In this post we will learn methods of adding different polynomials.

To understand the chapter you should have basic knowledge of polynomials, like terms and BODMAS rules.

**Methods of adding Polynomials**

Here we will learn two methods.

(a) **Horizontal addition** of polynomials

(b) **Vertical addition** of polynomials

**Both the method follow below rules**:

(1) In polynomials, **only like terms are added or subtracted**.

Like terms are the entity with the same variables.

(2) While adding like terms, **only coefficient is added, the variables remain the same**.

**For Example**

Add the below expression.

\mathtt{6x^{2} +\ 7x\ +\ 3x^{2} +\ 4}

Here \mathtt{6x^{2} \ \&\ \ 3x^{2}} are the like terms. Hence, only these two entities can be added.

\mathtt{\Longrightarrow \left( 6x^{2} \ +\ \ 3x^{2}\right) \ +\ 7x\ +\ 4}\\\ \\ \mathtt{\Longrightarrow \ 9x^{2} +\ 7x\ +\ 4}

Note that only coefficient of entity is added, the variable part remained the same.

Now let us study different methods of adding the polynomials.

**Horizontal addition of polynomials**

For adding the given polynomials, **follow the below steps**:

(a)** Identify the like terms** in the polynomial

(b) **Arrange the like terms under one bracket**

(c) ** Add the coefficients** of like terms and keep the variables constant.

Given below are some examples for your understanding.

**Examples of Adding Like terms**

**Example 01**

Add the below polynomials.

4x + 6y ; 3x ; 11x + 2y

Writing all the polynomials in addition form

4x + 6y + 3x + 11x + 2y

Now follow the below steps:

(a) **Identify the like terms**

(b) **Arrange the like terms **under one bracket

⟹ (4x + 3x + 11x) + (6y + 2y)

(c) **Add the coefficients** and keep the variables intact

⟹ 18x + 8y

Hence adding all the three expressions result in 18x + 8y

**Example 02**

Add the given three expressions.

\mathtt{3m^{2} -2mn\ +\ 6n^{2}}\\ \\ \mathtt{-m^{2} +6mn\ +\ n^{2}}\\ \\ \mathtt{8m^{2} +\ 4mn\ -\ n^{2}}

Solution

Writing all the expression in addition form.

Arranging the like terms under one bracket..

**Example 03**

Add the below expressions

\mathtt{2x^{3} +\ 2x^{2} +\ 4}\\ \\ \mathtt{3x^{2} +\ 6\ +\ 7x^{3}}

Solution

First write the expression in addition form.

\mathtt{\Longrightarrow \left( 2x^{3} +\ 2x^{2} +\ 4\right) +\ \left( 3x^{2} +\ 6\ +\ 7x^{3}\right)}

Arranging the like terms in single bracket.

\mathtt{\Longrightarrow \ \left( 2x^{3} \ +\ 7x^{3}\right) +\ \left( 2x^{2} +\ 3x^{2}\right) \ +\ ( 4\ +\ 6)}\\\ \\ \mathtt{\Longrightarrow \ 9x^{3} +\ 5x^{2} +\ 10}

**Example 04**

Add the below polynomials

7x + 5y + z ;

6x + 3y ;

11x + 9z

Solution

Writing all the expression in addition form.

⟹ (7x + 5y + z) + (6x + 3y) + (11x + 9z)

Arranging the like terms in bracket.

⟹ (7x + 6x + 11x) + (5y + 3y) + (z + 9z)

⟹ 24x + 8y + 10z

**Example 05**Add the below polynomials

\mathtt{-x^{3} +\ 6xy\ +\ x}\\ \\ \mathtt{y\ +\ 10xy}

Solution

Adding the polynomials

\mathtt{\Longrightarrow \ \left( -x^{3} +\ 6xy\ +\ x\ \right) \ +\ ( y\ +\ 10xy)}\\\ \\ \mathtt{\Longrightarrow \ -x^{3} +\ 6xy\ +\ 10xy\ +\ x+y}\\\ \\ \mathtt{\Longrightarrow \ -x^{3} +\ 16xy\ +\ x\ +\ y}

**Vertical Addition of Polynomial**

Given below are steps for vertically adding the polynomials.

(a) Find the like terms among the given polynomial.

(b) Vertically align the like terms so that they are in same column

(c) Add the coefficient of like terms and leave the variables as it is.

Given below are examples for further understanding.

**Vertically addition of Polynomials Examples**

**Example 01**

Add the given polynomials

x + 2yz + 3y;

2y + 9yz

Solution

(a) Find the like terms among given polynomial

(b) Vertically align the like terms in same column and add the entities

Hence, x + 11yz + 5y is the solution.

**Example 02**

Add the given polynomials.

\mathtt{x^{3} +\ 6xy\ -9x}\\ \\ \mathtt{x^{2} \ +\ 7xy}\\ \\ \mathtt{3x^{3} +\ 2x}

Solution

Vertically align the like terms and add the entities vertically

Hence, \mathtt{4x^{3} +\ x^{2} +13xy\ -\ 7x} is the solution

**Example 03**

Add the below polynomials

\mathtt{m^{2} +\ 5m^{3} +\ n^{2}}\\ \\ \mathtt{6mn\ +\ 7m^{2}}\\ \\ \mathtt{-4m^{2} +\ m^{3}}

Solution

Vertically align the like terms and do the addition.

Hence, \mathtt{6m^{3} +4m^{2} +\ n^{2} \ +6mn} is the solution.

**Frequently Asked Question – Polynomial addition**

**(01) Can you combine the below expression**

x + 4y and xy + yz

No!!

All the entities contain different variables.

x ⟹ variable x

4y ⟹ variable y

xy ⟹ variable xy

yz ⟹ variable yz

Addition is only possible between entities of same variables.

**(02) What are like terms? Can you share some examples.**

Entities with same variables are known as like terms.

Consider the equation

\mathtt{x^{2} y\ +\ 3y^{3} +\ 5x^{2} y}

Here the first and third entities are like variables.

**(03) Can we add entities with different power? **

No!! We cannot add entities with different exponents.

For example, \mathtt{x^{2} \ \&\ x^{3}} are entities with different power and cannot be added.