Let’s start this lesson with a few things that you’ll have to understand before we start adding and subtracting. A polynomial is an expression with more than one term. So what’s a term? A term can be a few things. It can be just a number like:

4                                 -23                              .5                                100,000,001

These all qualify as single terms. A term can also be a variable like:

x                                  b                                  g²                                y³

Lastly, a term can be a collection of the above. This would include:

xy²                              45ab²c                                   9dx³                           3.14r²

An addition or a subtraction sign separates terms.

This is one term, or a monomial:                                   6ab²

This would be two terms, or a binomial:                       4xy + 8z

This would be three terms, or a trinomial:                    d² + 4d – 3

How many terms does the following have?

-5x² – 14x³y + 4xy³ + 11x³

There are four terms!

Ok, got that out of the way! Now, let’s talk about something known as “like terms”. When terms are alike, they can be combined. Terms can be combined when the variables and the exponents of those variables are exactly alike. For example:

Alike:                       4x²y and -3x²y                      5a²b³c and 2a²b³c                    -4 and 6

Not Alike:                -3x²y and 2xy                       3a²b³c and 5a³b²c                   6yz and 6z

When terms are alike, we can add them together. Here is an example:

(-8x⁴ – 12x³y) + (-5x⁴ – 14x³y + 4xy³) + (-7xy³ – 4x³y)

Let’s highlight “like terms” with the same colors.

(-8x⁴ – 12x³y) + (-5x⁴ – 14x³y + 4xy³) + (-7xy³ – 4x³y)

We combine these by keeping the variables exactly the same but adding the coefficients (the number in front of the variable) and end up with:

-13x⁴ – 30x³y - 3xy³

Answer: -13x⁴ – 30x³y - 3xy³

And that’s all there is to adding polynomials! When we subtract polynomials, it adds one extra step. When there is a minus sign in front of the parenthesis, we can change it to addition as long as we change every sign in the parenthesis as well. We are technically distributing a “-1” into the parenthesis. For example:

(5x³ + 8x⁴y) – (-7x³ – 3x²y² + 12x⁴y⁴) + (-7x²y² – 11x³)

The second polynomial has a subtraction sign in front, so let’s change that.

(5x³ + 8x⁴y) + (7x³ + 3x²y² - 12x⁴y⁴) + (-7x²y² – 11x³)

Now, let’s locate “like terms”:

(5x³ + 8x⁴y) + (7x³ + 3x²y² - 12x⁴y⁴) + (-7x²y² – 11x³)

If a term doesn’t have any like terms, then it just stays the same value.

x³ + 8x⁴y - 4x²y² - 12x⁴y⁴

Answer: x³ + 8x⁴y - 4x²y² - 12x⁴y⁴

Below you can download some free math worksheets and practice.