Sum and Product of Roots | Symmetrical Functions of root

By symmetric function of roots, we mean that the function remains unchanged when the roots are interchanged.

Let α and β  be the roots.
A function in α  and β is said to be a symmetric function if  the function remains the same when α and β are interchanged.

Consider α+β.  When α is written as β, and  β is written as α, the function becomes β+α, which is same as α+β.

So α+β is a symmetric function of the roots.
αβ is also a symmetric function of the roots as αβ = βα

α-β is not a symmetric function as α-β is not equal to β-α unless if β=α

However, mod(α-β) is a symmetric function, as mod(α-β )= mod(β-α)

Value of a #symmetric #function of the #roots in terms of the coefficients of the quadratic equation

α +β = -b/a
αβ = c/a

See Proof:    • Relationship Between Roots and Coeffi...  

Value of a symmetric function of α and β can be obtained if α+β and αβ are known.

To find the value of the symmetric function of the roots, express the given function in terms of α+β and αβ.

You can use the following results:

 α²+β²   = (α +β)² - 2αβ

 α³+β³  = (α +β)³ - 3αβ(α+β)

 α⁴+β⁴ = (α³+β³)(α +β)-αβ(α²+β²)

(α+β)⁴= α⁴+6α³β+4α²β²+6αβ³+β⁴ 
= α⁴+β⁴+6αβ(α²+β²)+4α²β²
   
α⁵+β⁵ = (α³+β³)(α²+β²)-(α²β³+α³β²)
=(α³+β³)(α²+β²)-(α²β²)(α+β)

mod(α -β ) = √[(α +β)²-4αβ]

α²-β²= (α+β)(α-β)

α³-β³ = (α-β)³+3αβ(α-β)
= (α-β)[(α +β)²-αβ]

α⁴-β⁴ = (α²+β²)(α²-β²)
= (α+β)(α-β)(α²+β²)

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