CHAPTER 2 POLYNOMIALS

 Polynomial derived from the words “poly” which means “many” and the word “nominal” which means “term”. In maths, a polynomial expression consists of variables that are also known as indeterminates and coefficients. The coefficients involve the operations of subtraction, addition, non-negative integer exponents of variables, and multiplication. A detailed polynomials Class 9 notes are provided here along with some important questions so that students can understand the concept easily.

Polynomials Class 9 Topics

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
  • x2 + x + 1

  • Polynomials in One Variable

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

    Example of polynomials in one variable:

    • 3a
    • 2x2 + 5x + 15

    Polynomial Class 9 Notes

    To prepare for class 9 exams, students will require notes to study. These notes are of great help when they have to revise chapter 2 polynomials before the exam. The note here provides a brief of the chapter so that students find it easy to have a glance at once. The key points covered in the chapter have been noted. Go through the points and solve problems based on them.

    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 variables and numbers.
    • The degree of the polynomial is the highest power of the variable in a polynomial.
    • A polynomial of degree 1 is called 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 a 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

    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

    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:
      • 4x2 + 9y2 + 16z2 + 12xy – 24yx – 16xz
      • 2x2 + y2 + 8z2 – 2√2xy + 4√2yz – 8xz

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