Short Notes: Polynomials

# Short Notes: Polynomials Notes | Study Mathematics (Maths) Class 9 - Class 9

## Document Description: Short Notes: Polynomials for Class 9 2022 is part of Mathematics (Maths) Class 9 preparation. The notes and questions for Short Notes: Polynomials have been prepared according to the Class 9 exam syllabus. Information about Short Notes: Polynomials covers topics like Polynomial, Variable, Algebraic Expression and Short Notes: Polynomials Example, for Class 9 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Short Notes: Polynomials.

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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)(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 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.

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