Symmetric Functions of Roots

Symmteric Functions

Symmetric expressions are those in which the swapping of any two variables does not change the expression. This means that the expression treats all variables equally, and their order does not affect the result.

A simple symmetric expression in two variables could be \( x + y \). It's evident that exchanging \( x \) for \( y \) does not alter the expression—it will always sum to the same number regardless of the order of the variables.

Extending this to three variables, an example would be \( xyz + x + y + z \). No matter which of the variables \( x \), \( y \), or \( z \) you exchange, the expression as a whole remains the same. The product \( xyz \) includes all variables and doesn't change with their order, while the sum of the individual variables \( x + y + z \) is also unaffected by their order.

Another example with three variables would be \( x^2y + xy^2 + y^2z + yz^2 + z^2x + zx^2 \). In this expression, each term is a product of a squared variable and a different variable. Swapping any pair of variables simply rearranges the terms, but the overall expression does not change.

The combination of two symmetric expressions through addition, subtraction, multiplication, or division results in another symmetric expression. This means if you take any pair of symmetric expressions and combine them in these ways, their symmetry is preserved.

For instance, consider two symmetric expressions in variables \( x \) and \( y \), such as \( x^2 + y^2 \) and \( x + y \). Adding them gives \( x^2 + x + y^2 + y \), which is symmetric since swapping \( x \) and \( y \) leaves the expression unchanged. Similarly, the product \( (x^2 + y^2)(x + y) \) is symmetric because any interchange of \( x \) and \( y \) within each factor does not affect the overall expression. This holds true for subtraction and division as well.

Summation Notation

The expression \(x^2y + xy^2 + y^2z + yz^2 + z^2x + zx^2\) consists of terms that exhibit symmetry, with each term being of the same structural form. This symmetry can be succinctly expressed using the summation notation without explicitly denoting cyclic permutations. Therefore, to represent this expression in a more compact form, one can choose any single term that characterizes the entire sum and write:

\[ \sum x^2y \]

This notation implies the summation over all distinct permutations of the variables \(x, y,\) and \(z\) in the term \(x^2y\), yielding all possible expansions of the form \(x^2y\).

For symmetric expressions where each term follows a similar pattern, the summation notation \(\sum\) provides a concise way to represent these expressions. Here are more examples demonstrating how to express various symmetric expressions in short:

1. Expression Involving Linear Terms:

   • Original: \(x + y + z\)
   • Compact Form: \(\sum x\)

2. Expression Involving Squares and Products:

   • Original: \(x^2 + y^2 + z^2 + 2xy + 2yz + 2zx\)
   • Compact Form: \(\sum x^2 + 2\sum xy\)

3. Expression Involving Cubes and Triple Products:

   • Original: \(x^3 + y^3 + z^3 - 3xyz\)
   • Compact Form: \(\sum x^3 - 3xyz\)

4. Expression Involving Higher Powers and Mixed Terms:

   • Original: \(x^4 + y^4 + z^4 + 4x^2y^2 + 4y^2z^2 + 4z^2x^2\)
   • Compact Form: \(\sum x^4 + 4\sum x^2y^2\)

5. Expression with Complex Symmetric Terms:

   • Original: \(x^5y + y^5z + z^5x + x^2y^3 + y^2z^3 + z^2x^3\)
   • Compact Form: \(\sum x^5y + \sum x^2y^3\)

Symmetric functions of Roots of a polynomial

You must have already studied symmetric functions of roots of a quadratic expressions. In this section we are going to learn about dealing with symmetric functions in general.

Given \( \alpha, \beta, \gamma, \delta \) as roots of the polynomial \( p(x) = ax^4 + bx^3 - cx^2 + dx + e \), the expressions provided are symmetric functions of these roots:

(i) \( \Sigma(\alpha - \beta)^2 \) (ii) \( \Sigma(\alpha - \beta)^2\gamma\delta \) (iii) \( (\beta + \gamma - \alpha - \delta)(\gamma + \alpha - \beta - \delta)(\alpha + \beta - \gamma - \delta) \)

The sums and products of roots can be related to the coefficients of the polynomial by Vieta's formulas. For the given quartic polynomial, these relationships are:

1. \( \Sigma \alpha = -\frac{b}{a} \)
2. \( \Sigma \alpha\beta = \frac{c}{a} \)
3. \( \Sigma \alpha\beta\gamma = -\frac{d}{a} \)
4. \( \alpha\beta\gamma\delta = \frac{e}{a} \)

Using these relationships, we can find values of symmetric functions in roots. We are going to learn how to do this through various examples.

Example

Let \( \alpha, \beta, \gamma \) be the roots of the cubic polynomial \( ax^3 + bx^2 + cx + d \). Find the value of

a. \( \alpha^3 + \beta^3 + \gamma^3 \)

b. \( (\alpha^2 + 1)(\beta^2 + 1)(\gamma^2 + 1) \)

c. \( a^2[(\beta - \gamma)^2 + (\gamma - \alpha)^2 + (\alpha - \beta)^2] \)

d. \(a^2[\alpha(\beta - \gamma)^2 + \beta(\gamma - \alpha)^2 + \gamma(\alpha - \beta)^2]\)

Solution:

a. To find \( \alpha^3 + \beta^3 + \gamma^3 \), we start with the identity:

\[ \alpha^3 + \beta^3 + \gamma^3 = (\alpha + \beta + \gamma)(\alpha^2 + \beta^2 + \gamma^2 - \alpha\beta - \alpha\gamma - \beta\gamma) + 3\alpha\beta\gamma. \]

Using Vieta's formulas: - \( \alpha + \beta + \gamma = -\frac{b}{a} \), - \( \alpha\beta + \beta\gamma + \alpha\gamma = \frac{c}{a} \), - \( \alpha\beta\gamma = -\frac{d}{a} \).

Now, \( \alpha^2 + \beta^2 + \gamma^2 \) can be written as \( (\alpha + \beta + \gamma)^2 - 2(\alpha\beta + \alpha\gamma + \beta\gamma) \), which simplifies to:

\[ \left(-\frac{b}{a}\right)^2 - 2\left(\frac{c}{a}\right) = \frac{b^2}{a^2} - 2\frac{c}{a}. \]

Substituting these into the original identity, we get:

\[ \alpha^3 + \beta^3 + \gamma^3 = \left(-\frac{b}{a}\right)\left(\frac{b^2}{a^2} - 2\frac{c}{a} - \frac{c}{a}\right) + 3\left(-\frac{d}{a}\right). \]

Combining like terms, we have:

\[ \alpha^3 + \beta^3 + \gamma^3 = -\frac{b^3}{a^3} + 3\frac{bc}{a^2} - 3\frac{d}{a}. \]

Thus, the value of \( \alpha^3 + \beta^3 + \gamma^3 \) in terms of the coefficients of the given cubic polynomial is:

\[ \alpha^3 + \beta^3 + \gamma^3 = -\frac{b^3}{a^3} + 3\frac{bc}{a^2} - 3\frac{d}{a}. \]

b.

\[ (\alpha^2 + 1)(\beta^2 + 1)(\gamma^2 + 1) = \alpha^2\beta^2\gamma^2 + (\alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2) + (\alpha^2 + \beta^2 + \gamma^2) + 1. \]

Now let's find each term separately using identities and Vieta's formulas:

1. \( \alpha^2\beta^2\gamma^2 = (\alpha\beta\gamma)^2 = \left(-\frac{d}{a}\right)^2 \).
2. \( \alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2 \) can be written as \( (\alpha\beta + \beta\gamma + \gamma\alpha)^2 - 2\alpha\beta\gamma(\alpha + \beta + \gamma) \), which simplifies using Vieta's to \( \left(\frac{c}{a}\right)^2 - 2\left(-\frac{d}{a}\right)\left(-\frac{b}{a}\right) \).
3. \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to \( (\alpha + \beta + \gamma)^2 - 2(\alpha\beta + \alpha\gamma + \beta\gamma) \), which simplifies to \( \left(-\frac{b}{a}\right)^2 - 2\left(\frac{c}{a}\right) \).

Combining all these, we have:

\[ \begin{align*} &\left(\frac{d^2}{a^2}\right) + \left(\frac{c^2}{a^2} - 2\frac{bd}{a^2}\right) + \left(\frac{b^2}{a^2} - 2\frac{c}{a}\right) + 1 \\ &= \frac{b^2 - 2ac + c^2 - 2bd + d^2}{a^2} + 1 \\ &= \frac{b^2 - 2ac + c^2 - 2bd + d^2 + a^2}{a^2}. \end{align*} \]

Hence, the value of \( (\alpha^2 + 1)(\beta^2 + 1)(\gamma^2 + 1) \) in terms of the coefficients of the polynomial is \( \frac{a^2 + b^2 - 2ac + c^2 - 2bd + d^2}{a^2} = \frac{(a - c)^2 + (b - d)^2}{a^2}. \).

c.

\[ \begin{align*} a^2 & [(\beta - \gamma)^2 + (\gamma - \alpha)^2 + (\alpha - \beta)^2] \\ & = a^2 [\beta^2 - 2\beta\gamma + \gamma^2 + \gamma^2 - 2\gamma\alpha + \alpha^2 + \alpha^2 - 2\alpha\beta + \beta^2] \\ & = a^2 [2(\beta^2 + \gamma^2 + \alpha^2) - 2(\alpha\beta + \beta\gamma + \gamma\alpha)] \\ & = a^2 [2((\alpha + \beta + \gamma)^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha)) - 2(\alpha\beta + \beta\gamma + \gamma\alpha)] \\ & = a^2 [2\left(-\frac{b}{a}\right)^2 - 4\left(\frac{c}{a}\right) - 2\left(\frac{c}{a}\right)] \\ & = a^2 \left[\frac{2b^2}{a^2} - \frac{6c}{a}\right] \\ & = 2b^2 - 6ac. \end{align*} \]

Hence, the value of the expression is \( 2b^2 - 6ac \).

d.

\[ a^2[\alpha(\beta - \gamma)^2 + \beta(\gamma - \alpha)^2 + \gamma(\alpha - \beta)^2]. \]

Expanding the expression:

\[ a^2[\alpha(\beta^2 - 2\beta\gamma + \gamma^2) + \beta(\gamma^2 - 2\gamma\alpha + \alpha^2) + \gamma(\alpha^2 - 2\alpha\beta + \beta^2)] \]

\[ = a^2[\alpha\beta^2 - 2\alpha\beta\gamma + \alpha\gamma^2 + \beta\gamma^2 - 2\beta\gamma\alpha + \beta\alpha^2 + \gamma\alpha^2 - 2\gamma\alpha\beta + \gamma\beta^2] \]

Combine like terms:

\[ = a^2[\alpha\beta^2 + \beta\gamma^2 + \gamma\alpha^2 + \alpha\gamma^2 + \beta\alpha^2 + \gamma\beta^2 - 6\alpha\beta\gamma] \]

Now, let's use the fact that

\[ \color{red}(\alpha + \beta + \gamma)(\alpha\beta + \beta\gamma + \gamma\alpha) = \alpha^2\beta + \alpha\beta^2 + \beta^2\gamma + \beta\gamma^2 + \gamma^2\alpha + \gamma\alpha^2 + 3\alpha\beta\gamma \]

\[\implies \alpha^2\beta + \alpha\beta^2 + \beta^2\gamma + \beta\gamma^2 + \gamma^2\alpha + \gamma\alpha^2 = (\alpha + \beta + \gamma)(\alpha\beta + \beta\gamma + \gamma\alpha)-3\alpha\beta\gamma\]

Plugging these into our expanded expression, we get:

\[ = a^2[\alpha\beta^2 + \beta\gamma^2 + \gamma\alpha^2 + \alpha\gamma^2 + \beta\alpha^2 + \gamma\beta^2 - 6\alpha\beta\gamma] \]

\[ = a^2[(\alpha + \beta + \gamma)(\alpha\beta + \beta\gamma + \gamma\alpha)-3\alpha\beta\gamma-6\alpha\beta\gamma ]\]

\[ = a^2[(\alpha + \beta + \gamma)(\alpha\beta + \beta\gamma + \gamma\alpha)-9\alpha\beta\gamma] \]

\[ = a^2[-\frac{b}{a}\times\frac{c}{a}-9\times-\frac{d}{a}] \]

\[ = a^2\left[-\frac{bc}{a^2}+\frac{9d}{a}\right] \]

\[ = -bc + 9ad \]

Thus, the simplified value of the expression is \( -bc + 9ad \).