Adding and Subtracting Polynomials

Lesson Plan - Get It!

Audio:

Polynomials may appear large and terrifying when you first encounter them. Still, you can perform basic operations on polynomials (addition and subtraction) like on numbers.

You need to follow just one extra rule.

The rule is pretty simple. You must remember that you can only combine terms with like variables.

It may get a little more tricky than that. The variables must be the same when adding two polynomials.

• What does that mean?

It means that not only must the variable be the same, but if exponents are involved, those exponents must be the same as well.

For example, if you have x² + x², you can combine those terms because the variables are identical. Not only is the variable x on both sides of the addition sign, but they are also both squared.

On the other hand, if you have x + x², you cannot combine these variables.

Both variables are x but look at the exponents. The first variable has an implied one (not shown), and the other is squared. Since their exponents are different, they are considered unlike variables.

You cannot combine x² + y² because the variables (x and y) are different.

Fear not — feel free to combine your terms in any other situation. You can often add and subtract different terms in polynomials without any major cause for concern.

Check out these examples.

2x + 4x = 6x

Since the variables are the same, you add the coefficients, or numbers, attached to the variables.

8p - 5p = 3p

Since the variables are the same, you subtract the coefficients, or numbers, attached to the variables.

5n + 6n² = ?

Trick question! The variables aren't the same, so you can't add them together!

What happens when faced with unlike variables?

• Do you run in fear? Break pencils in rage? Quake in your boots and quit?

Sure, any of those may help to some extent, but to save yourself the agony and possible embarrassment, you could first try rearranging the variables so all like terms are together!

For example: 15m³ + 6m² + 2m³

Rearrange the terms so all like terms are together.

This gives you two coefficients with the variable m³: 15m³ and 2m³

Add the two like terms: 17m³.

• Can you go any further with this equation?

No. So, after combining like terms, you are left with the answer: 17m³ + 6m²

Try another example: 3x² + 5 - 7x² + 12.

Rearrange the terms so all like terms are together: 3x² - 7x² + 5 + 12

Combine the like terms together: -4x² + 17

One more example: 2x²y - x²y - x²y.

All terms are like terms so you can skip this step.

Combine like terms together: 0

What about negative signs?

The last potential area for difficulty when adding and subtracting polynomials comes when you need to distribute a negative sign.

You must distribute the negative sign to all terms in the parenthesis ().

This means that if a value is positive, it becomes negative. Likewise, if the value is negative, it becomes positive.

This is often called flipping the sign.

For example, you would approach 2x² + 6 - (4x²) as follows.

When you flip the sign in the parenthesis, the result is 2x² + 6 - 4x².

From here, you rearrange the polynomial so the like terms are together; you are left with 2x² - 4x² + 6.

Combine the like terms and simplify to -2x² + 6.

Try another: (a⁴ - 2a) - (3a⁴ - 3a + 1).

Distribute the negative and flip the sign to get rid of any parenthesis: (a⁴ - 2a) - (3a⁴ + 3a - 1)

You know what to do from here. Give it a try on your own, and then check your answer.

Nice job!

You have learned how to add and subtract polynomials by flipping the sign inside the parenthesis, rearranging the terms so like terms are together, and combining like terms.

Review with the video below for further clarification on adding and subtracting polynomials.

Now it's time to move to the Got It? section to practice these skills and see if you got it!