Relationship between Zeroes and Coefficients of a Polynomial

Introduction

The zeroes of a polynomial and its coefficients have a specific relationship that helps in analyzing polynomial equations without explicitly solving them. This relationship is particularly useful for quadratic and cubic polynomials, where the sum and product of the zeroes can be directly connected to the coefficients of the polynomial. Here, we explore this relationship with examples for better understanding.

Zero of a Linear Polynomial

For a linear polynomial of the form $\boldsymbol{ax + b}$ , the zero is given by:

$\displaystyle\boldsymbol{\text{Zero} = -\frac{b}{a}}$

This simple formula connects the coefficient of $\boldsymbol{x}$ ( $\boldsymbol{a}$ ) with the constant term ( $\boldsymbol{b}$ ).

Relationship for Quadratic Polynomials

For a quadratic polynomial $\boldsymbol{p(x) = ax^2 + bx + c}$ , where $\boldsymbol{a \neq 0}$ , let $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ be the zeroes. Then, the sum and product of the zeroes can be expressed in terms of the coefficients as follows:

Sum of Zeroes:
$\displaystyle\boldsymbol{\alpha + \beta = -\frac{b}{a}}$
Here, $\displaystyle\boldsymbol{-\frac{b}{a}}$ is the negative of the coefficient of $\boldsymbol{x}$ divided by the coefficient of $\boldsymbol{x^2}$ .
Product of Zeroes:
$\displaystyle\boldsymbol{\alpha \beta = \frac{c}{a}}$
This represents the constant term ( $\boldsymbol{c}$ ) divided by the coefficient of $\boldsymbol{x^2}$ .

Derivation of Sum and Product Formulas

In general, if $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ are the zeroes of the quadratic polynomial $\boldsymbol{p(x) = ax^2 + bx + c}$ , where $\boldsymbol{a \neq 0}$ , then $\boldsymbol{x - \alpha}$ and $\boldsymbol{x - \beta}$ are factors of $\boldsymbol{p(x)}$ . Thus, we can write:

$\boldsymbol{ax^2 + bx + c = k(x - \alpha)(x - \beta)}$

where $\boldsymbol{k}$ is a constant.

Expanding this, we get:

$\boldsymbol{ax^2 + bx + c = k[x^2 - (\alpha + \beta)x + \alpha \beta]}$

Comparing the coefficients of $\boldsymbol{x^2}$ , $\boldsymbol{x}$ , and the constant term on both sides, we obtain:

$\boldsymbol{a = k}$
$\boldsymbol{b = -k(\alpha + \beta)}$
$\boldsymbol{c = k \alpha \beta}$

Since $\boldsymbol{k = a}$ , dividing both equations by $\boldsymbol{a}$ gives:

$\displaystyle\boldsymbol{\alpha + \beta = -\frac{b}{a}}$ $\displaystyle\boldsymbol{\alpha \beta = \frac{c}{a}}$

Example for Quadratic Polynomials

Consider the polynomial $\boldsymbol{p(x) = 2x^2 - 8x + 6}$ .

Factorizing: We split the middle term to get: $\boldsymbol{2x^2 - 8x + 6 = 2(x - 1)(x - 3)}$
Zeroes: The zeroes are $\boldsymbol{x = 1}$ and $\boldsymbol{x = 3}$ .
Sum of Zeroes: $\displaystyle\boldsymbol{1 + 3 = 4 = -\frac{-8}{2} = -\frac{\text{Coefficient of } x}{\text{Coefficient of } x^2}}$
Product of Zeroes: $\displaystyle\boldsymbol{1 \times 3 = 3 = \frac{6}{2} = \frac{\text{Constant term}}{\text{Coefficient of } x^2}}$

This example confirms that the sum and product of the zeroes follow the derived relationships.

Further Examples for Quadratic Polynomials

Example 2: Consider $\boldsymbol{p(x) = x^2 + 7x + 10}$ .
- Zeroes: Solving, we find zeroes at $\boldsymbol{x = -2}$ and $\boldsymbol{x = -5}$ .
- Sum of Zeroes: $\displaystyle\boldsymbol{-2 + (-5) = -7 = -\frac{7}{1}}$
- Product of Zeroes: $\displaystyle\boldsymbol{(-2)(-5) = 10 = \frac{10}{1}}$
Example 3: For the polynomial $\boldsymbol{p(x) = x^2 - 3}$ , we find zeroes at $\boldsymbol{\sqrt{3}}$ and $\boldsymbol{-\sqrt{3}}$ .
- Sum of Zeroes: $\displaystyle\boldsymbol{\sqrt{3} + (-\sqrt{3}) = 0 = -\frac{0}{1}}$
- Product of Zeroes: $\displaystyle\boldsymbol{\sqrt{3} \cdot (-\sqrt{3}) = -3 = \frac{-3}{1}}$

Relationship for Cubic Polynomials

For a cubic polynomial $\boldsymbol{p(x) = ax^3 + bx^2 + cx + d}$ , let $\boldsymbol{\alpha}$ , $\boldsymbol{\beta}$ , and $\boldsymbol{\gamma}$ be the zeroes. The relationships between the zeroes and coefficients are as follows:

Sum of Zeroes:
$\displaystyle\boldsymbol{\alpha + \beta + \gamma = -\frac{b}{a}}$
Sum of Product of Zeroes (taken two at a time):
$\displaystyle\boldsymbol{\alpha \beta + \beta \gamma + \gamma \alpha = \frac{c}{a}}$
Product of Zeroes:
$\displaystyle\boldsymbol{\alpha \beta \gamma = -\frac{d}{a}}$

Example for Cubic Polynomial

Consider the polynomial $\boldsymbol{p(x) = 3x^3 - 5x^2 - 11x - 3}$ .

Given zeroes: $\displaystyle\boldsymbol{3}$ , $\boldsymbol{-1}$ , and $\boldsymbol{-\frac{1}{3}}$ .
Verifying relationships:
- Sum of Zeroes: $\displaystyle\boldsymbol{3 + (-1) + \left(-\frac{1}{3}\right) = \frac{-5}{3}}$
- Sum of Product of Zeroes (two at a time): $\displaystyle\boldsymbol{(3)(-1) + (-1)\left(-\frac{1}{3}\right) + \left(-\frac{1}{3}\right)(3) = \frac{11}{3}}$
- Product of Zeroes: $\displaystyle\boldsymbol{3 \cdot (-1) \cdot \left(-\frac{1}{3}\right) = 1 = -\frac{-3}{3}}$

This confirms the relationships for cubic polynomials as well.

General Formulas

For a quadratic polynomial $\boldsymbol{ax^2 + bx + c}$ with zeroes $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ :

$\displaystyle\boldsymbol{\alpha + \beta = -\frac{b}{a}}$
$\displaystyle\boldsymbol{\alpha \beta = \frac{c}{a}}$

For a cubic polynomial $\boldsymbol{ax^3 + bx^2 + cx + d}$ with zeroes $\boldsymbol{\alpha}$ , $\boldsymbol{\beta}$ , and $\boldsymbol{\gamma}$ :

$\displaystyle\boldsymbol{\alpha + \beta + \gamma = -\frac{b}{a}}$
$\displaystyle\boldsymbol{\alpha \beta + \beta \gamma + \gamma \alpha = \frac{c}{a}}$
$\displaystyle\boldsymbol{\alpha \beta \gamma = -\frac{d}{a}}$

FAQs

What are some practical applications of the relationship between zeroes and coefficients?admin2024-11-26T13:28:32+05:30

What are some practical applications of the relationship between zeroes and coefficients?

This concept is widely used in algebra, calculus, and even fields like physics and engineering. For example, in circuit analysis, certain electrical parameters can be modeled using polynomial equations, and understanding the relationships between zeroes and coefficients can help solve complex problems efficiently.

How does this concept apply to higher-degree polynomials?admin2024-11-26T13:28:14+05:30

How does this concept apply to higher-degree polynomials?

For higher-degree polynomials (beyond cubic), similar relationships exist. The sum of zeroes, the sum of products of zeroes taken two at a time, and so on, can be related to the coefficients. However, the exact relationships depend on the polynomial’s degree and are more complex as the degree increases.

Can this relationship be used to construct a polynomial if we know its zeroes?admin2024-11-26T13:27:57+05:30

Can this relationship be used to construct a polynomial if we know its zeroes?

Yes, knowing the zeroes and their relationships with the coefficients allows us to construct polynomials. For example, if the zeroes of a quadratic polynomial are given as $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ , we can write it as:
$\boldsymbol{p(x) = a(x - \alpha)(x - \beta)}$
Expanding this will provide a polynomial with the desired zeroes.

Why is the relationship between zeroes and coefficients important?admin2024-11-26T13:26:07+05:30

Why is the relationship between zeroes and coefficients important?

This relationship allows us to determine properties of a polynomial without fully solving it. It is useful in factoring polynomials, solving equations, and understanding the behavior of polynomial functions in graphing and analysis.

What is the zero of a linear polynomial?admin2024-11-26T13:25:47+05:30

What is the zero of a linear polynomial?

For a linear polynomial $\boldsymbol{p(x) = ax + b}$ , the zero is:
$\displaystyle\boldsymbol{x = -\frac{b}{a}}$

Is there a relationship involving the sum of products of zeroes taken two at a time in a cubic polynomial?admin2024-11-26T13:25:08+05:30

Is there a relationship involving the sum of products of zeroes taken two at a time in a cubic polynomial?

Yes, in a cubic polynomial $\boldsymbol{p(x) = ax^3 + bx^2 + cx + d}$ , the sum of the products of zeroes taken two at a time is:
$\displaystyle\boldsymbol{\alpha \beta + \beta \gamma + \gamma \alpha = \frac{c}{a}}$

How can we find the product of zeroes in a cubic polynomial?admin2024-11-26T13:24:28+05:30

How can we find the product of zeroes in a cubic polynomial?

For a cubic polynomial $\boldsymbol{p(x) = ax^3 + bx^2 + cx + d}$ , the product of the zeroes $\boldsymbol{\alpha}$ , $\boldsymbol{\beta}$ , and $\boldsymbol{\gamma}$ is given by:
$\displaystyle\boldsymbol{\alpha \beta \gamma = -\frac{d}{a}}$

How is the sum of zeroes related to the coefficients in a cubic polynomial?admin2024-11-26T13:24:02+05:30

How is the sum of zeroes related to the coefficients in a cubic polynomial?

For a cubic polynomial $\boldsymbol{p(x) = ax^3 + bx^2 + cx + d}$ , if $\boldsymbol{\alpha}$ , $\boldsymbol{\beta}$ , and $\boldsymbol{\gamma}$ are the zeroes, then:
$\displaystyle\boldsymbol{\alpha + \beta + \gamma = -\frac{b}{a}}$
This is the sum of zeroes expressed in terms of the coefficients of $\boldsymbol{x^3}$ and $\boldsymbol{x^2}$ .

What is the relationship between zeroes and coefficients of a quadratic polynomial?admin2024-11-26T13:17:54+05:30

What is the relationship between zeroes and coefficients of a quadratic polynomial?

For a quadratic polynomial $\boldsymbol{p(x) = ax^2 + bx + c}$ , if $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ are the zeroes, then:
$\displaystyle\boldsymbol{\alpha + \beta = -\frac{b}{a}}$ and $\displaystyle\boldsymbol{\alpha \beta = \frac{c}{a}}$
where $\boldsymbol{\alpha + \beta}$ represents the sum of zeroes, and $\boldsymbol{\alpha \beta}$ represents the product of zeroes.