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. Complete the following statements, and then determine who is correct.
A + B = ______ A − B = _____ A • B = 10x2 + ____ ____is correct.

Answer 1

A set of polynomials is closed under an operation if the result after operation is also a polynomials.

Given A = 5x − 1 and B = 2x

A + B

= 5x - 1 + 2x

= 7x -1, a polynomials

A - B

= 5x - 1 - 2x

= 3x - 1, a polynomials

A * B

=(5x - 1)*(2x)

= 10x2 + (-2x), a polynomials

All three operations give a result in polynomials. So Mariana is correct.

Answer 2

A = 5x − 1 and B = 2x

A + B = 5x - 1 + 2x

= 7x - 1

A − B = 5x - 1 - 2x

= 3x  - 1

A • B = (5x - 1) *2x

= 10x2 + (-2x)

all 3 are polynominals

so Mariana is correct.