Question

Side 1: 3x2 − 2x − 1

Side 2: 9x + 2x2 − 3
Part C: Do the answers for Part A and Part B show that the polynomials are closed under addition and subtraction? Justify your answer. (2 points) how do i figure out if its closed or not

Answer 1

Let me explain how a set of numbers is closed or open.
If a set of numbers is closed under an operation, that means that if you use elements of that set to do the operation, the solution must also be an element of the set.

Answer 2

Here is an explanation using specific sets of numbers and specific operations to explain more clearly what the statement above means.

Answer 3

Question: Is the set of integers closed under multiplication?

Answer: Here we are using specifically the set of integers, and we are using
multiplication as the operation.
If you multiply any two integers, is the product also an integer?
The answer is yes, so the set of integers is closed under multiplication.

The set of integers is also closed under addition and subtraction.
Any two integers added together equal an integer.
Any subtraction of integers is an integer.

Do you follow so far?

Answer 4

kinda i was wondering if i need to work out the problem to figure this out or if it needs to be worked out at all

Answer 5

@mathstudent55

Answer 6

hello

Answer 7

If you add two polynomials, do you always get a polynomial?
If you subtract two polynomials, is the difference always a polynomial?
If both answers are yes, then polynomials are closed under addition and subtraction.

Answer 8

you lost me with that

Answer 9

Here an example of a set of items not being closed under an operation.
The set of whole numbers is not closed under subtraction.
The set of whole numbers is {1, 2, 3, 4, 5, ...} (Notice, there are no negative numbers int eh set of whole numbers.)
Let's do this subtraction: 5 - 10 = -5
We did a subtraction of whole numbers, 5 and 10, and the answer is -5, not a whole number.
That shows that the set of whole numbers is not closed under subtraction.

Answer 10

Try adding any two polynomials.
Isn't the sum always going to be a polynomial?

Answer 11

im going to add the two they already gave me

Answer 12

5x2+11x-2 is what i got when i added them together

Answer 13

Ok. Isn't that a polynomial?
You cannot prove this from one single addition of polynomials, but if you think of what a polynomial is, then adding polynomials together will only produce other polynomials, no matter which polynomials you added together, so addition of polynomials is closed under addition.

Answer 14

Now think of subtraction of polynomials.
When you subtract polynomial B from polynomial A, you are simply adding polynomial A to the additive inverse of polynomial B. Polynomial B is a polynomial. The additive inverse of polynomial B is also a polynomial. Then when you add polynomial A to the additive inverse of polynomial B, you are just adding polynomials, and we already concluded above that the sum of polynomials will always be a polynomial, so polynomials are also closed under subtraction.

Answer 15

so that's it i'm done thank you so much

Answer 16

can i ask your help for another one

Answer 17

Yes, that is it.
Polynomials are closed under addition and subtraction.

This is not part of your problem, but polynomials are also closed under multiplication since the product of two polynomials is always a polynomial.
Polynomials are NOT closed under division because the quotient of two polynomials is not always a polynomial.

Answer 18

wait what do you mean this is not my problem

Answer 19

i thought it was addition and subtraction

Answer 20

When I wrote above "This is not part of your problem.", I meant what is below that line, dealing with multiplication and division of polynomials. The part above is exactly your problem. Replace "this" with "what follows."

Answer 21

oh ok well can you help me another one

Answer 22

@mathstudent55