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Class 9th Math
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Online Class For 9th Standard Students (CBSE) (English Medium)
About Lesson

The topics and subtopics covered in class 9 polynomials chapter 2 include:

  • Introduction
  • Polynomials in One Variable
  • Zeros of Polynomials
  • Remainder Theorem
  • Factorization of Polynomials
  • Algebraic Identities

Polynomial Definition

Polynomials are expressions with one or more terms with a non-zero coefficient. A polynomial can have more than one term. In the polynomial, each expression in it is called a term.  Suppose x2 + 5x + 2 is polynomial, then the expressions x2, 5x, and 2 are the terms of the polynomial.  Each term of the polynomial has a coefficient. For example, if 2x + 1 is the polynomial, then the coefficient of x is 2.

The real numbers can also be expressed as polynomials. Like 3, 6, 7, are also polynomials without any variables. These are called constant polynomials.  The constant polynomial 0 is called zero polynomial. The exponent of the polynomial should be a whole number. For example,  x-2 + 5x + 2, cannot be considered as a polynomial, since the exponent of x is -2, which is not a whole number.

The highest power of the polynomial is called the degree of the polynomial. For example, in x3 + y3 + 3xy(x + y), the degree of the polynomial is 3. For a non zero constant polynomial, the degree is zero. Apart from these, there are other types of polynomials such as:

  • Linear polynomial – of degree one
  • Quadratic Polynomial- of degree two
  • Cubic Polynomial – of degree three

This topic has been widely discussed in class 9 and class 10.

Example of polynomials are:

  • 20
  • x + y
  • 7a + b + 8
  • w + x + y + z
  • x+ x + 1
Quadratic Equation Algebraic Identities
Quadratic Formula & Quadratic Polynomial Degree Of A Polynomial
Class 9th Maths Polynomial

Polynomials in One Variable

Polynomials in one variable are the expressions which consist of only one type of variable in the entire expression.

Example of polynomials in one variable:

  • 3a
  • 2x2 + 5x + 15

Some important points in Polynomials Class 9 Chapter 2 are given below:

  • An algebraic expression p(x) = a0xn + a1xn-1 + a2xn-2 + … an is a polynomial where a0, a1, ………. an are real numbers and n is non-negative integer.
  • A term is either a variable or a single number or it can be a combination of variable and numbers.
  • The degree of the polynomial is the highest power of the variable in a polynomial.
  • A polynomial of degree 1 is called as a linear polynomial.
  • A polynomial of degree 2 is called a quadratic polynomial.
  • A polynomial of degree 3 is called a cubic polynomial.
  • A polynomial of 1 term is called a monomial.
  • A polynomial of 2 terms is called binomial.
  • A polynomial of 3 terms is called a trinomial.
  • A real number ‘a’ is a zero of a polynomial p(x) if p(a) = 0, where a is also known as root of the equation p(x) = 0.
  • A linear polynomial in one variable has a unique zero, a polynomial of a non-zero constant has no zero, and each real number is a zero of the zero polynomial.
  • Remainder Theorem: If p(x) is any polynomial having degree greater than or equal to 1 and if it is divided by the linear polynomial x – a, then the remainder is p(a).
  • Factor Theorem : x – c is a factor of the polynomial p(x), if p(c) = 0. Also, if x – c is a factor of p(x), then p(c) = 0.
  • The degree of the zero polynomial is not defined.
  • (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)
  • x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – xy – yz – zx)

Polynomials Class 9 Examples (Class 9th Maths Polynomial)

Example 1:

Write the coefficients of x  in each of the following:

  • 3x + 1
  • 23x2 – 5x + 1

Solution:

In 3x + 1, the coefficient of x is 3.

In 23x2 – 5x + 1, the coefficient of x is -5.

Example 2:

What are the degrees of following polynomials?

  1. 3a2 + a – 1
  2. 32x3 + x – 1

Solution: 

  1. 3a2 + a – 1 : The degree is 2
  2. 32x3 + x – 1 : The degree is 3

Polynomials Class 9 Important Questions(Class 9th Maths Polynomial)

  1. Find value of polynomial 2x2 + 5x + 1 at x = 3.
  2. Check whether x = -1/6 is zero of the polynomial p(a) = 6a + 1.
  3. Divide 3a2 + x – 1 by a + 1.
  4. Find value of k, if (a – 1) is factor of p(a) = ka2 – 3a + k.
  5. Factorise each of the following:
    1. 4x+ 9y2 + 16z2 + 12xy – 24yx – 16xz
    1. 2x2 + y2 + 8z2 – 2√2xy + 4√2yz – 8xz
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