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Vieta's general formulas

The general formulae of Viète allow the coefficients of a polynomial to be expressed in the sums and products of its roots.

 

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Explanation

A polynomial of the degree n

P(x) = a_nxⁿ  + a_n−1xⁿ⁻¹  + ··· + a₂x² + a₁x + a₀ = (x − x₁)(x − x₂) ··· (x − x_n)

with real or complex coefficients, in which a_n ≠ 0, has according to the fundamental theorem of algebra n zeros x₁, x₂, ... , x_n. The formulas arise by calculating

     for    j = 1, 2, ... , n

where

 

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Example 1

For a first-degree polynomial

x + p

it gives

x₁ = −p

 

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Example 2

For a second-degree polynomial

x² + px + q

it gives

x₁ + x₂ = −p
x₁x₂ = q

 

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Example 3

For a third-degree polynomial

x³ + px² + qx + r

it gives

x₁ + x₂ + x₃ = −p
x₁x₂ + x₁x₃  + x₂x₃ = q
x₁x₂x₃ = −r

 

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Example 4

For a fourth-degree polynomial

x⁴ +px³ + qx² + rx + s

it gives

x₁ + x₂ + x₃ + x₄ = −p
x₁x₂ + x₁x₃ + x₁x₄ + x₂x₃ + x₂x₄ + x₃x₄ = q
x₁x₂x₃ + x₁x₂x₄ + x₁x₃x₄ + x₂x₃x₄ = −r
x₁x₂x₃x₄ = s

 

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History The formulas were developed by the French mathematician Albert Girard (1595 - 1632).

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