If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Algebra 1 (Eureka Math/EngageNY)>Unit 1

Lesson 3: Topic B: Lesson 8: Adding and subtracting polynomials

# Polynomial subtraction

Learn how to subtract polynomials. Discover how distributing a negative sign changes the signs of all terms in a polynomial. Understand that when you subtract polynomials, you still get a polynomial, showing that the set of polynomials is 'closed' under subtraction.

## Want to join the conversation?

• What exactly is set closure?
• If you perform an operation on two elements of some set, the result of the operation will also be in the set . . . . .

Suppose we have some popcorn in a big paper bag - but it is not enough. so we add more popcorn to it. After adding more popcorn we look in the bag. It is still popcorn, right? We could say that popcorn is closed under addition, that is, if you add popcorn to popcorn, you get popcorn, not licorice or chocolate.

It is the same with closed sets. If a set is closed, that means if you add, or subtract or multiply members of the set with each other, the result will also be a member of the set. In this case, we performed subtraction on two elements from the set of polynomials and the result was another polynomial - that is because the set of polynomials is closed under subtraction.

Whether a set is closed or not becomes very important in later math. There are sets of objects that are not closed under some operations, for example, the positive integers are not closed under subtraction: 3 is a positive integer, 2 is a positive integer, but 2-3 is not a positive integer.

You may want to spend some free time researching "Abelian groups" and "algebraic fields" . This is all very abstract, but an investigation and understanding of it will pay off in your future mathematical studies.
• Algebra 2 is so much easier than algebra 1! Upvote if you agree!!
• I agree. The concepts are much easier to understand!
• would you get a polynomial if you added two polynomials (just like subtraction)?
• A polynomial by definition consists of several terms. When you add polynomials, the result will still be multiple terms, so it fits the definition of a polynomial. To summarize, yes, you would get a polynomial.
• Does anyone else's brain hurt when watching this? It hurts!
• the PaaaAAAAaaaIiiIIIiIIINNNnnNNNN
• What's the difference between just subtracting polynomials and subtracting polynomials with set closure?
And what doe sit mean by set closure? I don't whats the difference between just subtracting the polynomials....
• When subtracting polynomials with set closure, you are subtracting two polynomials from one another to get another polynomial as your answer. And on the topic of set closure, someone else asked what it was so I suggest you check out the answer to that if you need help.
• at why not change the signs of 6x squared + 3x - 9
• For your reference, I will use ^2 to mean squared. * means multiplied by.
The reason why the sign does not change is because there is no reason for it too. Sal changes the sign of (-2x^2 + 4x - 1) because he is multiplying it by negative 1. Any number multiplied by 1 is equal to itself, so there could be an "invisible" 1 * in front of every number. So 6 could be written as 1*6. When Sal has 6x^2 + 3x - 9 - (-2x^2 + 4x -1), he wants to combine them, but he must get rid of the parenthesis first. So, he changes 6x^2 + 3x - 9 - (-2x^2 + 4x -1) to 6x^2 + 3x - 9 -1 * (-2x^2 + 4x -1). He then distributes the -1 * to every term inside the parenthesis, and since a negative times a negative is a positive, and a negative times a positive is a negative, this changes all of the signs inside the parenthesis. Since there is no -1 to be distributed to 6x^2 + 3x -9, the signs do not change.
• Can someone please define a polynomial?
• A polynomial has multiple terms that are separate by addition and subtraction signs. Usually there are some specific names. One term is monomial. Two terms in binomial. Three terms is trinominal. 4 Terms in quartic. And i believe 5 terms is called "quartic". Dont quote me on this :p You can usually call an expression a polynomial when the expression has two or more terms like mentioned above/
• why did the variable "x" turned into "q"?? I don't understand
• x never changes to q. Just like how you usually see f(x), Sal used q to define a function. He had to use q because one of the other polynomials was already defined as f(x). I hope this helps. :)