Adding Polynomials

Polynomials consist of non-negative exponential values and other elements like coefficients, operators, and a constant. However, each element does not 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 as a polynomial. For example, 4×2 + 3x + 2.

Adding polynomials

To perform the addition of polynomials, we have to first identify like terms. Then, we can add those like terms to get the result. We can perform the addition in two ways, either by adding the polynomials vertically or by adding the polynomials horizontally.

Like terms are the same variables, their coefficient can differ in value, but the variable should be identical. 

Illustration 1

17xy+6x+8y and 2xy+16x+44y

As we can see in the above statement, 17xy and 2xy have different coefficients. Still, their variable is the same, so they are like terms, but if we observe 6x and 2xy, both have x in their variable, but they will not qualify as like terms because 2xy has y in its variable, so the variables are not identical.

Horizontal addition 

This is a very simple type of addition. First, we have to identify like terms, and then we have to arrange them in one spot. Then we simplify all the like terms by adding all the values of like terms. After simplification of all like terms, we will have the solution.

We can better understand this concept with an illustration.

Illustration

Add these two polynomials 14x + 22y + 3 and 8x + 13y + 4.

Solution: 

To add these polynomials, we first write these polynomials this way:

 14x + 22y + 3 + 8x + 13y + 4

Now we arrange the like terms together like this:

14x + 8x + 22y + 13y + 3 + 4

Now we perform a simplification of the expression, and we get the solution.

22x + 35y + 7

Illustration

Add these two polynomials 11x + 3y + 9 and 9x + 7y + 1.

Solution:

To add these polynomials, we first write these polynomials this way:

11x + 3y + 9 + 9x + 7y + 1

Now, we arrange the like terms together like this:

11x + 9x + 3y + 7y + 9 + 1

Now, we perform a simplification of the expression, and we get the solution.

20x + 10y + 10

Illustration

Add these two polynomials 2x – 9y and 78x + 6y.

Solution:

To add these polynomials, we first write these polynomials like this:

 2x – 9y + 78x + 6y

Now, we arrange the like terms together like this:

2x + 78x + 6y – 9y

Now, we perform a simplification of the expression, and we get the solution.

22x – 3y

We can use the same process while adding more than two polynomials. We can learn this by following the illustration below.

Illustration

Add the polynomials 14x + 22y + 3 , 8x + 13y + 4 , 22x + 5y – 8 , xy + 6y.

Solution:

To add these polynomials, we first write these polynomials like 

 14x + 22y + 3 + 8x + 13y + 4 + 22x + 5y – 8 + xy + 6y 

Now, we arrange the like terms together like this

xy + 14x + 8x + 22x + 22y + 13y + 5y + 6y + 3 – 8

Now, we perform a simplification of the expression, and we get the solution.

xy + 44x + 46y + 5

Vertical addition

Vertical addition follows the same procedure of horizontal addition; we gather the like terms together, then add the like terms, and then simplify them until only one value is left of every type of variable in the expression. The major difference is that it is performed vertically instead of horizontally. We can understand this concept by observing the following illustrations.

Illustration

Add these two polynomials 2x + 3y + 14 and 9x + 6y + 12.

Solution:

To add these polynomials, we must first write them as shown below.

We now have terms aligned with each other. We can simplify them and get the result.

Illustration

Add these two polynomials 4x + 7y and 77x + 2y + 1.

Solution: To add these polynomials, we must first write them as shown below:

We now have like terms aligned with each other. We can simplify them and get the result.

Illustration

Add these two polynomials 8x + 4y + 66 and 99x + 56y + 3

Solution: 

To add these polynomials, we must first write them like shown below.

We now have like-terms aligned with each other. We can simplify them and get the result.

Conclusion

To add polynomials, we have to first identify the like terms and then gather them together and add them to obtain the solution. Like terms may have different coefficients, but their variables are identical. If they have different exponential power or have any other variables attached to them, they are not like terms.