**Polynomials** can be defined as algebraic expressions that contain variables and constants. Variables are also sometimes called indeterminates.

## What are Polynomials?

Polynomials are like the heart of mathematics. They are basically used to express numbers in the field of mathematics and are referred to as very important in some branches of mathematics, such as calculus and algebra.

Polynomial is made up of two words:-

**Poly = “many”**

**Nominal = “terms.”**

The terms are combined using mathematical operations such as addition, subtraction, multiplication, and division.

For example = 32x + 2 and 3x + 5y are **Polynomials**.

1. 32x + 2 = In this example, there is one variable x and two terms 32x and 12.

2. Similarly, 3x + 5y. In this example, there are two variables x and y.It contains two terms 3x and 5y.

## The Standard form of Polynomial

**Definition 1 =** The standard form of a polynomial basically deals with how to write a polynomial in the descending power of the variable.

**Definition 2 =** The Standard form for writing a polynomial is to write the terms with the highest degree first.

Let us understand with an example:-

**Convert the polynomial 8 + 2x**^{3}+ x^{2 }**in the standard form.**

**Step 1 =** First, check the degree of the polynomial. In the given polynomial, the degree is 3.

**Step 2** = Write the term with degree 3.

**Step 3** = Now, check if there is a term with an exponent of variable less than 3 which is 2 and 1, and note it down next.

**Step 4** = At last, write the term with the exponent of the variable as 0, which is the constant term.

**Step 5 = **2x ^{3} + x^{ }^{2} + 8 is the standard form of the polynomial.

Read More: Remember Multiplication Tables of Any Number Using Mental Mathematics

## Degree of a Polynomial

**Definition** = The highest exponent of the variable in a polynomial can be defined as the degree of a polynomial.

(Exponent is a method of expressing large numbers in terms of powers, for example, when 3 is multiplied by itself 4 times, which is 3 × 3 × 3 × 3. This can be written as 34.)

The degree is used to tell the maximum number of solutions a polynomial equation can have.

Degree = 3 means the equation can have 3 solutions.

Let us understand with the help of some examples:-

- A polynomial 3x
^{5}+ 7 has a degree equal to five. - 2x
^{2}+ 8 has a degree equal to two.

Now, let us calculate the degree of a polynomial having more than one variable, simply add the powers of all the variables in a term. So, we will get the final degree of the given polynomial p.

The degree of the polynomial 2x ^{2} y ^{3} + 7x ^{4} y is 10.

Read More : METHODS FOR SOLVING QUADRATIC EQUATION

## Types of Polynomials

Polynomials can be divided on the basis of their degree and power.

- Based on the Number of Terms.
- Based on Number of Degrees.

### A. Polynomials Types: Base on the Number of Terms

According to the number of terms, polynomials can be categorized into three types:-

1. Monomials

2. Binomials

3. Trinomials

1.** Monomial**

A Monomial can be defined as an expression containing a single term. ( Note – The single term must be non-zero. ) For example = x, 5xy, 4y^{2} and 6y^{3}

**2. Binomial **

A Binomial can be defined as an expression containing two terms. For example = Y+5, x^{3}+9 and 3x^{4}-2

**3. Trinomial**

A Trinomial can be defined as an expression that contains three terms. For example =

3x^{4} + 8x – 5, x + y + 2z and 3x + y – 5.

Read More : Sum of cubes of n natural numbers

### Polynomials Types: Based on the Number of Degrees

According to the number of degrees, polynomials can be categorized into four major types:-

**Zero Polynomial or Constant polynomial****Linear polynomial****Quadratic polynomial****Cubic polynomial**

**1. Zero Polynomial**

A zero polynomial can be defined as a polynomial whose degree is equal to zero. All the constants are examples of zero polynomials. For example = 3, 6 and 9.

### 2. Linear Polynomial

Linear polynomials can be defined as polynomials whose degree is equal to one. For example = x – 4.

### 3. Quadratic Polynomial

Quadratic polynomials can be defined as polynomials whose degree is equal to two.For example = 2q^{2} – 7.

### 4. Cubic polynomials

Cubic polynomials can be defined as polynomial whose degree is equal to three.For example = 7x^{3} – y – 4

Read More : Fractions From Basics. Understand What is Fraction, Types with Examples

## Operations on Polynomials

Any basic operation that can be performed on numbers can be performed on the polynomials too. So, there are four types of operations that can be performed on polynomials:-

1. Addition of polynomials

2. Subtraction of polynomials

3. Multiplication of polynomials

4. Division of polynomials

## 1. Addition of Polynomials

It is one of the basic algebraic operations which is used to increase and decrease the value of the polynomials.

The rule to add numbers and polynomials is exactly the same.

To add the polynomials, you should simply add on the like terms (like terms means the terms having the same variable and same power).

For eg –

**a. Add 2x ^{2} + 4x + 5 and 3x^{2}**

= 5x^{2} + 4x + 5

b.** Add 4x ^{3}+ x^{2} + 8 and 2x^{3} + 4x^{2} + 4x**

= 6x^{3} + 5x^{2} + 4x + 8

Read More : Subtraction of Fractions

**Subtraction of Polynomials**

It is also one of the basic algebraic operations which are used to increase and decrease the value of the polynomials.

The rules to subtract numbers and polynomials are also similar.

To subtract the polynomials, you should simply subtract the like terms (like terms mean the terms having the same variable and same power).

For eg –

**a. Subtract 4x ^{2} **+ 4x + 5 and 3x

^{2}

= x^{2} + 4x + 5

**b. Subtract **4x^{3}+ x^{2} + 8 and 2x^{3} + 4x^{2} + 4x

= 2x^{3} – 3x^{2} + 4x + 8

Read More : Multiplication of Fractions

### 3. Multiplication of Polynomials

In the multiplication of polynomials, two or more two polynomials are multiplied and always give a result with a higher degree.

In polynomials, you can simply multiply the like and unlike terms both together. (like terms mean the terms having the same variable and same power and vice versa…).

For eg –

**a.** Multiply 2x^{2} and 3x^{2}

= 6x^{4}

**b. Multiply 4x ^{3} and 2x^{3} + 4x^{2} + 4x**

**Step 1 =** firstly multiply 4x^{3} with 2x^{3}= 8x^{6 }(in multiplication powers add on)

**Step 2 =** then, multiply 4x^{3} with 4x^{2} = 16x^{5}

**Step 3** = At last, multiply 4x^{3} with 4x = 16x^{4}

Now, combine the terms Answer is 8x^{6} + 16x^{5} + 16x^{4}.

Read More : Multiplication of Fractions

### 4. Division of Polynomials

When we are dividing the polynomials, two or more than two polynomials can be divided and always give a result with a smaller degree.

In **Polynomials** you can simply divide the like and unlike terms both together. (like terms mean the terms having the same variable and same power and vice versa…).

For eg –

**a. Divide 2x ^{2} and x **

= 2x

In this blog, we have studied the basic concepts of **Polynomials**, their types, degrees, and operations on polynomial expressions. We will cover the simplification of polynomials in the next session.

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## Practice Question:

**1. Add: **2x^{2}+3x and 4x^{2}

**2. Add: **4x^{2}+2x+7 and 3x^{2}+3x+7

**3. Subtract: **5x^{2}+3x from 7x^{2}

**4. Multiply: **4x^{2}(3x^{2}+5x)

** 5. Divide: ** 8x^{2}/2x

## Quizzes for you:-

**Question 1: Which of the following is an example of a monomial?**

- 2x
^{2} **3x ( Correct Answer )**- 4x
^{3}

**Question 2: In broadly, how many types of polynomials based on the degree are there?**

**Options**

- 4 ( correct answer )
- 5
- 2
- 1

**Question 3: What is the sum of **3x^{2}+5x and 2x^{2}+6x?

**Options**

- 4x
^{2}+6x - x
^{2}-x **5x**^{2}+11x ( Correct answer )- -x
^{2}+11x