Short Notes - Polynomials Class 9 Notes | EduRev

Mathematics (Maths) Class 9

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Class 9 : Short Notes - Polynomials Class 9 Notes | EduRev

The document Short Notes - Polynomials Class 9 Notes | EduRev is a part of the Class 9 Course Mathematics (Maths) Class 9.
All you need of Class 9 at this link: Class 9

Facts that Matter 

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

VARIABLE
A symbol which 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, Short Notes - Polynomials Class 9 Notes | EduRev is called an algebraic expression. 

Example: Short Notes - Polynomials Class 9 Notes | EduRev 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 Short Notes - Polynomials Class 9 Notes | EduRev

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

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 co-efficient. 

DEGREE OF A POLYNOMIAL

I . In case of a polynomial involving in 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 case of a polynomial involving in 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 number of its 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.

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