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Roots Symmetric Functions

Let , , be the roots of a cubic, then the expressions of the types + + ,

+ + , ( - )

are the symmetric functions of , , but - = ( - ) is not a symmetric function of , , .

Moreover, the expression for the symmetric functions can be written by symmetry by considering different permutations of the roots of an equation, if a single term of the function is given. The symmetric function is generally represented by writing

S

are called elementary symmetric functions.

^{2}+^{2}+^{2}, + + are called the functions of the roots. By symmetric function of the roots of an equation, we mean those functions which remain invariant (unaltered) in values when any two of the roots are changed cyclically. For example, the functions+ + , ( - )

^{2}+ ( - )^{2}+ ( - )^{2},^{2}+^{2}+^{2}are the symmetric functions of , , but - = ( - ) is not a symmetric function of , , .

Moreover, the expression for the symmetric functions can be written by symmetry by considering different permutations of the roots of an equation, if a single term of the function is given. The symmetric function is generally represented by writing

**Σ**, (sigma) before one of the terms of the functions. If the degree of each term in a symmetric function is same, then the symmetric function is called homogeneous, otherwise non-homogeneous. The number of factors in each term of a homogeneous symmetric function is called its weight, and the highest power in which each root enters the function is called its order. For example, the symmetric function Σx^{4}^{2}^{2}is of order 4 and its weight is**4 + 3 + 2 = 9**.**Elementary symmetric functions:**The symmetric functions occurring in**δ, 4, 7**, viz,S

_{1}= Σ, S_{2}= Σ_{i}_{j}, …. S_{n}=_{1}_{2}….._{n}are called elementary symmetric functions.

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