Theory of Equations: Roots, Coefficients & Symmetric Functions

1. The Fundamental Theorem of Algebra

Theorem (d'Alembert–Gauss): Every non-constant single-variable polynomial with complex coefficients has at least one complex root.
As an immediate corollary, any polynomial of degree $n \ge 1$: $$P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0, \quad a_n \ne 0, \; a_i \in \mathbb{C}$$ can be completely factored into linear factors over $\mathbb{C}$: $$\mathbf{P(x) = a_n (x - \alpha_1)(x - \alpha_2)\cdots(x - \alpha_n) = a_n \prod_{i=1}^n (x - \alpha_i)}$$ where $\alpha_1, \alpha_2, \dots, \alpha_n \in \mathbb{C}$ are the $n$ roots (counted with multiplicity).

2. The Conjugate Pairs Theorem for Real Polynomials

Theorem: If $P(x)$ is a polynomial with real coefficients ($a_i \in \mathbb{R}$) and $\alpha = u + iv$ is a complex root ($v \ne 0$), then its complex conjugate $\bar{\alpha} = u - iv$ is also a root of $P(x)$ of identical multiplicity.
Proof: Since $P(\alpha) = \sum_{k=0}^n a_k \alpha^k = 0$, taking complex conjugates yields: $$\overline{P(\alpha)} = \overline{\sum_{k=0}^n a_k \alpha^k} = \sum_{k=0}^n \bar{a}_k (\bar{\alpha})^k = \sum_{k=0}^n a_k (\bar{\alpha})^k = P(\bar{\alpha}) = \bar{0} = 0$$ Thus $P(\bar{\alpha}) = 0$. $\blacksquare$

Consequently, every complex root pair yields an irreducible real quadratic factor: $$(x - \alpha)(x - \bar{\alpha}) = x^2 - 2u x + (u^2 + v^2) \in \mathbb{R}[x]$$ This guarantees that any real polynomial can be factored over $\mathbb{R}$ into linear and irreducible quadratic factors. In particular, every real polynomial of odd degree has at least one real root.

1. General Formulation for Degree $n$

François Viète discovered the universal algebraic relations connecting the roots $\alpha_1, \dots, \alpha_n$ of a monic polynomial $x^n + p_1 x^{n-1} + p_2 x^{n-2} + \dots + p_n = 0$ to its coefficients: $$\prod_{i=1}^n (x - \alpha_i) = x^n - \left(\sum \alpha_i\right)x^{n-1} + \left(\sum_{i < j} \alpha_i \alpha_j\right)x^{n-2} - \dots + (-1)^n (\alpha_1 \cdots \alpha_n)$$ Equating coefficients of identical powers of $x$: $$\mathbf{p_k = (-1)^k e_k(\alpha_1, \dots, \alpha_n), \quad k = 1, 2, \dots, n}$$ where $e_k$ is the $k$-th elementary symmetric polynomial.

2. Explicit Relations for the Cubic Equation

For the cubic polynomial $x^3 + p x^2 + q x + r = 0$ with roots $\alpha, \beta, \gamma$:

3. Explicit Relations for the Quartic Equation

For the quartic polynomial $x^4 + p x^3 + q x^2 + r x + s = 0$ with roots $\alpha, \beta, \gamma, \delta$:

1. Definition of Symmetric Polynomials

A polynomial $f(x_1, x_2, \dots, x_n)$ is called symmetric if it remains strictly invariant under every permutation $\sigma \in S_n$ of its variables: $$f(x_{\sigma(1)}, x_{\sigma(2)}, \dots, x_{\sigma(n)}) = f(x_1, x_2, \dots, x_n)$$

2. The Elementary Symmetric Polynomials

The elementary symmetric polynomials $e_1, e_2, \dots, e_n$ in $n$ variables are defined as: $$e_1 = \sum_{1 \le i \le n} x_i, \quad e_2 = \sum_{1 \le i < j \le n} x_i x_j, \quad \dots, \quad e_n = x_1 x_2 \cdots x_n$$ with generating function: $$\prod_{i=1}^n (1 + t x_i) = 1 + e_1 t + e_2 t^2 + \dots + e_n t^n = \sum_{k=0}^n e_k t^k$$

3. The Fundamental Theorem of Symmetric Polynomials

Theorem: Every symmetric polynomial $f(x_1, \dots, x_n)$ with coefficients in a ring $R$ can be written uniquely as a polynomial in the elementary symmetric polynomials $e_1, \dots, e_n$ with coefficients in $R$: $$\mathbf{f(x_1, \dots, x_n) = P(e_1, e_2, \dots, e_n)}$$ Significance: Any symmetric combination of the roots of a polynomial equation can be evaluated purely in terms of the polynomial's given coefficients without ever explicitly solving for the roots!

1. Classical Symmetric Sums for Cubic Roots

Let $\alpha, \beta, \gamma$ be the roots of $x^3 + p x^2 + q x + r = 0$, so $e_1 = -p$, $e_2 = q$, $e_3 = -r$. We express classical symmetric combinations in terms of $p, q, r$:

• Sum of Squares: $$\mathbf{\sum \alpha^2 \equiv \alpha^2 + \beta^2 + \gamma^2 = e_1^2 - 2e_2 = p^2 - 2q}$$
• Product-Cross Sum: $$\mathbf{\sum \alpha^2 \beta = e_1 e_2 - 3e_3 = -pq + 3r}$$
• Sum of Cubes: $$\mathbf{\sum \alpha^3 = e_1^3 - 3e_1 e_2 + 3e_3 = -p^3 + 3pq - 3r}$$
• Sum of Squares of Differences: $$(\alpha - \beta)^2 + (\beta - \gamma)^2 + (\gamma - \alpha)^2 = 2\sum \alpha^2 - 2\sum \alpha\beta = 2(p^2 - 2q) - 2q = \mathbf{2p^2 - 6q}$$

2. The Polynomial Discriminant

The discriminant of a polynomial $P(x)$ of degree $n$ with roots $\alpha_1, \dots, \alpha_n$ is defined by: $$\mathbf{\Delta \equiv a_n^{2n-2} \prod_{1 \le i < j \le n} (\alpha_i - \alpha_j)^2}$$ Since $\Delta$ is symmetric in the roots, it is a polynomial in the coefficients. For the depressed cubic $x^3 + px + q = 0$, its roots satisfy: $$\mathbf{\Delta = -4p^3 - 27q^2}$$ If $\Delta > 0$, the cubic has 3 distinct real roots; if $\Delta = 0$, it has repeated roots; if $\Delta < 0$, it has 1 real root and 2 non-real conjugate roots.

1. Definition of Power Sums

For a monic polynomial $P(x) = x^n + p_1 x^{n-1} + \dots + p_n = 0$ with roots $\alpha_1, \dots, \alpha_n$, we define the $k$-th power sum as: $$\mathbf{s_k \equiv \sum_{i=1}^n \alpha_i^k = \alpha_1^k + \alpha_2^k + \dots + \alpha_n^k}$$ For $k = 0$, $s_0 = n$.

2. The Newton-Girard Recurrence Relations

Isaac Newton and Albert Girard derived the recursive relations connecting power sums $s_k$ directly to the polynomial coefficients $p_1, \dots, p_n$:

3. Derivation via Logarithmic Differentiation

Writing $P(x) = \prod_{i=1}^n (x - \alpha_i)$, taking the formal logarithmic derivative: $$\frac{P'(x)}{P(x)} = \sum_{i=1}^n \frac{1}{x - \alpha_i} = \frac{1}{x} \sum_{i=1}^n \frac{1}{1 - \alpha_i/x} = \sum_{i=1}^n \sum_{k=0}^\infty \frac{\alpha_i^k}{x^{k+1}} = \sum_{k=0}^\infty \frac{s_k}{x^{k+1}}$$ Multiplying both sides by $P(x) = x^n + p_1 x^{n-1} + \dots + p_n$ and equating coefficients of corresponding powers of $x$ establishes the Newton-Girard identities for all $k \ge 1$. $\blacksquare$