Overview

A polynomial in several variables is symmetric if it is unchanged under any permutation of its variables. For example, $r^2 + s^2$ is symmetric (swapping gives $s^2 + r^2$, which is identical), but $r + 2s$ is not (swapping gives $s + 2r \neq r + 2s$). Symmetric polynomials arise naturally from polynomial roots: every expression Vieta's formulas produce is symmetric in the roots by construction. The basic idea is that since we are able to switch up various variables without changing it, we can use various constructions to our advantage

Elementary Symmetric Polynomials

The elementary symmetric polynomials $S_k$ in variables $x_1, \ldots, x_n$ are defined as the sum of all distinct products of exactly $k$ variables:

For three variables $x, y, z$ explicitly:

The number of terms in $S_k$ is $\binom{n}{k}$, since each term chooses $k$ variables from $n$.

Generating function interpretation. The elementary symmetric polynomials are the coefficients in the expansion

Fundamental Theorem of Symmetric Polynomials

Theorem. Every symmetric polynomial with integer (or rational, or real) coefficients can be written uniquely as a polynomial in $S_1, S_2, \ldots, S_n$.

Key Techniques

The $s$–$p$ Substitution (Two Variables)

For two variables $x, y$, set $s = x + y$ and $p = xy$. Then:

Expression	In terms of $s, p$
$x^2 + y^2$	$s^2 - 2p$
$x^3 + y^3$	$s^3 - 3sp = s(s^2 - 3p)$
$x^4 + y^4$	$s^4 - 4s^2p + 2p^2$
$x^2 y + xy^2$	$sp$
$(x-y)^2$	$s^2 - 4p$

The key is $(x+y)^k$ expands with cross terms that must be subtracted; never guess the formula, always expand.

For three variables, the analogous substitution is $u = x+y+z$, $v = xy+yz+zx$, $w = xyz$.

Palindromic (Even Symmetric) Polynomials

For a degree-$2n$ palindromic polynomial such as $ax^4 + bx^3 + cx^2 + bx + a = 0$, divide by $x^n$ and substitute $y = x + x^{-1}$:

where $T_k$ satisfies the same recurrence $T_k = y \cdot T_{k-1} - T_{k-2}$. This reduces a degree-$2n$ polynomial to a degree-$n$ polynomial in $y$.

Constructing Polynomials with Shifted Roots

To evaluate $(r_1 + c)(r_2 + c) \cdots (r_n + c)$ where $r_i$ are roots of $f(x)$: substitute $x = y - c$ to get a new polynomial $g(y) = f(y-c)$ whose roots are $r_i + c$. The product equals $\pm g(0)/a_n$ by Vieta's.

Worked Examples

Example 1. Express $x^2 + y^2 + z^2$ in terms of $S_1, S_2, S_3$.

Expand $S_1^2 = (x+y+z)^2 = x^2+y^2+z^2 + 2(xy+yz+zx) = x^2+y^2+z^2 + 2S_2$.

Rearranging: $\boxed{x^2+y^2+z^2 = S_1^2 - 2S_2}$.

Example 2. Given $a+b = 1$ and $a^2+b^2 = 2$, find $a^3+b^3$.

Set $s = a+b = 1$ and $p = ab$. From the sum of squares:

Now apply the sum of cubes identity:

Example 3. Let $a, b, c$ be roots of $f(x) = x^3 + 8x + 5 = 0$. Find $(1+a)(1+b)(1+c)$.

Do not use $S_k$ directly — instead construct a polynomial with roots $1+a, 1+b, 1+c$. Substitute $x = y - 1$:

By Vieta's for $g$, the product of its roots is $-(-4)/1 = \boxed{4}$.

Example 4. Solve $x^4 + x^3 + x^2 + x + 1 = 0$.

The coefficients are symmetric (palindromic). Divide by $x^2$:

Group: $(x^2 + x^{-2}) + (x + x^{-1}) + 1 = 0$. Let $u = x + x^{-1}$, so $x^2 + x^{-2} = u^2 - 2$:

For each value of $u$, solve $x + x^{-1} = u$, i.e., $x^2 - ux + 1 = 0$, giving the four roots of the original.

Example 5 (Discriminant). Determine whether a polynomial has a repeated root using only its coefficients.

A polynomial $f(x)$ with roots $x_1, \ldots, x_n$ has a repeated root iff $\Delta = \prod_{i < j}(x_i - x_j)^2 = 0$. Since $\Delta$ is symmetric in the roots, the Fundamental Theorem guarantees it is a polynomial in $S_1, \ldots, S_n$. By Vieta's, each $S_k$ is a coefficient of $f$ up to sign. Therefore $\Delta$ — the discriminant — is computable purely from the coefficients of $f$ using arithmetic operations.

Common Pitfalls

Asymmetric expressions. An expression must be unchanged under every variable permutation, not just one specific swap. $r + 2s$ is not symmetric and cannot be expressed in $S_k$ alone.

Sign errors in Vieta's. $S_1 = -a_{n-1}/a_n$ (negative when $a_{n-1} > 0$); $S_2 = +a_{n-2}/a_n$ (positive); signs alternate. This is the most common arithmetic error.

Wrong $s$–$p$ expansions. $x^3+y^3 \neq s^3$. The correct identity is $x^3+y^3 = s^3 - 3sp$. Always expand carefully.

Palindromic substitution cross-term. When substituting $y = x + x^{-1}$, do not write $x^2 + x^{-2} = y^2$. The correct identity is $x^2 + x^{-2} = y^2 - 2$ (the cross term $2 \cdot x \cdot x^{-1} = 2$ must be subtracted).

Root multiplicity. If every root of $g(x)$ is also a root of $f(x)$, this does not mean $g \mid f$. You must verify that the multiplicity of each shared root in $g$ does not exceed its multiplicity in $f$.

Practice Problems