Adding and Subtracting Polynomials

Polynomials consist of non-negative exponential values and other coefficients, operators, and a constant. However, each element doesn’t need to be present. To call a statement a polynomial, it should not have a non-negative exponential, or else it will not be considered a polynomial. For example 5x + 2.

We can perform all the arithmetic operations such as addition, subtraction, multiplication and division on polynomials. This article aims to provide an understanding of the addition and subtraction of polynomials.

Types of polynomials

Before learning about the multiplication of polynomials, let us understand the types of polynomials on the basis of the number of terms.

• Monomial: Polynomials that have only one term are known as monomials. For example, 3x, 7xy, 5, 4x².

• Binomial: Polynomials that have two terms are known as binomial. For example, 3x + 4, 4x²+3, 5z + 3x.

• Trinomial: Polynomials that have three terms are known as trinomials. For example, 3x + 5y + 6, x²+3x+3, 5z + 3x + 3.

Polynomial, in general, is used for expressions having more than one algebraic term.

The highest exponent of the variable in a polynomial is the degree of the polynomial. Look at the example below:

8x³+10x²+6x+5: the degree is 3.

5x²+4x+4: the degree is 2.

Adding or subtracting two polynomials

Adding or subtracting two or more polynomials can be done in two ways—horizontal and vertical methods. Now let us go through horizontal and vertical addition and subtraction in detail.

Horizontal addition and subtraction

The horizontal method of adding two or more polynomials is very simple.

First, we have to identify the like terms and group them. After that, we can add or subtract coefficients of the like terms.

Illustration: Add these two polynomials 10x-8y and 4x+9y.

Solution:

To add these polynomials, we first write these polynomials as

10x-8y+4x+9y

Arrange the like terms together like this:

10x+4x+9y-8y

Now, we can simplify the expression easily:

(10+4)x + (9-8)y

We get

14x+y

Illustration: Subtract 4x + 6y + 8 from 5x + 2y + 10.

Solution:

We will need to write the polynomial in this form:

5x+2y+10-(4x+6y+8 )

On opening the bracket, the sign will change all the terms in the bracket:

5x+2y+10 – 4x – 6y- 8

Now, we will arrange the like terms:

5x-4x+2y-6y+10-8

Now, we will simplify the expression:

 x-4y+2

Vertical addition and subtraction

This method is approximately the same in horizontal addition and subtraction. As in this method, we also have to group or take the like terms together with them.

The difference between horizontal and vertical addition and subtraction is that in the horizontal method, the polynomials are horizontally aligned in a row, while in the vertical method, the polynomials are arranged in columns.

Illustration: Add 10x + 3y + 30 and 80x + 60y + 10

Solution: 

We must first arrange them in columns and at the same time group them together.

10x + 3y + 30

80x + 60y + 10

We now have like terms aligned with each other. Therefore, we can start adding. 

Illustration: Subtract x + 3y + 14 from 9x + 6y + 12.

Solution:

We must first write them as shown below.

4x+2y+3

6x+4y+5

We now have terms aligned with each other. We have to change the sign of all the terms of the second polynomial.

2x + 2y + 2

Taking 2 as common, we get 2 (x + y +1).

Solved examples

1. Solve:

a) 25y²+15y²

b) 16pq³-(-7pq³)

Solution:

a) Adding the like terms: 25y²+15y²=40y²

b) Subtracting like terms: 16pq³-(-7pq³)=23pq³

2. Find the sum: (7y²-2y+9)+(4y²-8y-7)

Solution:

Identifying and rearranging the like terms: 7y²-4y²-2y-8y+9-7

Adding and subtracting the like terms, we get:

11y²-10y+2

3. Find the difference: (9w²-7w+5)-(2w²-4)

Solution:

On grouping the like terms, we get:

(9w²-2w²-7w+5+4)

Adding and subtracting the like terms, we get:

 7w²-7w+9

Conclusion

Polynomials are mathematical expressions containing any number of algebraic terms combined by addition or subtraction. Through this article, we studied all types of polynomials and the degree of polynomials. We can perform all kinds of arithmetic operations on polynomials. There are two significant methods for adding or subtracting polynomials—horizontal and vertical methods. We studied both methods in this article.