Question

Under which of the following operations are the polynomials 8a + 14 and
3b - 9 not closed?
A. Subtraction
B. Multiplication
C. Addition
D.Division

Answer 1

The answer is division. Here is why:

First lets understand what a polynomial is (you may already know this, but it's important in order to understand why division is "not closed".

Polynomials are a set (or group) of numbers exhibiting the same traits. That trait is that they are algebraic expressions which contain variables such as 8a+14 and 3b-9.

Two expressions are considered "Closed" under an operation if when you perform the operation the answer belongs to the same set. In this case, the answer MUST result in a Polynomial.

Here is a basic way to think about it. If you had a set of quadrilaterals {Square, Rectangle, Rhombus} and added a Trapezoid to it, the set would remain closed since the addition of trapezoid would still mean the entire set is filled with quadrilaterals. Make sense?

So knowing what a polynomial is, and knowing what "closed" means lets take a look.

If we add the two expressions above do we get a polynomial?

8a+14+3b-9 = 8a+3b+5 YES, so the set is closed under addition, the answer is still a polynomial!

I'll let you complete subtraction and addition, but I assure you that you will have yourself a polynomial at the end of each!

Now division has one property which the other operations don't have. It can break your calculator if you divide by 0.

So WHAT IF you found out that b=3? Well, then 3b-9=0. This would result in an undefined answer.

I'm sure someone else could substantiate further and go into a deeper explanation, but I hope this helps you understand the concept of closure a little better and why the Polynomials are not closed under division.