Some Basic Definitions

Polynomials

An expression of the form

in which a_i are numbers belonging to some number system and 'n' is a non-negative integer, is called a polynomial. a_i are called the coefficients of if a_o ≠ 0, the polynomial is said to be of degree n.

The value of the polynomial

If x is replaced by a number from a number system to which a_i belong, we get a number called a value of a polynomial.

Notation

Usually a polynomial is denoted by P(x) and if k is any number then P(k) denotes the value of P(x) at x=k. Then a polynomial can be used to define a function with x or y or any letter of the alphabet. The coefficient of the polynomial may belong to any system of numbers.

The degree of a polynomial

If a_o ≠ 0, then the polynomial P(x) or f(x) is of degree 'n' i.e., it is the highest power of the variable x in the rational and integral polynomial. A polynomial of degree 2 is called a quadratic polynomial.

Polynomial equation

An equation in a single variable is called a polynomial equation of degree 'n' if it is an equation of the form

In other words, an equation is a statement of equality between two algebraic expressions. It is satisfied by a limited number of values of the variable.

Example:

i) 3x + 5 = 8, is satisfied by the value of x = 1.

ii) x² - 4 = 0, is satisfied by the value of x = 2 or x = -2.

Algebraic identity

An algebraic identity is a statement of equality between two algebraic expressions, but it is satisfied for all values of the variable.

Example:

is satisfied for all the values of x. The sign is used to distinguish an identity from an equation.

Quadratic equation

An equation of the type

where a, b, c are constants is called a quadratic equation in the variable x or an equation of the second degree.

In the above equation,

a is the coefficient of x²

b is the coefficient of x

and c is the constant term or absolute term.

Examples:

are called Pure quadratic equation.

are called adfected quadratic equations.

are called adfected quadratic equations.

ax² + bx + c = 0 is called a general quadratic equation, a, b, c are

Methods of solving quadratic equations

There are four methods of solving quadratic equations.

i) By factorization

ii) By completing the squares

iii) By using the formula

iv) By graphing

Roots of a quadratic equation

A root of the equation f(x) = 0 is that value or values of x which make f(x) = 0. In other words, x = a or x = b are said to be the root of f(x) = 0, if f(a) = 0, and f(b) = 0

i.e., in f(x) = 0, replace x either by a or by b.

Solution of an equation

The determination of all the roots of a given equation is called the solution of the equation.