Polynomials Class 9 Notes Maths Chapter 2

Polynomial

• An algebraic expression, in which the variables involved have only whole number powers, is called a polynomial.
• Polynomials of degree 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively.
• Polynomials containing one, two and three non-zero terms are called monomial, binomial and trinomial respectively.
• If f(x) be a polynomial of degree n ≥ 1 and let ‘a’ be any real number, then f(a) is the remainder for ‘f(x) being divided by (x – a)’.
• Let f(x) be a polynomial of degree n ≥ 1, then (x – a) is a factor of f(x) provided f(a) = 0. Also if (x + a) is a factor of f(x), then f(–a) = 0

Algebraic Identities

• x³ + y³ = (x + y)(x² – xy + y²)
• x³ – y³ = (x – y)(x² + xy + y²)
• x³ + y³ + z³ – 3xyz = (x + y + z)(x² + y² + z² – xy – yz – zx)
• If x + y + z = 0, then x³ + y³ + z³ = 3xyz
• (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx
• (x + y)³ = x³ + y³ + 3xy (x + y)
• (x – y)³ = x³ – y³ – 3xy (x – y)

Variable

A symbol that can be assigned different numerical values is known as a variable. Variables are generally denoted by x, y, z, p, q, r, s, etc.

Constant

A symbol having a fixed value is called a constant, e.g. 8, 5, 9, p, a, b, c, etc. are constants.

Note: The values of constants remain the same throughout a particular situation, but the value of a variable can keep changing. 

Algebraic Expression

A combination of constants and variables, connected by some or all the basic operations +, –, x,  is called an algebraic expression. 

Example:  is an algebraic expression.

Terms

Various parts of an algebraic expression separated by (+) or (–) operations are called terms.
Examples: (i) In the above algebraic expression, terms are 7, 8x, –6x²y and

(ii) Various terms of 3p⁴ – 6q² + 8r³s – 2pq + 6s³ are: 3p⁴, –6q², 8r³s, –2pq and 6s³ 

Algebraic Expression

Polynomial

An algebraic expression in which the variables involved have non-negative integral powers is called a polynomial.

Examples: (i) x³ + x² – 4x – 7;
(ii) 3p³ + 5p – 9;
(iii) x² + 2x; etc. are all polynomials

Note: I. Each variable in a polynomial has a whole number as its exponent.

II. Each term of a polynomial has a coefficient. 

Degree of a Polynomial

I. In the case of a polynomial involving one variable, the highest power of the variable is called the degree of the polynomial.
Example: The degree of x⁵ – 2x³ + x is 5.

II. In the case of a polynomial involving more than one variable, the highest sum of exponents of variables in any term is called the degree of the polynomial.
Example: The degree of p² – 6p⁶q + 5p²q³ – 3q⁴ is 7
(∵ The term –p⁶q has the sum of exponents of p and q as 6 + 1, i.e. 7)

Types of Polynomials

A [On the basis of the number of terms]

(i) Monomial: A polynomial containing only one non-zero term is called a monomial.

(ii) Binomial: A polynomial containing two non-zero terms is called a binomial.

(iii) Trinomial: A polynomial containing three non-zero terms is called a trinomial.

Note: I. A monomial containing a constant only is called a constant polynomial.

II. A monomial containing its term as zero only is called a zero polynomial.

III. If we add polynomials, we get a polynomial.

IV. If we multiply polynomials, we get a polynomial.

B [On the basis of its degree] 

(i) Linear Polynomial: A polynomial of degree 1 is called a linear polynomial.

(ii) Quadratic Polynomial: A polynomial of degree 2 is called a quadratic polynomial.

(iii) Cubic Polynomial: A polynomial of degree 3 is called a cubic polynomial.

(iv ) Biquadratic Polynomial: A polynomial of degree 4 is called a biquadratic polynomial.