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Symmetric Functions

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Any expression f(a,b) involving two numbers a and b is said to be symmetric if it remains unchanged when a and b are interchanged.

[i.e. if f(a,b) = f(b,a)].

Some of the symmetric functions of a and b are

All symmetric functions of a and b can be expressed in terms of two symmetric functions a+b and ab.

For instance,

We thus observe, without actually solving the quadratic equation

(a) We can find the value of every symmetric function involving the roots of the equation in terms of the coefficients of the equation.

(b) We can find a quadratic equation whose roots are any one of the following pairs of numbers.

where a and b are the roots of the given equation.

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