Symmetric Functions Math Example 2

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Example 2

medium
Express x2+y2x^2 + y^2 in terms of the elementary symmetric polynomials e1=x+ye_1 = x + y and e2=xye_2 = xy.

Solution

  1. 1
    Recall that (x+y)2=x2+2xy+y2(x+y)^2 = x^2 + 2xy + y^2.
  2. 2
    Therefore x2+y2=(x+y)2−2xy=e12−2e2x^2 + y^2 = (x+y)^2 - 2xy = e_1^2 - 2e_2.
  3. 3
    Verify with x=3,y=1x = 3, y = 1: 9+1=109 + 1 = 10 and (4)2−2(3)=16−6=10(4)^2 - 2(3) = 16 - 6 = 10. ✓

Answer

x2+y2=e12−2e2x^2 + y^2 = e_1^2 - 2e_2
The elementary symmetric polynomials in two variables are e1=x+ye_1 = x + y and e2=xye_2 = xy. By the fundamental theorem of symmetric polynomials, every symmetric polynomial can be expressed in terms of these building blocks. This connection is used extensively in solving polynomial equations via Vieta's formulas.

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 3 medium

If [formula] and [formula], find [formula].

Example 4 hard

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

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