Relationship of Zeros and Coefficients of Polynomials

Consider a polynomial x² + 5x + 6
Zeros of x² + 5x + 6 are -2 and -3
Here, coefficient of x² = 1
coefficient of x = 5
constant term = 6
Therefore, a = 1, b = 5, c = 6
Here, α = -2 and β = -3
$\text{Sum of zeros}=–\frac{\text{coefficient of x}}{{\text{coefficient of x}}^2}$
$\mathrm{\alpha \; +\; \beta }=\frac{–b}{a}$
$\mathrm{(-2)\; +\; (-3)}=\frac{–5}{1}$
-5 = -5
$\text{Product of zeros}=\frac{\text{constant of term}}{{\text{constant of term}}^2}$
$\mathrm{\alpha \; \beta }=\frac{c}{a}$
$\mathrm{(-2)(-3)}=\frac{6}{1}$
6 = 6

Consider a cubic polynomial x³ + 2x² – 5x – 6
Zeros of x³ + 2x² – 5x – 6 are -1, 2 and -3
Here, coefficient of x³ = 1
coefficient of x² = 2
coefficient of x = -5
constant term = -6
Therefore, a = 1, b = 2, c = -5, d = -6
Here, α = -1, β = 2 and γ=-3
$\text{Sum of zeros}=–\frac{{\text{coefficient of x}}^2}{{\text{coefficient of x}}^3}$
$\mathrm{\alpha \; +\; \beta \; +\; \gamma }=\frac{–b}{a}$
$\mathrm{(-1)\; +\; (2)\; +\; (-3)}=–\frac{2}{1}$
-2 = -2
Sum of product of zeros taken two at a time $=\frac{\text{coefficient of x}}{{\text{coefficient of x}}^9}$
$\mathrm{\alpha \; \beta +\beta \; \gamma +\alpha \; \gamma }=\frac{c}{a}$
$\mathrm{(-1)(2)\; +\; (2)(-3)\; +\; (-3)(-1)}=\frac{\mathrm{(-5)}}{1}$
-2 -6 + 3 = -5
-5 = -5
$\text{Product of zeros}=–\frac{\text{constant of term}}{{\text{coefficient of x}}^3}$
$\mathrm{\alpha \; \beta \; \gamma }=–\frac{d}{a}$
$\mathrm{(-1)(2)(-3)}=–\frac{\mathrm{(-6)}}{1}$
6 = 6

Let x⁴ – 6x³ – 4x² + 54x – 45 be a biquadratic polynomial
Zeros of x⁴ – 6x³ – 4x² + 54x – 45 are 1, 3, 5 and -3
Here, coefficient of x⁴ = 1
coefficient of x³ = -6
coefficient of x² = -4
coefficient of x = 54
constant term = -45
∴ a = 1, b = -6, c = -4, d = 54, e = -45
Let zeros be, α = 1, β = 3, γ=5 and δ=-3
∴ $\text{Sum of zeros}=–\frac{{\text{coefficient of x}}^3}{{\text{coefficient of x}}^4}$
$\mathrm{\alpha \; +\; \beta \; +\; \gamma \; +\; \delta }=\frac{–b}{a}$
$\mathrm{(1)\; +\; (3)\; +\; (5)\; +\; (-3)}=–\frac{\mathrm{(-6)}}{1}$
6 = 6
Sum of product of zeros taken two at a time $=\frac{{\text{coefficient of x}}^2}{{\text{coefficient of x}}^4}$
$\mathrm{\alpha \; \beta \; +\; \beta \; \gamma \; +\; \delta \; \gamma \; +\; \alpha \; \delta \; +\; \delta \; \beta \; +\; \gamma \alpha }=\frac{c}{a}$
(1)(3) + (3)(5) + (5)(-3) + (-3)(1) + (-3)(3) + (5)(1) $=\frac{-4}{1}$
3 + 15 – 15 – 3 – 9 + 5 = -4
-4 = -4
Sum of product of zeros taken three at a time $=–\frac{\text{coefficient of x}}{{\text{coefficient of x}}^4}$
$\mathrm{\alpha \beta \gamma \; +\; \beta \gamma \delta \; +\; \alpha \beta \delta \; +\; \alpha \gamma \delta }=–\frac{d}{a}$
(1)(3)(5)+(3)(5)(-3)+(1)(3)(-3)+(1)(5)(-3) $=–\frac{54}{1}$
$\mathrm{15-45-9-15}=–\frac{54}{1}$
-54 = -54
$\text{Product of zeros}=\frac{\text{constant of term}}{{\text{coefficient of x}}^3}$
$\mathrm{\alpha \; \beta \; \gamma \; \delta }=\frac{e}{a}$
$\mathrm{(1)(3)(5)(-3)}=–\frac{45}{1}$
-45 = -45