Vieta Formula, Quadratic and Cubic Equation

Vieta Formula: Algebra, a fundamental field in mathematics, enclose various key components, among which polynomials hold significant importance. Vieta's formula, an integral concept within algebra, specifically pertains to polynomials. This formula establishes a relationship between the sum and product of roots and the coefficients of a polynomial, forming a crucial aspect in algebraic computations.

Vieta Formula

Vieta formulas serve as fundamental tools that establish connections between the sum and product of roots of a polynomial and the coefficients within that polynomial. This formulation enables the expression of the polynomial coefficients through the combined values of its roots and their products.

Vieta formula addresses the relationship between the sum and product of a polynomial's roots and the coefficients, serving as a pivotal method when solving for a polynomial given its roots, or when determining the sum or product of these roots.

Vieta Formula for Quadratic Equations

Consider a quadratic equation, f(x) = ax ² + bx + c, with roots α and β. Vieta’s formula defines the following relationships:

Sum of roots (α + β) = -b/a

Product of roots (αβ) = c/a

Given the sum and product of roots, the quadratic equation can be constructed as follows:

x ² – (sum of roots)x + (product of roots) = 0

Vieta’s Formula for Cubic Equations

Vieta’s Formula: Cubic Equations Consider a cubic equation, f(x)=ax ³ +bx ² +cx+d, with roots α, β, and γ. Vieta’s formula reveals the following relationships:

Sum of roots (α + β + γ) = -b/a

Sum of the product of two roots (αβ + αγ + βγ) = c/a

Product of roots (αβγ) = -d/a

Given the sum and product of roots, the cubic equation can be formulated as:

x ³ −(sum of roots)x ² +(sum of product of two roots)x−(product of roots)=0

Vieta Formula for the Generalized Equation

Vieta Formula for Polynomials in General Form For a generalized equation

f(x)=a _n ​ x _n +a _n−1 ​ x ⁿ⁻¹ +a _n−2 ​ x ⁿ⁻² +⋯+a ₂ ​ x ² +a ₁ ​ x+a ₀ ​ with roots r ₁ ​ ,r ₂ ​ ,r ₃ ​ ,…,r _n−1 ​ ,r _n ​ , Vieta’s formula unveils the connections between these roots and the coefficients as follows:

Sum of all roots

r ₁ ​ +r ₂ ​ +r ₃ ​ +⋯+r _n−1 ​ +r _n ​ = − ​ a _n−1 ​/ a _n ​

Sum of products taken two at a time:

(r ₁ ​ r ₂ ​ +r ₁ ​ r ₃ ​ +⋯+r ₁ ​ r _n ​ )+(r ₂ ​ r ₃ ​ +r ₂ ​ r ₄ ​ +⋯+r ₂ ​ r _n ​ )+⋯+r _n−1 ​ r _n ​ = ​ a _n−2 ​ / a _n ​

The pattern continues similarly, involving the sum of products taken three at a time, four at a time, and so on, ultimately leading to the product of all roots:

r ₁ ​ r ₂ ​ …r _n ​ =(−1) ⁿ (​ a ₀ ​/ a _n ​ )

Vieta Formula Solved Example

Example 1: Find the sum of the squares of the roots and the product of the roots of the quadratic equation

3x ² −7x+4=0.

Solution: Given the quadratic equation 3x ² −7x+4=0.

Using Vieta's Formulas: For the equation 3x ² −7x+4=0:

α+β=−b/a=−(−7)/3=7/3

αβ=c/a=4/3

Calculating

(α ² +β ² )=(α+β) ² −2αβ = =(7/3) 2 −2×(4/3) = 49 / 9 − 8 / 3 =49/9−8/3 (α ² +β ² )=49/9−24/9=25/9

Calculating the product of the roots squared  α ² β ² =(αβ) 2 = ( 4 / 3 ) 2 = 16 / 9 =(4/3) 2 =16/9

Hence, the sum of the squares of the roots (α ² +β ² ) is 25/9 and the product of the roots squared α ² β ² is 16/9.

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