Use Sophia to knock out your gen-ed requirements quickly and affordably. Learn more
×

Adding and Subtracting Polynomials

Author: Sophia

what's covered
In this lesson, you will learn how to add or subtract two polynomial expressions. Specifically, this lesson will cover:

Table of Contents

1. Combining Like Terms

When adding and subtracting polynomials, it really is an exercise in combining like terms. Like terms have the same variables, which are raised to the same power.

EXAMPLE

2 x squared and 7 x squared are like terms, because they both contain x raised to the second power.

EXAMPLE

5 x cubed and 5 x squared are NOT like terms. Although they both start with 5 and also contain the variable x, their exponents are different: one is x cubed, and the other is x squared.

big idea
Like terms can be combined by adding their coefficients. For instance, combining 2 x squared and 7 x squared gives us 9 x squared.

If coefficients are negative, we can think of it as subtraction: combining 2 x squared and short dash 7 straight x squared gives us short dash 5 straight x squared.


2. Identifying Like Terms in Polynomials

Polynomials have several terms, and likely contain several different types of terms. When we add and subtract polynomials, it is important to identify like terms and organize the terms by their type, so that we can easily add or subtract their coefficients.

EXAMPLE

Rewrite the following polynomial by combining like terms:

2 x y plus 4 x squared plus 3 x squared y minus x squared

Let's first examine the variables and exponents to determine which terms are like terms and can be combined. When looking at the term 2 x y, we scan the other terms to see if we have other terms with an x and a y. We see 3 x squared y, but is this a like term to 2 x y? It actually isn't. This is because 3 x squared y has two factors of x (the exponent of 2 is attached to the x), whereas 2 x y only has 1 factor of x.

We do however have the like terms short dash x squared and 4 x squared. These two terms can be combined as 3 x squared because -1 and 4 add to 3.

Let's rewrite this polynomial with the combined like terms:

3 x squared y plus 3 x squared plus 2 x y

hint
It is standard to write the terms in order of their degree, from highest to lowest. In the example above, 3 x squared y is written first because its degree is 3 (since the x variable had an exponent of 2 and there is an implied exponent of 1 for y), whereas the other terms have a degree of 2.


3. Adding Polynomials

When adding polynomials, it helps to write the problem vertically, so that you can align the like-terms, making the addition of the coefficients easier to do in your head. We'll see in the section below that this is the same process for subtraction.

EXAMPLE

Add the following two polynomials:

open parentheses 3 x cubed plus 2 x minus 5 close parentheses plus open parentheses 2 x cubed plus 5 x squared minus 7 x close parentheses

Let's first set this up by writing the problem vertically:

table attributes columnalign left end attributes row cell space space space left parenthesis 3 x cubed plus 2 x minus 5 right parenthesis end cell row cell stack plus left parenthesis 2 x cubed plus 5 x squared minus 7 x right parenthesis with bar below end cell end table

Before we complete the addition, let's take a closer look at the vertical set up. The whole point of writing the addition vertically is to line up the like terms. But we noticed that there were some terms in one polynomial that were not in the other, so like terms are not lining up together. In this case, it helps to write a term with a coefficient of zero, as a placeholder to keep everything vertically aligned.

table attributes columnalign left end attributes row cell space space space left parenthesis 3 x cubed plus 0 x squared plus 2 x minus 5 right parenthesis end cell row cell stack plus left parenthesis 2 x cubed plus 5 x squared minus 7 x plus 0 right parenthesis with bar below end cell end table

Now we can add the two polynomials together, combining the coefficients of like terms:

table attributes columnalign left end attributes row cell table attributes columnalign left end attributes row cell space space space left parenthesis 3 x cubed plus 0 x squared plus 2 x minus 5 right parenthesis end cell row cell stack plus left parenthesis 2 x cubed plus 5 x squared minus 7 x plus 0 right parenthesis with bar below end cell end table end cell row cell space space space space space 5 x cubed plus 5 x squared minus 5 x minus 5 end cell end table


4. Subtracting Polynomials

When subtracting polynomials, we follow the same procedure, only we subtract coefficients rather than add them. The only tricky thing to watch out for is subtracting a negative number. This should be thought of as adding a positive number.

EXAMPLE

Subtract the following two polynomials:

open parentheses 6 x cubed minus 4 close parentheses minus open parentheses 2 x squared minus 7 x minus 3 close parentheses

Once again, we'll write the problem vertically and add terms with coefficients of 0 to keep everything aligned. It also helps to group each polynomial in parentheses, so that we can get the subtraction out in front and still be aware of the signs of each term within the polynomial.

table attributes columnalign left end attributes row cell space space space open parentheses 6 x cubed plus 0 x squared plus 0 x minus 4 close parentheses end cell row cell negative stack open parentheses 0 x cubed plus 2 x squared minus 7 x minus 3 close parentheses with bar below end cell end table

Now we can subtract the two polynomials, evaluating for the coefficients of like terms:

table attributes columnalign left end attributes row cell space space space open parentheses 6 x cubed plus 0 x squared plus 0 x minus 4 close parentheses end cell row cell stack negative open parentheses 0 x cubed plus 2 x squared minus 7 x minus 3 close parentheses with bar below end cell row cell space space space space space 6 x cubed minus 2 x squared plus 7 x minus 1 end cell end table

hint
Was any part of that subtraction tricky? It was probably easy enough to get the coefficients of the first two terms:

6 x cubed minus 0 x cubed equals 6 x cubed
0 x squared minus 2 x squared equals short dash 2 x squared

However, sometimes it can be difficult to remember signs with subtraction, especially for the last two coefficients:

0 x minus open parentheses short dash 7 x close parentheses equals 0 x plus 7 x equals 7 x
short dash 4 minus open parentheses short dash 3 close parentheses equals short dash 4 plus 3 equals short dash 1

summary
When combining like terms, add the coefficients and keep the variables and powers the same. Identifying like terms in polynomials is where the terms have the same variables and associated variable power. When adding polynomials, a term with an addition sign or no sign indicates that the coefficient for the term is positive. When subtracting polynomials, a term with a subtraction sign before it signifies that the coefficient for the term is negative.

Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License