# Relations between Roots and Coefficients

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:

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$$={{\left( -1 \right)}^{3}}\frac{d}{a}=-\frac{d}{a}$$.

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

$$\,\,\,\,\,\,={{\left( -1 \right)}^{4}}\frac{e}{a}=\frac{e}{a}$$