Let the roots of an "n" degree equation are \(\alpha_1, \alpha_2, \alpha_3.........\alpha_n.\) then the equation $$ \alpha_0 \ x^n + \alpha_1 \ x^{n - 1} + \alpha_2 \ x^{n - 2}\\............+ \alpha_{n - 1} \ x + \alpha_n = 0 $$ The given formula can be used to find the relation between roots and their coefficients of an equation.

Taking P number of roots at a time.

The sum of roots \(= (-1)^P \ \frac{Coefficient \ of \ (P - 1) \ term}{Coefficient \ of \ first \ term}\)

**(1):** Let the roots of a two-degree equation are \(\alpha\), and \(\beta\) then.

Sum of roots \((\alpha + \beta)\) = \(- \frac{b}{a}\)

Product of roots \(\alpha \times \beta\) = \(\frac{c}{a}\)

**(2):** Let the roots of a three-degree equation are \(\alpha\), \(\beta\), and \(\gamma\) then.

Sum of roots \((\alpha + \beta + \gamma)\) = \(- \frac{b}{a}\)

Sum of roots if taking two roots at a time \((\alpha \beta + \beta \gamma + \gamma \alpha)\) = \(\frac{c}{a}\)

Product of roots \(\alpha \times \beta \times \gamma\) = \(- \frac{d}{a}\)

**(3):** Let the roots of a four-degree equation are \(\alpha\), \(\beta\), \(\gamma\), and \(\delta\) then.

Sum of roots \((\alpha + \beta + \gamma + \delta)\) = \(- \frac{b}{a}\)

Sum of roots if taking two roots at a time \((\alpha \beta + \alpha \gamma + \alpha \delta +\) \(\beta \gamma + \beta \delta + \gamma \delta)\) = \(\frac{c}{a}\)

Sum of roots if taking three roots at a time \((\alpha \beta \gamma + \alpha \beta \delta +\) \(\alpha \gamma \delta + \beta \gamma \delta)\) = \(- \frac{d}{a}\)

Product of roots \(\alpha \times \beta \times \gamma \times \delta\) = \(\frac{e}{a}\)

Let the roots of an equation are \(\alpha_1, \alpha_2, \alpha_3.......\alpha_n\) then the factors of the equation will be \((x - \alpha_1), (x - \alpha_2), (x - \alpha_3),\) \(..........(x - \alpha_n)\) hence equation will be $$ (x - \alpha_1), (x - \alpha_2), \\ (x - \alpha_3),..........(x - \alpha_n) = 0 $$

**Note:** If the roots of a two-degree equation are \(\alpha\) and \(\beta\) then equation will be $$ (x - \alpha) \ (x - \beta) = 0 $$ $$ x^2 - (\alpha + \beta) x + \alpha \beta = 0 $$ $$ x^2 - (sum \ of \ roots) x + \\ (product \ of \ roots) = 0 $$

If a function made by the roots of a quadratic equation does not change after interchanging the position of roots then it is called the symmetric function of roots.

**Example:** If the roots of a quadratic equation are \(\alpha\) and \(\beta\) then the fuctions \((\alpha + \beta)\) \((\alpha^3 + \beta^3)\), \((\alpha^4 + \beta^4)\)....etc and if we interchange the position of \(\alpha\) and \(\beta\), it does not change function.

Lec 1: Introduction
Lec 2: Equation and Identity
Lec 3: The Relation between Roots and its Coefficients