The addition of polynomials is the easiest mathematical operation that we can perform with polynomials. To solve additions of polynomials, we simply have to combine like terms.
Here, we will look at a summary of polynomial addition along with the process used to solve these types of problems. In addition, we will explore various examples with answers in order to fully master this topic.
Summary of addition of polynomials
The addition of polynomials can be done simply by combining like terms and considering the order of operations. The only thing we have to take into account is to distinguish the “plus” and “minus” signs in each polynomial.
We can follow the following steps to add to the polynomials:
Step 1: Remove all parentheses. It is recommended to write the problem vertically as this makes the next steps easier to visualize. When adding, we have to distribute the positive sign, which does not change any of the signs.
Step 2: Combine like terms. This is easier if we have the expressions written vertically. Remember that to combine like terms, the variables and powers of each term must be the same.
Addition of polynomials – Examples with answers
The following examples can be used to master the topic of the addition of polynomials. Each example has a detailed solution that indicates the process and reasoning used.
EXAMPLE 1
Solve the addition $latex (3x+4y)+(2x-2y)$.
Solution
We start by removing the parentheses. This is easy when we add polynomials since we don’t have to change the signs. Then, we will group like terms according to their variables, and finally, we simplify:
$latex (3x+4y)+(2x-2y)$
$latex =3x+4y+2x-2y$
$latex =3x+2x+4y-2y$
$latex =5x+2y$
The two terms we obtained are not like terms since they have different variables, therefore we cannot combine them.
EXAMPLE 2
Add the polynomials $latex (3x+4y)$ and $latex (2x-2y)$ vertically.
Solution
To add polynomials vertically, we place each variable in its own column. In this case, the first column will be x and the second will be y:
$latex 3x+4y$
$latex 2x-2y$
___________
$latex 5x+2y$
We see that we got the same answer as when we added horizontally. The format used simply depends on your preference. You can solve the sum of polynomials with the format that you feel most comfortable with.
Generally, for simple additions, the horizontal format is easier, but for longer and more complicated polynomials, adding vertically can make solving easier.
EXAMPLE 3
Solve the addition of polynomials: $$(2{{x}^3}+5{{x}^2}-4x+5)+(4{{x}^3}+2{{x}^2}+3x-6)$$
Solution
We can do the addition horizontally. Therefore, we remove the parentheses and combine like terms:
$$(2{{x}^3}+5{{x}^2}-4x+5)+(4{{x}^3}+2{{x}^2}+3x-6)$$
$$=2{{x}^3}+5{{x}^2}-4x+5+4{{x}^3}+2{{x}^2}+3x-6$$
$$=2{{x}^3}+4{{x}^3}+5{{x}^2}+2{{x}^2}-4x+3x+5-6$$
$latex =6{{x}^3}+7{{x}^2}-x-1$
We can also perform this sum vertically. We place each variable with a different exponent in its own column:
$latex 2{{x}^3}+5{{x}^2}-4x+5$
$latex 4{{x}^3}+2{{x}^2}+3x-6$
______________
$latex 6{{x}^3}+7{{x}^2}-x-1$
EXAMPLE 4
Solve the addition: $latex (2{{x}^2}+7x-6)+(4{{x}^2}-2x-3)$ $latex +(-3{{x}^2}+4x+5)$.
Solution
We start by doing the addition horizontally. We are going to remove the parentheses to combine like terms:
$latex (2{{x}^2}+7x-6)+(4{{x}^2}-2x-3)$ $latex +(-3{{x}^2}+4x+5)$
$latex =2{{x}^2}+7x-6+4{{x}^2}$ $latex -2x-3-3{{x}^2}+4x+5$
$latex =2{{x}^2}+4{{x}^2}-3{{x}^2}+7x$ $latex -2x+4x-6-3+5$
$latex =3{{x}^2}+9x-4$
Now, we perform this sum vertically. We assign a different column to each exponent of the variable:
$latex 2{{x}^2}+7x-6$
$latex 4{{x}^2}-2x-3$
$latex -3{{x}^2}+4x+5$
______________
$latex 3{{x}^2}+9x-4$
EXAMPLE 5
Solve the addition of polynomials: $latex (2{{x}^3}+4{{x}^2}-5x)+(3{{x}^3}+3x+4)$ $latex +(-2{{x}^2}+4x-5)$.
Solution
Again, we start by doing the addition horizontally. For that, we need to remove the parentheses and combine like terms:
$latex (2{{x}^3}+4{{x}^2}-5x)+(3{{x}^3}+3x+4)$ $latex +(-2{{x}^2}+4x-5)$
$latex =2{{x}^3}+4{{x}^2}-5x+3{{x}^3}$ $latex +3x+4-2{{x}^2}+4x-5$
$latex =2{{x}^3}+3{{x}^3}+4{{x}^2}$ $latex -2{{x}^2}-5x+3x+4x+4-5$
$latex =5{{x}^3}+2{{x}^2}+2x-1$
Now, we can do the addition vertically. We separate the variables with different exponents in different columns and leave a space if a polynomial does not have an exponent:
$latex 2{{x}^3}+4{{x}^2}-5x$
$latex 3{{x}^3}~~~~~~~+3x+4$
$latex 2{{x}^2}+4x-5$
________________
$latex 5{{x}^3}+2{{x}^2}+2x-1~~$
Addition of polynomials – Practice problems
Test your knowledge about the addition of polynomials with the following problems. Find the result of the additions and choose your answer. Click “Check” to verify your answer.
See also
Interested in learning more about operations with polynomials? Take a look at these pages:
Jefferson Huera Guzman
Jefferson is the lead author and administrator of Neurochispas.com. The interactive Mathematics and Physics content that I have created has helped many students.
