
For P(x)=(x-r₁)(x-r₂)...(x-rₙ): Σrᵢ=-aₙ₋₁, Σrᵢrⱼ=aₙ₋₂, ..., Πrᵢ=(-1)ⁿa₀
Excel Formula
=ForP(x)=(x-r₁)(x-r₂)...(x-rₙ):Σrᵢ=-aₙ₋₁,Σrᵢrⱼ=aₙ₋₂,...,Πrᵢ=(-1)ⁿa₀
Variables
- Calculate — Choose the Calculate option used by the Vieta's Formulas Calculator.
- Input (roots or coefficients) — Enter the Input (roots or coefficients) text used by the Vieta's Formulas Calculator.
How the Vieta's Formulas Calculator Works
Vieta's formulas, named after French mathematician François Viète (1540-1603), establish elegant relationships between polynomial coefficients and roots. These formulas state that for a monic polynomial (leading coefficient 1) of degree n, the coefficients are elementary symmetric polynomials of the roots. Specifically, the coefficient of x^(n-1) equals the negative sum of roots, the coefficient of x^(n-2) equals the sum of products of roots taken two at a time, and so on, with the constant term being ±product of all roots. This remarkable result allows us to extract information about roots without actually finding them, forms the foundation of the theory of equations, and connects polynomial algebra to symmetric function theory. Vieta's formulas are essential tools in algebra, appearing in proofs, problem-solving techniques, and applications ranging from optimization to dynamical systems.
The core relationship is For P(x)=(x-r₁)(x-r₂)...(x-rₙ): Σrᵢ=-aₙ₋₁, Σrᵢrⱼ=aₙ₋₂, ..., Πrᵢ=(-1)ⁿa₀. Typical inputs include Calculate, Input (roots or coefficients).
Enter your values in the vieta's formulas calculator above, review the step-by-step solution, and compare against the worked examples below so you can see how each input changes the result. This free online mathematics tool is built for homework, design checks, and professional verification.
Vieta's Formulas Calculator Theory & Explanation
Vieta's Formulas for General Polynomials
For a monic polynomial of degree n with roots r_1, r_2, ..., r_n: P(x) = x^n - (r_1+r_2+·s+r_n)x^n-1 + (Σ_i<jr_ir_j)x^n-2 - ·s + (-1)^n r_1r_2·s r_n More compactly using elementary symmetric polynomials e_k: P(x) = Σ_k=0^n (-1)^k e_k(r_1,...,r_n) x^n-k where e_0=1, e_1=Σ r_i, e_2=Σ_i<j r_ir_j, etc. For a general (non-monic) polynomial a_nx^n + a_n-1x^n-1 + ·s + a_0, divide by a_n first. These formulas appear simple but encode deep connections between roots and coefficients.
P(x) = \prod_i=1^n(x-r_i) = x^n - e_1x^n-1 + e_2x^n-2 - ·s + (-1)^n e_n
Quadratic and Cubic Cases
For quadratic x^2 + bx + c with roots r, s: r + s = -b, \quad rs = c For cubic x^3 + bx^2 + cx + d with roots r, s, t: r+s+t = -b, \quad rs+rt+st = c, \quad rst = -d For quartic x^4 + bx^3 + cx^2 + dx + e with roots r_1, r_2, r_3, r_4: Σ r_i = -b, \quad Σ_i<j r_ir_j = c, \quad Σ_i<j<k r_ir_jr_k = -d, \quad r_1r_2r_3r_4 = e The alternating signs (-1)^k are crucial. These low-degree cases appear constantly in algebra, and memorizing them is valuable for problem-solving.
x^2 + bx + c: \quad r+s=-b, \quad rs=c
Applications to Root Properties
Vieta's formulas allow analyzing root properties without solving. For x^2-5x+6=0: sum of roots is 5, product is 6. Both positive, so either both roots positive or both negative. Product positive means same sign; sum positive means both positive. Indeed, roots are 2 and 3. For x^3-6x^2+11x-6=0: root sum 6, pairwise sum 11, product 6. Trying small integers: x=1 works, so factor out (x-1): remaining quadratic x^2-5x+6 has roots 2,3. Vieta shortcuts finding rational roots. Another use: if roots of x^2+px+q are α, β, then roots of x^2-px+q are -α, -β (sum flips sign, product unchanged).
\textSum > 0, \text product > 0 \Rightarrow \textboth roots positive
Constructing Polynomials from Roots
Given roots, Vieta's formulas construct the polynomial. Roots 1, 2, 3: sum=6, pairwise sum=1·2+1·3+2·3=11, product=6. Polynomial: x^3-6x^2+11x-6. This is essentially expanding (x-1)(x-2)(x-3) but faster using Vieta. For roots 2± i (complex conjugate pair), r_1+r_2=4, r_1r_2=(2+i)(2-i)=5. Polynomial: x^2-4x+5. For higher degrees with many roots, Vieta's method (computing elementary symmetric polynomials) is more systematic than expanding products. This construction appears in interpolation, coding theory, and cryptography.
\textRoots r_1,...,r_n \xrightarrow\textVieta P(x) = \prod(x-r_i)
Newton's Identities and Power Sums
Newton's identities relate power sums p_k = Σ r_i^k to elementary symmetric polynomials. For monic polynomial with roots r_i and elementary symmetric polynomials e_k: p_k - e_1p_k-1 + e_2p_k-2 - ·s + (-1)^k-1e_k-1p_1 + (-1)^k ke_k = 0 This allows computing power sums from coefficients without finding roots. Example: x^3-6x^2+11x-6 has e_1=6, e_2=11, e_3=6. Then p_1=6, p_2=e_1p_1-2e_2=36-22=14, p_3=e_1p_2-e_2p_1+3e_3=84-66+18=36. Check: 1^2+2^2+3^2=14 ✓, 1^3+2^3+3^3=36 ✓. Newton's identities extend Vieta's relationships to arbitrary symmetric functions.
p_k = e_1p_k-1 - e_2p_k-2 + ·s + (-1)^k-1ke_k
Connection to Discriminant and Resultant
The discriminant Δ of polynomial P(x) equals (up to sign) the square of the product of all pairwise root differences: Δ = a_n^2n-2\prod_i<j(r_i-r_j)^2 It can be expressed as a polynomial in the coefficients using Vieta's formulas and symmetric polynomial theory. For quadratic ax^2+bx+c: Δ = b^2-4ac. For cubic: Δ = -4b^3d + b^2c^2 + 18bcd - 4c^3 - 27d^2 (for x^3+bx^2+cx+d). The discriminant vanishes iff the polynomial has repeated roots. Its sign (when real coefficients) determines the nature of roots. Vieta's formulas underlie these calculations, converting root properties to coefficient properties.
Δ = a_n^2n-2\prod_i<j(r_i-r_j)^2
Applications and Extensions
Vieta's formulas extend beyond basic polynomials. In Galois theory, they explain why elementary symmetric functions of roots are in the base field (they equal coefficients). In algebraic number theory, they relate conjugates of algebraic numbers (roots of minimal polynomials). In combinatorics, generating functions for sequences use Vieta-like relationships. In control theory, characteristic polynomials of matrices have Vieta formulas relating eigenvalues to matrix trace, determinant, and other invariants. For matrix A, \texttr(A) = Σ\lambda_i, \det(A) = \prod\lambda_i (Vieta for characteristic polynomial). Numerical analysts use Vieta to check computed roots—sum should equal -coefficient, catching errors. The formulas are fundamental throughout mathematics.
\texttr(A) = Σ \lambda_i, \quad \det(A) = \prod \lambda_i \quad \text(Vieta for matrices)
Vieta's Formulas Calculator Worked Examples
Worked Example
Inputs
- direction: rootsToCoeffs
- input: 2, 3, 5
Result: P(x) = x³ - 10x² + 31x - 30
Explanation
Roots 2,3,5 give: sum=10, pairwise=2·3+2·5+3·5=31, product=30, so P(x)=x³-10x²+31x-30
Quadratic from Roots
Inputs
- direction: rootsToCoeffs
- input: 3, 7
Result: P(x) = x² - 10x + 21
Explanation
Roots 3,7: sum=10, product=21, giving x²-10x+21
Common Vieta's Formulas Calculator Use Cases
- Homework and exam practice
- Engineering and science coursework
- Quick verification of hand calculations
- Vieta's Formulas homework and study
- Vieta's Formulas design and analysis
Vieta's Formulas Calculator FAQs
What are Vieta's formulas?
Vieta's formulas relate polynomial coefficients to sums and products of roots. For quadratic x²+bx+c with roots r and s: r+s=-b and rs=c. For cubic x³+bx²+cx+d with roots r,s,t: r+s+t=-b, rs+rt+st=c, rst=-d. Generally, coefficients are elementary symmetric polynomials of roots. These formulas let you find relationships between roots without actually solving for them, check solutions, and construct polynomials from given roots. They're named after François Viète, who discovered them in the 16th century.
How do I use Vieta's formulas to check my polynomial solutions?
After finding roots, verify using Vieta: sum all roots and compare to -b (coefficient of x^(n-1) in monic polynomial). Compute product and compare to ±constant term. For x²-5x+6 with claimed roots 2,3: check 2+3=5 ✓ (matches -b with b=-5), and 2·3=6 ✓ (matches c). This catches arithmetic errors. For higher degrees, check all symmetric combinations. If any fail, you made a mistake. This is faster than substituting each root back into the polynomial and is a standard verification technique.
Can I find roots using only Vieta's formulas?
Not directly—Vieta gives relationships but doesn't solve for individual roots. However, combined with other information, Vieta helps. If you know the roots are integers summing to 6 with product 11: try small integers. Or if one root is known, Vieta constrains the others. For x³-6x²+11x-6 and you guess x=1 is a root: the remaining two roots sum to 5 (from 1+r+s=6) and multiply to 6 (from 1·r·s=6), so they satisfy t²-5t+6=0, giving t=2 or 3. Vieta transforms n equations for n roots into simpler problems.
What is the connection between Vieta's formulas and symmetric polynomials?
Vieta's formulas state that polynomial coefficients are elementary symmetric polynomials of roots. The coefficient of x^(n-k) (up to sign) equals eₖ(r₁,...,rₙ), the k-th elementary symmetric polynomial. This is the Fundamental Theorem of Symmetric Polynomials in action: coefficients (which are in the base field) must be symmetric in the roots. Any permutation of roots gives the same polynomial, so coefficients are invariant—hence symmetric. This deep connection underlies Galois theory and explains solvability by radicals.
How do Vieta's formulas work for polynomials with complex roots?
Vieta's formulas work identically for complex roots. For x²-4x+5 with complex roots 2±i: sum = (2+i)+(2-i)=4=-b ✓, product = (2+i)(2-i)=5=c ✓. Real polynomials have complex roots in conjugate pairs, so even though individual roots are complex, their symmetric combinations (sums, products) are real, matching the real coefficients. This explains why complex roots don't "show" in coefficients—they're hidden in symmetric combinations. Vieta provides a window into complex root structure without explicit calculation.
What is the historical significance of Vieta's formulas?
Vieta's formulas (1591) were revolutionary—the first general relationships between polynomial coefficients and roots. Before Vieta, solving equations meant finding specific numerical roots. Vieta introduced symbolic algebra, using letters for unknowns and coefficients. His formulas showed that roots relate systematically to coefficients, paving the way for modern algebra. They influenced Descartes, Newton, and eventually Galois. The formulas revealed that algebra studies relationships and structure, not just numerical solutions. This philosophical shift created modern mathematics.
Can Vieta's formulas help with optimization problems?
Yes! When optimizing sum/product of variables under constraints. Example: maximize x+y given xy=12. By Vieta, x and y are roots of t²-St+12 where S=x+y. For real roots, discriminant S²-48≥0, so S≥4√3. Maximum S with constraint is S=4√3, achieved at x=y=2√3. Or minimize x²+y²+z² given x+y+z=6, xyz=6. Using Vieta and Newton's identities: x²+y²+z²=p₂=e₁²-2e₂=36-2e₂. For fixed e₁=6, e₃=6, minimize by maximizing e₂, subject to algebraic constraints. Vieta converts constrained optimization to polynomial analysis.
How are Vieta's formulas used in numerical analysis?
Numerical algorithms use Vieta as error checks. After computing polynomial roots numerically, sum them and compare to -aₙ₋₁/aₙ. Large discrepancies indicate numerical instability or algorithm failure. Iterative refinement uses Vieta: if approximate roots don't satisfy Vieta within tolerance, improve them. In companion matrix methods for root-finding, Vieta formulas verify that eigenvalues (which are the roots) match coefficient relationships. Vieta also appears in deflation: after finding one root, construct the deflated polynomial using Vieta to ensure remaining roots sum correctly. These checks are standard in robust numerical software.