If α_{1, } α_{2,}α_{3}_{ ... } α_{n}_{ }are the roots of the equation f(x)= a _{0}x^{n }+a_{1}x^{n-1 }+a_{2}x^{n-2} +...+a_{n-1}x + a_{n} =0, thenf(x)= a _{0} (x-α_{1)}(x-α_{2)}(x-α_{3)... }(x-α_{n)}Equating both the RHS terms we get, a _{0}x^{n }+a_{1}x^{n-1 }+a_{2}x^{n-2} +...+a_{n-1}x + a_{n = }a_{0}(x-α_{1)}(x-α_{2)}(x-α_{3)... }(x-α_{n)Comparing coefficients of }x^{n-1 }on both sides, we get^{ S}_{1} = α_{1 + } α_{2}+α_{3}_{ +... }+ α_{n} = ∑α_{i }= -a_{1/} a_{0} or, S _{1}= - coeff. of x^{n-1}/coeff. of x^{n} _{Comparing coefficients of }x^{n-2 }on both sides, we get^{ S}_{2} = α_{1} α_{2}+ α_{1}α_{3}_{ +... } = ∑α_{i} α_{j}_{ }= (-1)^{2}a_{2/} a_{0} i≠ j or, S _{2}= (-1)^{2} coeff. of x^{n-2}/coeff. of x^{n} _{Comparing coefficients of }x^{n-3 }on both sides, we get^{ S}_{3} = α_{1} α_{2}α_{3}+ α_{2}α_{3}α_{4}_{ +... } = ∑α_{i} α_{j}_{ }α_{k}_{ }_{ }= (-1)^{3}a_{3/} a_{0} i≠ j≠ k or, S3= (-1) ^{3} coeff. of x^{n-3}/coeff. of x^{n} ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... S _{n}=α_{1}α_{2}α_{3... }α_{n =}(-1)^{n}a_{n/} a_{0= }_{ } (-1)^{n} constant term/coeff. of x^{n} Here, S _{k} denotes the sum of the products of the roots taken k at a time.For example, S _{3} denotes the sum of the product of roots taken 3 at a time.PARTICULAR CASES: Quadratic Equation:If α and β are roots of the quadratic equation ax ^{2} + bx + c=0, thenα + β = -b/a α * β = c/a Cubic Equation:If α , β, γ are roots of a cubic equation ax ^{3} + bx^{2} + cx + d=0, thenα + β + γ = -b/a α β +β γ + γα = c/a α βγ = -d/a Biquadratic equation :If α , β, γ, δ are roots of a cubic equation ax ^{4}+ bx^{3} + cx^{2} + dx +e=0, thenα + β + γ + δ = -b/a α β +β γ + γδ +αγ + αδ+ βδ = c/a α β γ + αγδ +αβδ + βγδ = -d/a αβγδ = e/a |