Polynomial comes from the word “poly” which means “many” and the word “nomial” which means “term”. In maths, a polynomial expression consists of variables which 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 **polynomial Class 9 notes** are provided here along with some important questions so that students can understand the concept in an easy manner.

**Also Read:**

Quadratic Equation | Algebraic Identities |

Quadratic Formula & Quadratic Polynomial | Degree Of A Polynomial |

## Definition of Polynomials

Polynomials are expressions with one or more terms with a non-zero coefficient. A polynomial can have more than one number of terms. 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 polynomials.** 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 **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 – 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
^{2 }+ 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
- 2x
^{2}+ 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 do revision for 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 Class 9 Chapter 2 Polynomials are given below:**

- An algebraic expression p(x) = a
_{0}x^{n}+ a_{1}x^{n-1}+ a_{2}x^{n-2 }+ … a_{n}is a polynomial where a_{0}, a_{1}, ………. 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 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}= x^{2}+ y^{2}+ z^{2}+ 2xy + 2yz + 2zx - (x + y)
^{3}= x^{3}+ y^{3}+ 3xy(x + y) - (x – y)
^{3}= x^{3}– y^{3}– 3xy(x – y) - x
^{3}+ y^{3}+ z^{3}– 3xyz = (x + y + z) (x^{2}+ y^{2}+ z^{2}– xy – yz – zx)

### Polynomials Class 9 Examples

**Q.1 Write the coefficients of x in each of the following:**

**3x+1****23x**^{2}-5x + 1

Solution: In first case, 3x+1. the coefficients of x is 3.

In second case, the coefficients of x is -5.

**Q.2: What are the degrees of following polynomials?**

**3a**^{2}+ a – 1**32x**^{3}+ x – 1

**Solution: **

- 3a
^{2}+ a – 1 : The degree is 2 - 32x
^{3}+ x – 1 : The degree is 3

### Important Questions on Polynomials

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

- 4x

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