The Addition of Polynomials is the process of adding different terms present in the polynomial. Check out the various problems and procedures on how do we add polynomials in this article. Know the different rules in addition of polynomials consisting of different exponents. Students of 6th Grade Math can get a strong grip on the Polynomial Addition by practicing the Addition of Polynomials Examples over here.

Read more:

- Types of Algebraic Expressions
- Worksheet on Addition and Subtraction of Polynomials
- Worksheet on Addition of Polynomials

## What is Polynomial Addition?

The addition of two polynomials is the process of combining like terms present in the two polynomials. Like terms are the terms those having the same variable and same exponent.

## How to Solve Addition of Polynomials? | Rules in Addition of Polynomials

We have given the process to add polynomials below. Follow the procedure given here to solve all the addition of polynomial problems.

(i) Arrange each polynomial along with their terms and also with the highest degree in decreasing order of degree.

(ii) Next, group the like terms whose variables and exponents are the same.

(iii) Finally, simplify by combining like terms.

#### Horizontal Addition of Polynomials

Go through the below procedure to do the Horizontal addition of polynomials. Let’s check the Addition of Polynomials horizontally with a simple example.

**Example:** Find the sum of the following two polynomials:

12m^{4} + 3m^{3} + 8m – 5, 10m^{4} – 2m^{3} + 6m^{2} – 3m + 2.

**Solution:
**Given polynomials are 12m

^{4}+ 3m

^{3}+ 8m – 5 and 10m

^{4}– 2m

^{3}+ 6m

^{2}– 3m + 2.

The first polynomial is 12m

^{4}+ 3m

^{3}+ 8m – 5 and the second is 10m

^{4}– 2m

^{3}+ 6m

^{2}– 3m + 2.

Now, add the given polynomials horizontally.

(12m

^{4}+ 3m

^{3}+ 8m – 5) + (10m

^{4}– 2m

^{3}+ 6m

^{2}– 3m + 2)

Group the same variables with the same exponents. Terms that are not like terms cannot be added.

12m

^{4}and 10m

^{4}are the like terms.

3m

^{3}and 2m

^{3}are the like terms.

8m and 3m are the like terms.

5 and 2 are constants.

Now, add the given polynomials with the like terms.

(12m

^{4}+ 3m

^{3}+ 8m – 5) + (10m

^{4}– 2m

^{3}+ 6m

^{2}– 3m + 2) = 12m

^{4}+ 3m

^{3}+ 8m – 5 + 10m

^{4}– 2m

^{3}+ 6m

^{2}– 3m + 2

12m

^{4}+ 10m

^{4}+ 3m

^{3}– 2m

^{3}+ 6m

^{2}+ 8m – 3m – 5 + 2 = 22m

^{4}+ m

^{3}+ 6m

^{2}+ 5m – 3.

Therefore, the addition of given polynomials is 22m^{4} + m^{3} + 6m^{2} + 5m – 3.

#### Vertical Addition of Polynomials

The step-by-step process to Add Polynomials vertically is given here. Let’s check the Addition of Polynomials Vertically with a simple example.

**Example:**

Find the sum of the following two polynomials:

12m^{4} + 3m^{3} + 8m – 5, 10m^{4} – 2m^{3} + 6m^{2} – 3m + 2.

**Solution:
**Given polynomials are 12m

^{4}+ 3m

^{3}+ 8m – 5 and 10m

^{4}– 2m

^{3}+ 6m

^{2}– 3m + 2.

The first polynomial is 12m

^{4}+ 3m

^{3}+ 8m – 5 and the second is 10m

^{4}– 2m

^{3}+ 6m

^{2}– 3m + 2.

In the first term, we don’t have the m

^{2}term. So, we can take it as 0.

Now, add the given polynomials vertically.

(12m

^{4}+ 3m

^{3}+ 0 + 8m – 5)

+ (10m

^{4}– 2m

^{3}+ 6m

^{2}– 3m + 2)

————————————

22m

^{4}+ m

^{3}+ 6m

^{2 }+ 5m – 3

Therefore, the addition of given polynomials is 22m^{4} + m^{3} + 6m^{2 }+ 8m – 3.

### Examples of Addition of Polynomials with Answers

**Question 1.**

Add: 6x + 4y, 5x – 5y + 2z and -2x + 6y + 3z

**Solution:**

Given polynomials are 6x + 4y, 5x – 5y + 2z and -2x + 6y + 3z.

The first polynomial is 6x + 4y and the second is 5x – 5y + 2z and the third polynomial is -2x + 6y + 3z.

Now, add the given polynomials horizontally.

(6x + 4y) + (5x – 5y + 2z) + (-2x + 6y + 3z)

Group the same variables with the same exponents. Terms that are not like terms cannot be added.

6x, 5x, and 2x are the like terms.

4y, 5y, and 6y are the like terms.

2z and 3z are the like terms.

Now, add the given polynomials with the like terms.

(6x + 4y) + (5x – 5y + 2z) + (-2x + 6y + 3z) = 6x + 4y + 5x – 5y + 2z -2x + 6y + 3z

6x + 5x -2x+ 4y – 5y + 6y + 2z + 3z = 9x + 5y + 8z.

Therefore, the addition of given polynomials is 9x + 5y + 8z.

**Question 2.**

Add: 6a^{2} + 2ab – 2b^{2}, -2a^{2} + 4ab + 6b^{2} and 6a^{2} – 20ab + 8b^{2}

**Solution:**

Given polynomials are 6a^{2} + 2ab – 2b^{2}, -2a^{2} + 4ab + 6b^{2} and 6a^{2} – 20ab + 8b^{2}.

The first polynomial is 6a^{2} + 2ab – 2b^{2}, and the second is -2a^{2} + 4ab + 6b^{2} and the third polynomial is 6a^{2} – 20ab + 8b^{2}.

Now, add the given polynomials horizontally.

(6a^{2} + 2ab – 2b^{2}) + (-2a^{2} + 4ab + 6b^{2}) + (6a^{2} – 20ab + 8b^{2})

Group the same variables with the same exponents. Terms that are not like terms cannot be added.

6a^{2}, -2a^{2}, and 6a^{2} are the like terms.

2ab, 4ab, and 20ab are the like terms.

2b^{2}, 6b^{2}, and 8b^{2} are the like terms.

Now, add the given polynomials with the like terms.

(6a^{2} + 2ab – 2b^{2}) + (-2a^{2} + 4ab + 6b^{2}) + (6a^{2} – 20ab + 8b^{2}) = 6a^{2} + 2ab – 2b^{2} -2a^{2} + 4ab + 6b^{2} + 6a^{2} – 20ab + 8b^{2}

6a^{2} – 2a^{2} + 6a^{2 }+ 2ab + 4ab – 20ab – 2b^{2} + 6b^{2} + 8b^{2} = 10a^{2} – 18ab – 12b^{2}

Therefore, the addition of given polynomials is 10a^{2} – 18ab – 12b^{2}.

**Question 3.**

Add: 9a + 7b, 8a – 8b + 5c and -7a + 9b + 6c

**Solution:**

Given polynomials are 9a + 7b, 8a – 8b + 5c and -7a + 9b + 6c.

The first polynomial is 9a + 7b and the second is 8a – 8b + 5c and the third polynomial is -7a + 9b + 6c.

In the first term, we don’t have the c term. So, we can take it as 0.

Now, add the given polynomials vertically.

9a + 7b + 0

8a – 8b + 5c

-7a + 9b + 6c

——————

10a +8b + 11c

Therefore, the addition of given polynomials is 10a +8b + 11c.

**Question 4.**

Add: 5x^{3} – 7x^{2} + 10x + 20, 17x^{3} – 4x – 25, 11x^{2} – 2x + 17 and -10x^{3} + 4x^{2} – 9x.

**Solution:**

Given polynomials are 5x^{3} – 7x^{2} + 10x + 20, 17x^{3} – 4x – 25, 11x^{2} – 2x + 17 and -10x^{3} + 4x^{2} – 9x.

The first polynomial is 5x^{3} – 7x^{2} + 10x + 20 and the second is 17x^{3} – 4x – 25, and the third polynomial is 11x^{2} – 2x + 17 also the fourth polynomial is -10x^{3} + 4x^{2} – 9x.

Where the variable is not present we can take it as 0.

Now, add the given polynomials vertically.

5x^{3} – 7x^{2} + 10x + 20

17x^{3} + 0 – 4x – 25

0 + 11x^{2} – 2x + 17

-10x^{3} + 4x^{2} – 9x + 0

———————–

12x^{3} + 8x^{2} – 5x + 12

Therefore, the addition of given polynomials is 12x^{3} + 8x^{2} – 5x + 12.

### FAQs on Addition of Polynomials

**1. What is a linear Polynomial?**

The linear polynomial is a polynomial that has the degree 1.

** 2. How do we add polynomials?**

We find similar terms first then we add the coefficients of the terms to add those polynomials.

** 3. Is it possible to add different terms with different exponents?**

No, generally we add similar terms having the same exponents.

**4. Add 3x + 2y and 4x + 3y?**

By adding the given polynomials, we can get 7x + 5y.

**5. Add 2a + 9b and 3a?**

By adding the given polynomials, we can get 5a.

### Conclusion

The complete article will help you to learn the addition of polynomials easily. Without missing any part, read the complete concept and gain the knowledge.