**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 x^{2} + 5x + 2 is polynomial, then the expressions x^{2}, 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 x^{3} + y^{3} + 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 â€“ of degree one
- Quadratic Polynomial- of degree two
- Cubic Polynomial â€“ 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
^{2 }+ x + 1

Quadratic Equation | Algebraic Identities |

Quadratic Formula & Quadratic Polynomial | Degree Of A 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
- 2x
^{2}+ 5x + 15

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

- An algebraic expression p(x) = a
_{0}x^{n}+ a_{1}x^{n-1}+ a_{2}x^{n-2 }+ â€¦ a_{n}is a polynomial where a_{0}, a_{1}, â€¦â€¦â€¦. 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 â€˜aâ€™ 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 â€“ a, then the remainder is p(a).**Factor Theorem**: x â€“ c is a factor of the polynomial p(x), if p(c) = 0. Also, if x â€“ c is a factor of p(x), then p(c) = 0.- The degree of the zero polynomial is not defined.
- (x + y + z)
^{2}= x^{2}+ y^{2}+ z^{2}+ 2xy + 2yz + 2zx - (x + y)
^{3}= x^{3}+ y^{3}+ 3xy(x + y) - (x â€“ y)
^{3}= x^{3}â€“ y^{3}â€“ 3xy(x â€“ y) - x
^{3}+ y^{3}+ z^{3}â€“ 3xyz = (x + y + z) (x^{2}+ y^{2}+ z^{2}â€“ xy â€“ yz â€“ zx)

Polynomials Class 9 Examples (Class 9th Maths Polynomial)

**Example 1:**

Write the coefficients of x in each of the following:

- 3x + 1
- 23x
^{2}â€“ 5x + 1

**Solution:**

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

In 23x^{2} â€“ 5x + 1, the coefficient of x is -5.

**Example 2:**

What are the degrees of following polynomials?

- 3a
^{2}+ a â€“ 1 - 32x
^{3}+ x â€“ 1

**Solution: **

- 3a
^{2}+ a â€“ 1 : The degree is 2 - 32x
^{3}+ x â€“ 1 : The degree is 3

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

- Find value of polynomial 2x
^{2}+ 5x + 1 at x = 3. - Check whether x = -1/6 is zero of the polynomial p(a) = 6a + 1.
- Divide 3a
^{2}+ x â€“ 1 by a + 1. - Find value of k, if (a â€“ 1) is factor of p(a) = ka
^{2}â€“ 3a + k. - Factorise each of the following:
- 4x
^{2 }+ 9y^{2}+ 16z^{2}+ 12xy â€“ 24yx â€“ 16xz

- 2x
^{2}+ y^{2}+ 8z^{2}â€“ 2âˆš2xy + 4âˆš2yz â€“ 8xz

- 4x