Adding and Subtracting Polynomials

Contributor: Mason Smith. Lesson ID: 11548

Polynomials seem complex enough on their own, so why add and subtract them to make things worse? Actually, it's not that bad, and you will learn the simple rules that make them friendly and useful!


Algebra I, Expressions and Equations

learning style
personality style
Grade Level
High School (9-12)
Lesson Type
Quick Query

Lesson Plan - Get It!


Teacher: What is b + b?

Shakespeare: Is it 2b or not 2b?

Polynomials may appear large and terrifying when you first encounter them, but you can perform basic operations on polynomials (addition and subtraction) in the same way that you perform operations on numbers.

If you have not yet done so, please review the first lesson in the Polynomials series found in the right-hand sidebar under Related Lessons.

There is just one extra rule that you need to follow.

The rule is pretty simple, really. All you need to remember is that you can only combine terms with like variables.

Okay, maybe it does get a little more tricky than that. The variables need to be exactly the same when you add two polynomials.

What does that mean?

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

For example, if you have x2 + x2, you can most definitely combine those terms together 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.

Yes, both variables are x, but take a 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 either, because the variables (x and y) themselves are different.

Fear not — in any other situation, feel free to combine your terms. In many cases, you can add and subtract different terms in polynomials without any major cause for concern.

For example:

2x + 4x = 6x

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

8p - 5p = 3p

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

5n + 6n2 = ?

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

  • So what happens when we are faced with unlike variables?

Do we run in fear? Break pencils in rage? Quake in our 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

Let's add our two like terms, giving us 17m3.

  • Can we go any further with this equation?

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

Let's 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 we can skip this step.

Combine like terms together: 0

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 referred to as "flipping the sign."

For example, take 2x2 + 6 - (4x2), which you would approach like this:

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

From here, you simply 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.

Let's 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)

I bet you know what to do from here. Give it a try on your own, and then check your answer:

For further clarification on adding and subtracting polynomials, take a few minutes to watch Beginning Algebra & Adding Subtracting Polynomials by Bill White:

We have covered 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.

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

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