Question

Mariana and Bethany disagree about whether the set of polynomials is closed under addition, subtraction, and multiplication, as the set of integers is. To examine their two positions, they consider the polynomials A = 5x − 1 and B = 2x. Mariana thinks these polynomials are closed under all three operations. Bethany argues that these polynomials are closed under only two operations

Answer 1

If the question is asking who is right and why -

In order for polynomials to be closed under addition, that means any polynomials added together must also result in another polynomial

Same logic with subtraction, division, etc.

So you need to add the 2 polynomials, subtract the 2 polynomials, multiply the two polynomials, and divide the two polynomials, to demonstrate that they’re closed under these operations

It’s a lot of arithmetic unfortunately but that’s the logic behind it

Answer 2

\(\color{#0cbb34}{\text{Originally Posted by}}\) @Vocaloid
If the question is asking who is right and why -

In order for polynomials to be closed under addition, that means any polynomials added together must also result in another polynomial

Same logic with subtraction, division, etc.

So you need to add the 2 polynomials, subtract the 2 polynomials, multiply the two polynomials, and divide the two polynomials, to demonstrate that they’re closed under these operations

It’s a lot of arithmetic unfortunately but that’s the logic behind it
\(\color{#0cbb34}{\text{End of Quote}}\)
thank you!