Objective Learn how to add and subtract polynomials.

This lesson is not difficult conceptually, but it is important that you get sufficient practice in addition and subtraction of polynomials.

Let's begin with an example in order to see how to add polynomials.

Example 1

Add x 3 + x + 1 and 3x 3 + x 2 + 2x .

Solution

First, write the polynomials side by side.

( x 3 + x + 1 ) + ( 3x 3 + x 2 + 2x )

The monomials x 3 and x each occur twice, so group the terms together and combine them.

 x 3 + x + 1 + 3x 3 + x 2 + 2x = ( x 3 + 3x 3 ) + x 2 + ( x + 2x ) + 1 = 4x 3 + x 2 + 3x + 1

This polynomial is simpler than the original two polynomials written side by side.

## Like Terms

To be explicit about the steps taken to add polynomials, let's talk a little bit about like terms.

Definition of Like Terms

When we are given two polynomials, we say the monomials in the polynomials are like terms if they contain exactly the same number of occurrences of each variable.

Example 2

Name the like terms in x 4 + 4x 3 + 6x 2 + 4 x + 1 and x 3 + x 2 + 1.

Solution

There are several pairs of like terms, shown in the diagram below.

The pairs of like terms are connected by arrows. So, the like terms are 4x 3 and x 3 , 6x 2 and x 2 , 1 and 1.

Example 3

Name the like terms in ab + 2a + 2b and ab - a + 1.

Solution

In this case, there are two pairs of like terms, ab and ab , 2a and -a , shown in the diagram below.

Whenever there are like terms, collect them and add them together to get a single term. In Example 2, add 4x 3 and x 3 together to get 5x 3 . In the same way, add 6x 2 and x 2 together to get 7x 2 , and finally, 1 + 1 = 2.

When we do this, we get a simpler polynomial.

( x 4 + 4x 3 + 6x 2 + 4 x + 1 ) + ( x 3 + x 2 + 1 ) = x 4 + 5x 3 + 7x 2 + 4x + 2

In Example 3, collect the ab and a terms to get

( ab + 2a + 2b ) + ( ab - a + 1) = 2ab + a + 2b + 1.

Key Idea

Sums of polynomials can be simplified by adding together like terms. In the same way, differences of polynomials can be simplified by collecting together all pairs of like terms.

## Subtracting Polynomials

Example 4

Simplify ( x 3 - 3x 2 + 3 x - 1) - ( x 2 + 2x + 1).

Solution

( x 3 - 3x 2 + 3 x - 1) - ( x 2 + 2x + 1) = ( x 3 - 3x 2 + 3 x - 1) + ( -x 2 - 2x - 1)

To simplify the difference, collect all pairs of like terms. These are shown in the diagram below.

Collect the terms to get the following polynomial.

x 3 + ( - 3 - 1) x 2 + (3 - 2) x + ( - 1 - 1) = x 3 - 4x 2 + x - 2

There is another method of adding and subtracting polynomials that is similar to adding and subtracting numbers. Write the polynomials one over the other, with like terms lined up in columns. To find the difference, either subtract the coefficients in each of the columns, or add the additive inverse.

Example 5

Subtract 4x 3 + 5x 2 + 6x + 7 and 2x 3 + 3x + 4.

Solution

Write the polynomials with the like terms in columns.

Notice that a zero coefficient is added for each missing term. Now add the coefficients in each column.