Symmetric Functions and Polynomial Equations

Symmetric Functions and Polynomial Equations

This lesson covers the following key concepts:

• Symmetric functions of roots
• Higher degree polynomial equations
• Relation between roots and coefficients for cubic, quartic
• Sum of powers of roots
• Product of differences of roots
• Transformation of equations
• Newton's identities for symmetric functions

Important Formulas

• $$\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$$
• $$\alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta)$$
• $$\alpha^4 + \beta^4 = (\alpha^2 + \beta^2)^2 - 2(\alpha\beta)^2$$
• $$\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta}$$
• $$\text{Cubic: } x^3 + px^2 + qx + r = 0, \alpha + \beta + \gamma = -p$$
• $$\alpha\beta + \beta\gamma + \gamma\alpha = q, \alpha\beta\gamma = -r$$
• $$|\alpha - \beta| = \frac{\sqrt{D}}{|a|}$$