Adding and Subtracting Polynomials

A polynomial looks like this:

polynomial example
example of a polynomial
this one has 3 terms

To add polynomials we simply add any like terms together ... so what is a like term?

Like Terms

Like Terms are terms whose variables (and their exponents such as the 2 in x2) are the same.

In other words, terms that are "like" each other.

Note: the coefficients (the numbers you multiply by, such as "5" in 5x) can be different.


7x x -2x πx

are all like terms because the variables are all x


(1/3)xy2 -2xy2 6xy2 xy2/2

are all like terms because the variables are all xy2

Example: These are NOT like terms because the variables and/or their exponents are different:

2x 2x2 2y 2xy

Adding Polynomials

Two Steps:

Example: Add   2x2 + 6x + 5   and   3x2 - 2x - 1


Start with:2x2 + 6x + 5   +   3x2 − 2x − 1
Place like terms together:2x2+3x2   +   6x−2x   +   5−1
Which is:(2+3)x2  +   (6−2)x   +   (5−1)
Add the like terms:5x2  +   4x   +   4

Here is an animated example:

(Note: there was no "like term" for the -7 in the other polynomial, so we didn't have to add anything to it.)

Adding in Columns

We can also add them in columns like this:

Adding Several Polynomials

We can add several polynomials together like that.

Example: Add     (2x2 + 6y + 3xy)  ,   (3x2 - 5xy - x)   and   (6xy + 5)

Line them up in columns and add:

2x2 + 6y + 3xy
3x2      - 5xy - x
           6xy     + 5

5x2 + 6y + 4xy - x + 5

Using columns helps us to match the correct terms together in a complicated sum.

Subtracting Polynomials

To subtract Polynomials, first reverse the sign of each term we are subtracting (in other words turn "+" into "-", and "-" into "+"), then add as usual.

Like this:

Note: After subtracting 2xy from 2xy we ended up with 0, so there is no need to mention the "xy" term any more.