Symmetric Functions Math Example 3

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Example 3

medium
If x+y=7x + y = 7 and xy=10xy = 10, find x3+y3x^3 + y^3.

Solution

  1. 1
    Use the identity x3+y3=(x+y)3−3xy(x+y)x^3 + y^3 = (x+y)^3 - 3xy(x+y).
  2. 2
    x3+y3=73−3(10)(7)=343−210=133x^3 + y^3 = 7^3 - 3(10)(7) = 343 - 210 = 133.

Answer

x3+y3=133x^3 + y^3 = 133
The identity x3+y3=(x+y)3−3xy(x+y)x^3 + y^3 = (x+y)^3 - 3xy(x+y) expresses the power sum p3=x3+y3p_3 = x^3 + y^3 in terms of e1e_1 and e2e_2. These Newton's identities connect power sums to elementary symmetric polynomials and are useful when you know the sum and product of two numbers.

About Symmetric Functions

A symmetric function is one that remains unchanged (or changes in a predictable way) under specific variable transformations. Even functions satisfy f(−x)=f(x)f(-x) = f(x) and are mirror-symmetric about the y-axis; odd functions satisfy f(−x)=−f(x)f(-x) = -f(x) and have 180-degree rotational symmetry about the origin.

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More Symmetric Functions Examples

Example 1 easy

Is the function [formula] symmetric in [formula] and [formula]?

Example 2 medium

Express [formula] in terms of the elementary symmetric polynomials [formula] and [formula].

Example 4 hard

Determine whether [formula] is a symmetric function of three variables.

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