Vieta's Formulas
Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots.
Includes examples for quadratic and cubic equations.
The Formula
Cubic: x³ + ax² + bx + c = 0 with roots r₁, r₂, r₃ r₁ + r₂ + r₃ = −a r₁r₂ + r₁r₃ + r₂r₃ = b r₁ × r₂ × r₃ = −c
Vieta's formulas, named after French mathematician François Viète (1540–1603), express elegant relationships between the roots of a polynomial and its coefficients. You can determine the sum, product, and other symmetric combinations of the roots directly from the coefficients — without ever solving for the roots themselves.
For a quadratic equation x² + bx + c = 0, the two roots r₁ and r₂ always add up to −b and always multiply to c. This is not a coincidence — it follows directly from the factored form (x − r₁)(x − r₂) = x² − (r₁+r₂)x + r₁r₂, which must equal x² + bx + c.
Vieta's formulas generalize to polynomials of any degree. For a degree-n polynomial, there are n such relationships, each involving an elementary symmetric polynomial of the roots. These formulas are invaluable for solving competition math problems, verifying roots, and building polynomials with specified root properties.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| r₁, r₂, ... | Roots of the polynomial | dimensionless |
| a, b, c, ... | Coefficients of the polynomial (with leading coefficient 1) | dimensionless |
| n | Degree of the polynomial (number of roots) | dimensionless |
Example 1
The quadratic x² − 5x + 6 = 0 has roots r₁ and r₂. Find their sum and product without solving the equation.
Write in the form x² + bx + c: here b = −5, c = 6
By Vieta: r₁ + r₂ = −b = −(−5) = 5
By Vieta: r₁ × r₂ = c = 6
Sum = 5, Product = 6. (The actual roots are 2 and 3: 2+3 = 5 and 2×3 = 6. Confirmed.)
Example 2
Build a quadratic equation whose roots are 4 and −7.
Sum of roots: 4 + (−7) = −3, so −b = −3, meaning b = 3
Product of roots: 4 × (−7) = −28, so c = −28
The polynomial is x² + 3x − 28 = 0
x² + 3x − 28 = 0 has exactly the roots 4 and −7. Vieta allows you to construct polynomials from roots without multiplying out factors by hand.
When to Use It
Use Vieta's formulas when:
- Finding the sum or product of roots without solving the full equation
- Constructing a polynomial that has specified roots
- Verifying whether two values are the roots of a given polynomial
- Solving competition math problems that ask for expressions like r₁² + r₂²
- Analyzing the behavior of systems described by characteristic polynomials in engineering