Question

The set of polynomials is closed under the operation of subtraction.

A: Which equation illustrates this concept?

B: Which statement correctly explains this concept?

Select one answer for question A and one answer for question B.

B: The sum of the polynomials (3x+4) and (5x−5) is a polynomial, 8x−1, which shows closure under subtraction.
A: (3x+4)−(5x−5)=81
B: The difference of the polynomials (3x+4) and (5x−5) is not a polynomial. It is 81, which does not show there is closure under subtraction.
A: (3x−4)+(2x+2)=5x−2
A: (3x+4)+(5x−5)=8x−1
B: The product of the polynomials (3x−4) and (2x+2) is a polynomial, 6x2−2x−8, which shows closure under subtraction.
B: The sum of the polynomials (3x−4) and (2x+2) is a polynomial, 5x−2, which shows closure under subtraction.
A: (3x−4)−(2x+2)=x−6
A: (3x−4)(2x+2)=6x2−2x−8
B: The difference of the polynomials (3x−4) and (2x+2) is a polynomial, x−6, which shows closure under subtraction.

Answer 1

See below.

Explanation:

A. (3x - 4) - (2x + 2) = x - 6 Shows closure under subtraction.

B. The difference of the polynomials (3x−4) and (2x+2) is a polynomial, x−6, which shows closure under subtraction.

Answer 2

A: (3x−4)−(2x+2)=x−6

B: The difference of the polynomials (3x−4) and (2x+2) is a polynomial, x−6, which shows closure under subtraction.

Explanation:

Only two of the answer choices have anything to do with differences, and only one of those correctly computes the difference. That answer choice is the correct one in each case.

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Often, these questions are mostly about choosing an answer that is responsive to the question. Answer choices in left-field can be ignored immediately.

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When a set (polynomials) is closed under an operation (subtraction), it means that the result of that operation on any two members of the set is also a member of the set.

(Any sets containing 0 are not closed under division because division by 0 is undefined.)