SYMMETRIC FUNCTIONS OF ROOTS - PUNJAB-CURRICULUM-AND-TEXTBOOK-BOARD-PCTB / 10TH-CLASS / MATHEMATICS / MATHEMATICS-10-10TH-CLASS-NBF-2024 / THEORY-OF-QUADRATIC-EQUATIONS / SYMMETRIC-FUNCTIONS-OF-ROOTS

2.4 Symmetric functions of the roots of a quadratic equation.

2.4.1 Define symmetric functions of the roots of a quadratic equation

Definition: Symmetric functions are those functions in which the roots involved are such that the value of the expressions involving them remain unaltered, when roots are interchanged. For example, if $f (α, β) = α^{2} + β^{2}$ , then $f (β, α) = β^{2} + α^{2} = α^{2} + β^{2}$ $∴ β^{2} + α^{2} = α^{2} + α^{2}$ $= f (α, β)$

Example: Find the value of $α^{2} + β^{3} + 3 α β$ , if $α = 2$ , $β = 1$ . Also find the value of $α^{3} + β^{3} + 3 α β$ if $α = 1$ , $β = 2$ .

Solution: When $α = 2$ and $β = 1$ , $α^{3} + β^{3} + 3 α β = (2)^{3} + (1)^{3} + 3 (2) (1)$ $= 8 + 1 + 6 = 15$

When $α = 1$ and $β = 2$ , $α^{3} + β^{3} + 3 α β = (1)^{3} + (2)^{3} + 3 (1) (2)$ $= 1 + 8 + 6 = 15$

The expression $α^{3} + β^{3} + 3 α β$ represents a symmetric function of $α$ and $β$ .

2.4.2. Evaluate a symmetric function of roots of a quadratic equation in terms of its co- coefficients

If $α$ , $β$ are the roots of the quadratic equation $ax^{2} + bx + c = 0$ , $(a \neq 0)$ (i)