What is the answer to this question ?

The answer is D. Polynomials must be closed under the operations of addition, subtraction, and multiplication because under these processes, you will most likely/highly need to distribute something.

1) For example, in addition and subtraction, you will need to distribute the sign over the parenthesis.

Let's take B for example:

x^3 + 4x^2 + 2x - 5 - (x^2 + 3x + 1)

When you have a negative sign in front of a parenthesis, you will need to distribute that negative sign to all the terms inside the parenthesis.

x^3 + 4x^2 + 2x - 5 - (x^2 + 3x + 1) will become:

x^3 + 4x^2 + 2x - 5 - x^2 - 3x - 1

Note how the sign changes.

2) For multiplication, it's very important to be closed because you will need to multiply each term in the first parenthesis by each term in the second parenthesis.

So, ultimately, the parenthesis is not needed when dividing polynomials because signs are not being distributed. Dividing polynomials are only solved by long division and by synthetic division.

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