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Maths Query > Unit > Algebra > Polynomials

# Relationship of Zeros and Coefficients of Polynomials

## Introduction

The detailed discussion about zeros of polynomials and how to represent a polynomial on a graph can be found in the chapter Zeros of Polynomial and its Geometrical Meaning. Also, we learnt how to find the value of zeros of polynomial graphically for a linear polynomial, a cubic polynomial and a biquadratic polynomial.

This chapter is about the relationships between zeros and coefficients of a quadratic polynomial, zeros and coefficients of cubic polynomial along with zeros and coefficients of a biquadratic polynomial. The zeros and coefficients of a polynomial is related to each other in finding sum and product of roots of the polynomial.

## Sum and product of zeros of quadratic polynomial

Consider a quadratic polynomial ax2 + bx + c. Let α and β are two zeros of the polynomial. Then, we can find the sum of zeros and product of zeros from coefficients of x and x2 with the following formulas for sum of zeros and product of zeros.

Formula

$\text{Sum of zeros}=–\frac{\text{coefficient of x}}{{\text{coefficient of x}}^{2}}$
$\mathrm{\alpha + \beta }=\frac{–b}{a}$
$\text{Product of zeros}=\frac{\text{constant of term}}{{\text{constant of term}}^{2}}$
$\mathrm{\alpha \beta }=\frac{c}{a}$

Example

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
$\text{Sum of zeros}=–\frac{\text{coefficient of x}}{{\text{coefficient of x}}^{2}}$
$\mathrm{\alpha + \beta }=\frac{–b}{a}$
$\left(-2\right) + \left(-3\right)=\frac{–5}{1}$
-5 = -5
$\text{Product of zeros}=\frac{\text{constant of term}}{{\text{constant of term}}^{2}}$
$\mathrm{\alpha \beta }=\frac{c}{a}$
$\mathrm{\left(-2\right)\left(-3\right)}=\frac{6}{1}$
6 = 6

## Sum and product of zeros of cubic polynomial

Now, consider a cubic polynomial p(x)=ax3 + bx2 + cx + d. Let α, β and γ are three zeros of polynomial. Sum of zeros and product of zeros can be found from the coefficients of x2 and x3 with the following formulas for sum of zeros and product of zeros.

Formula

$\text{Sum of zeros}=–\frac{{\text{coefficient of x}}^{2}}{{\text{coefficient of x}}^{3}}$
$\mathrm{\alpha + \beta + \gamma }=\frac{–b}{a}$
Sum of product of zeros taken two at a time $=\frac{\text{coefficient of x}}{{\text{coefficient of x}}^{9}}$
$\mathrm{\alpha \beta +\beta \gamma +\alpha \gamma }=\frac{c}{a}$
$\text{Product of zeros}=–\frac{\text{constant of term}}{{\text{coefficient of x}}^{3}}$
$\mathrm{\alpha \beta \gamma }=–\frac{d}{a}$

Example

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
$\text{Sum of zeros}=–\frac{{\text{coefficient of x}}^{2}}{{\text{coefficient of x}}^{3}}$
$\mathrm{\alpha + \beta + \gamma }=\frac{–b}{a}$
$\left(-1\right) + \left(2\right) + \left(-3\right)=–\frac{2}{1}$
-2 = -2
Sum of product of zeros taken two at a time $=\frac{\text{coefficient of x}}{{\text{coefficient of x}}^{9}}$
$\mathrm{\alpha \beta +\beta \gamma +\alpha \gamma }=\frac{c}{a}$
$\left(-1\right)\left(2\right) + \left(2\right)\left(-3\right) + \left(-3\right)\left(-1\right)=\frac{\left(-5\right)}{1}$
-2 -6 + 3 = -5
-5 = -5
$\text{Product of zeros}=–\frac{\text{constant of term}}{{\text{coefficient of x}}^{3}}$
$\mathrm{\alpha \beta \gamma }=–\frac{d}{a}$
$\left(-1\right)\left(2\right)\left(-3\right)=–\frac{\left(-6\right)}{1}$
6 = 6

## Sum and product of zeros of biquadratic polynomial

Polynomial p(x)=ax4 + bx3 + cx2 + dx + e, where a ≠ = 0 is a biquadratic polynomial. The graph of y = ax4 + bx3 + cx2 + dx + e intersects the x-axis. These coordinates are the only zeros of the biquadratic polynomial.
Consider a polynomial ax4 + bx3 + cx2 + dx + e. Let α, β, γ and δ are four zeros of the polynomial.
Sum of zeros and product of zeros can be found from the coefficients of x3 and x4 with the following formulas for sum of zeros and product of zeros.

Formula

$\text{Sum of zeros}=–\frac{{\text{coefficient of x}}^{3}}{{\text{coefficient of x}}^{4}}$
$\mathrm{\alpha + \beta + \gamma + \delta }=–\frac{b}{a}$
Sum of product of zeros taken two at a time $=\frac{{\text{coefficient of x}}^{2}}{{\text{coefficient of x}}^{4}}$
$\mathrm{\alpha \beta + \beta \gamma + \delta \gamma + \alpha \delta + \delta \beta + \gamma \alpha }=\frac{c}{a}$
Sum of product of zeros taken three at a time $=–\frac{\text{coefficient of x}}{{\text{coefficient of x}}^{4}}$
$\mathrm{\alpha \beta \gamma + \beta \gamma \delta + \alpha \beta \delta + \alpha \gamma \delta }=–\frac{d}{a}$
$\text{Product of zeros}=\frac{\text{constant of term}}{{\text{coefficient of x}}^{3}}$
$\mathrm{\alpha \beta \gamma \delta }=\frac{e}{a}$

Example

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
$\text{Sum of zeros}=–\frac{{\text{coefficient of x}}^{3}}{{\text{coefficient of x}}^{4}}$
$\mathrm{\alpha + \beta + \gamma + \delta }=\frac{–b}{a}$
$\left(1\right) + \left(3\right) + \left(5\right) + \left(-3\right)=–\frac{\left(-6\right)}{1}$
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) $=–\frac{54}{1}$
$15-45-9-15=–\frac{54}{1}$
-54 = -54
$\text{Product of zeros}=\frac{\text{constant of term}}{{\text{coefficient of x}}^{3}}$
$\mathrm{\alpha \beta \gamma \delta }=\frac{e}{a}$
$\left(1\right)\left(3\right)\left(5\right)\left(-3\right)=–\frac{45}{1}$
-45 = -45

### 1) What is the relationship between coefficient of a quadratic polynomial and its zero?

$\text{Sum of zeros}=-\frac{\text{coefficient of x}}{{\text{coefficient of x}}^{2}}$
$\text{Product of zeros}=\frac{\text{constant of term}}{{\text{constant of term}}^{2}}$

## Solved Examples

### 1) Verify its relationship between coefficient of polynomial and its zeros for a polynomial x2 - 9.

Solution

Here p(x) = x2 - 9
compare it with ax2 + bx + c
p(x) = x2 + 0x - 9
Here a = 1, b = 0, c = -9
$\mathrm{Sum of zeros}=\frac{-b}{a}$
-3 + 3 = 0
$=-\frac{0}{1}$
$=-\frac{b}{a}$
$\mathrm{Product of zeros}=\frac{c}{a}$
= (-3)(3) = -9
$=\frac{-9}{1}$
$=\frac{c}{a}$

### 2) Form a quadratic polynomial whose sum of zeros is 5 and product of zeros is 6.

Solution

Sum of zeros = 5
Product of zeros = 6
As we know, quadratic polynomial is the form of x2 - (sum of zeros)x + product of zeros
By putting the above values, it becomes x2 - 5x + 6
Hence, x2 - 5x + 6 is a quadratic polynomial.

### 3) Verify relationship between coefficient of polynomial and its zeros if x3 - 9x2 - 12x + 20 has zeros -2, 1 and 10.

Solution

Here, compare x3 - 9x2 - 12x + 20 with ax3 + bx2 + cx + d
a = 1, b = -9, c = -12, d = 20
Zeros are -2, 1 and 10 (given)
α = -2
β = 1
γ = 10
$\text{Sum of zeros}=-\frac{{\text{coefficient of x}}^{2}}{{\text{coefficient of x}}^{3}}$
∴ α + β + γ = -2 + 1 + 10
= 9
$=-\frac{\left(-9\right)}{1}$
$=-\frac{b}{a}$
Sum of product of zeros taken two at a time $=\frac{\text{coefficient of x}}{{\text{coefficient of x}}^{9}}$
αβ + βγ + γα = (-2)(1) + (1)(10) + (10)(-2)
= -2 + 10 -20
= -12
$=\frac{\left(-12\right)}{1}$
$=\frac{c}{a}$
$\text{Product of zeros}=-\frac{\text{constant of term}}{{\text{coefficient of x}}^{3}}$
αβγ = (-2)(1)(10)
= -20
$=-\frac{20}{1}$
$=-\frac{d}{a}$

Last updated on: 15-06-2024