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 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 + a₁x^n-1 + a₂x^n-2 + … a_n is a polynomial where a₀, a₁, ………. a_n 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. 4x² + 9y² + 16z² + 12xy – 24yx – 16xz
   1. 2x² + y² + 8z² – 2√2xy + 4√2yz – 8xz