Relations between Roots and Coefficients
If α₁, α₂, … αn are roots of the equation f(x) = aₒ xⁿ + a₁ x ⁿˉ¹ + a₂ x ⁿˉ² +. . . + an = 0 then f(x) = aₒ(x – α₁) (x – α₂) ….. (x – αn)
aₒ xⁿ + a₁ x ⁿˉ¹ + a₂ x ⁿˉ² +. . . + an
= aₒ(x – α₁) (x – α₂) ….. (x – αn)
Comparing the coefficient of like powers of x on both sides
We get,
S₁ = α₁ + α₂ + ……. + αn
Or, \(\Sigma {{\alpha }_{i}}=-\frac{{{a}_{1}}}{{{a}_{0}}}\)
S₂ = α₁ α₂ + α₁ α₂ + …….
Or, \(\sum\limits_{i=j}{{{\alpha }_{i}}{{\alpha }_{j}}}={{\left( -1 \right)}^{2}}\frac{{{a}_{2}}}{{{a}_{0}}}\)
Or, \({{S}_{2}}=\frac{{{\left( -1 \right)}^{2}}coeff\,of\,{{x}^{n-2}}}{coeff\,of\,{{x}^{n}}}\)
S₃ = α₁ α₂ α + α₂ α₂ α₂ + …….
Or , \(\sum\limits_{i\ne j\ne k}{{{\alpha }_{i}}{{\alpha }_{j}}{{\alpha }_{k}}}={{\left( -1 \right)}^{3}}\frac{{{a}_{3}}}{{{a}_{0}}}\)
Or, \({{S}_{3}}=\frac{{{\left( -1 \right)}^{3}}coeff\,of\,{{x}^{n-3}}}{coeff\,of\,{{x}^{n}}}\)
………….. …………….. …………….
………….. …………….. …………….
Sn = α₁ α₂ α₃ …. αn \(={{\left( -1 \right)}^{n}}\frac{{{a}_{n}}}{{{a}_{0}}}={{\left( -1 \right)}^{n}}\frac{const\,term}{coeff\,of\,{{x}^{n}}}\)
Here, Sk denotes the sum of the products of the roots taken k at a time.
Particular cases:
Quadratic equation:
If α, β are roots of the quadratic equation ax² + bx + c = 0, then
α + β \(=-\frac{b}{a}\) and α β \(=\frac{c}{a}\)
Cubic equation:
If α, β, γ are roots of a cubic equation
ax³ + bx² + cx +d = 0, then
α + β + γ \(=-\frac{b}{a}\)
αβ + βγ + γδ \(={{\left( -1 \right)}^{2}}\frac{c}{a}=\frac{c}{a}\)
And, αβγ\(={{\left( -1 \right)}^{3}}\frac{d}{a}=-\frac{d}{a}\).
Biquadratic equation:
If α, β, γ, δ are roots of the biquadratic equation ax⁴ + cx²+dx + e = 0, then
S₁ = α + β + γ + δ
= (-1)
S₂ = αβ + βγ + αδ + βγ + βδ + γδ
\(\,={{\left( -1 \right)}^{2}}\frac{c}{a}=\frac{c}{a}\)
S₂ = (α + β) (γ + δ) + αβ + γδ
\(\,\,\,=\frac{c}{a}\)
S₃ = αβγ + βγδ + γδα + αβγ
\(\,={{\left( -1 \right)}^{3}}\frac{d}{a}=-\frac{d}{a}\)
S₃ = αβ(γ + δ) + γδ(α + β)
\(\,\,\,\,\,\,=-\frac{d}{a}\)
S₄ = αβγδ
\(\,\,\,\,\,\,={{\left( -1 \right)}^{4}}\frac{e}{a}=\frac{e}{a}\)