Class 10 Mathematics Chapter 2 - Polynomials Relationship between Zeroes and Coefficients of a Polynomial

Class 10 Mathematics  Chapter 2 - Polynomials  Relationship between Zeroes and Coefficients of a Polynomial Notes and explanation

Relationship between Zeroes and Coefficients of a Polynomial

Polynomials consist of variables and constants raised to whole number exponents. Algebraic expressions can be categorized according to their degree, including linear, quadratic and cubic polynomial. The degree of a polynomial provides straightforward method for determine the polynomial's zeros which is equal to the number of zeros in its equation,

Relationship between Zeroes and Coefficients of a Linear Polynomial

If α is the zeros of a quadratic polynomial ax+b, then

The relationship between the zeros and coefficient of a linear polynomial are:

The zero of a linear polynomial is determined by solving the equation ax + b = 0 and It is denoted as “x₀”.

The sum of the reciprocals of the zeros:

A linear polynomial has only one zeros which is α. So, the sum of the reciprocals of the zeros is simply the reciprocal of that single zeros: 1/ α

Relationship between Zeroes and Coefficients of a Quadratic Polynomial

The relationship between the zeroes and the coefficients of a quadratic polynomial is defined by the quadratic formula and Viète's formulas. The general form of a quadratic polynomial is:

P(x) = ax² + bx + c

Here, a, b and c are coefficients.

We can express relationship between the zeroes (α and β) and the coefficients of a quadratic polynomial by two methods:

(1) By using Viète's Formulas:

If α and β are the zeros of a quadratic polynomial ax² + bx + c

(i) The sum of the zeroes is equal to the negation of the coefficient of the linear term (b) divided by the coefficient of the quadratic term (a):

(ii) The product of the zeroes is equal to the constant term (c) divided by the coefficient of the quadratic term (a):

These formulas allow you to directly relate the values of α and β to the coefficients a, b, and c.

(2) By using the Quadratic Formula:

Zeroes of a Quadratic Polynomial:

A quadratic polynomial can have zero, one or two real zeroes which depend on the value inside the square root of the quadratic equation which called Discriminant. The zeroes are denoted as ‘α’ and ‘β’ and they are the values of 'x' for which P(x) = 0.

The quadratic formula is used to directly find the values of the roots (zeroes) of the quadratic polynomial:

For a quadratic polynomial ax² + bx + c this formula provides the values of x where the polynomial equals zero.

The plus/minus sign indicates that there are two solutions, corresponding to the two roots, α and β.

Discriminant:

The discriminant of a quadratic polynomial, denoted as 'D' is calculated as:

D = b² — 4ac, is used to determine the nature and number of solutions:

1. If D > 0, the quadratic polynomial has two distinct real zeroes; α and β
2. If D = 0, the quadratic polynomial has one real zero; α = β
3. If D < 0, the quadratic polynomial has no real zeroes and the zeroes α and β are complex.

Difference between both methods:

Viète's formulas provide relationships between the coefficients a, b, and c and the roots α and β, while the quadratic formula directly provides the values of α and β based on the coefficients.

If α and β are the zeros of a quadratic polynomial:

ax² + bx + c

Where,

a, b and c - are the coefficients of the polynomial.

x - represents the variable.

α and β are the solution of the quadratic equation ax² + bx + c = 0

We want to find the sum of the reciprocals of these zeros,

Relationship between Zeroes and Coefficients of a Cubic Polynomial

If α, β and γ are the zeros of a cubic polynomial:

(i) The sum of the zeros is equal to the negation of the coefficient of the quadratic term (b) divided by the coefficient of the cubic term (a):

(ii) The product of the zeros taken two at a time is equal to the negation of the coefficient of the linear term (c) divided by the coefficient of the cubic term (a):

(iii) The product of all three zeros is equal to the negation of the constant term (d) divided by the coefficient of the cubic term (a):

These formulas allow to relate the values of α, β, and γ to the coefficients a, b, c and d.

If the sum and product of zeros are given then, the cubic equation is given by:

The sum of the reciprocals of the zeros:

If α, β and γ are the zeros of a cubic polynomial:

ax³ + bx² + cx + d

Where,

a, b, c and d - are the coefficients of the polynomial.

x - Represents the variable.

α, β and γ are the solution of the quadratic equation ax³ + bx² + cx + d = 0

We want to find the sum of the reciprocals of these zeros,

Example 1:

Find the zeroes of the quadratic polynomial x² + 7x + 10, and verify the relationship between the zeroes and the coefficients.

Solution: Given Polynomial

x² + 7x + 10

In this equation coefficients are:  a = 1 ; b = 7 ; c = 10

Finding zeros:

Example-2:

Find the zeroes of the polynomial x² – 3 and verify the relationship between the zeroes and the coefficients.

Solution: We know that a² — b² = ( a — b ) ( a + b )

Example-3:

Find a quadratic polynomial, the sum and product of whose zeroes are – 3 and 2, respectively.

Solution: