# Relation between Roots and Coefficients:

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 $$\alpha \times \beta \times \gamma \times \delta$$ = $$\frac{e}{a}$$

#### Forming equation by the given roots:

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 $$(x - \alpha) \ (x - \beta) = 0$$ $$x^2 - (\alpha + \beta) x + \alpha \beta = 0$$ $$x^2 - (sum \ of \ roots) x + \\ (product \ of \ roots) = 0$$

#### Symmetric functions of roots:

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.