# Adding and Subtracting Polynomials

A polynomial looks like this:

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 x^{2}) 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.

### Example:

7x |
x |
-2x |
πx |

are all **like terms** because the variables are all **x**

### Example:

(1/3)xy^{2} |
-2xy^{2} |
6xy^{2} |
xy/2^{2} |

are all **like terms** because the variables are all **xy ^{2}**

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

2x |
2x^{2} |
2y |
2xy |

## Adding Polynomials

Two Steps:

- Place
**like terms**together - Add the like terms

Example: Add **2x ^{2} + 6x + 5** and

**3x**

^{2}- 2x - 1

Start with: | 2x + ^{2} + 6x + 53x^{2} - 2x - 1 |
||||

Place like terms together: | 2x^{2} + 3x^{2} |
+ |
6x - 2x |
+ |
5 - 1 |

Add the like terms: | (2+3)x^{2} |
+ |
(6-2)x |
+ |
(5-1) |

= **5x ^{2} + 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 **(2x ^{2} + 6y + 3xy)** ,

**(3x**and

^{2}- 5xy - x)**(6xy + 5)**

Line them up in columns and add:

2x^{2} + 6y + 3xy

3x^{2} - 5xy - x

__ 6xy + 5__

5x^{2} + 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.*