Question

Which operation is NOT closed for polynomials ?

Answer 1

Dividing two binomials will not always result in a polynomial. For instance, divide (x+2) over (x+3) and we won't get a polynomial. The result is known to be a rational expression but not a polynomial. So that's why division is not closed for polynomials.

Contrast that with addition, subtraction and multiplication. For any of those operations taking two polynomials and adding them, subtracting them, or multiplying them will lead to another polynomial. That's why these operations are closed for polynomials.

Answer: Choice B

Answer 2

The operation that is not closed for polynomial is $\boxed{{\text{Dividing binomials}}}.$ Option (B) is correct.

Further explanation:

Given:

The given options are as follows,

(A). Adding binomials

(B). Dividing binomials

(C). Subtracting binomials

(D). Multiplying binomials

Explanation:

Consider the two polynomials $x + 1$ and $2x + 4.$

Add the two polynomials $x + 1$ and $2x + 4.$

$\begin{aligned}{\text{Add}} &= x + 1 + 2x + 4\\&= 3x + 5\\\end{aligned}$

The result after adding is a polynomial. Therefore, adding binomials is a closed for polynomials.

After dividing the two polynomials $x + 1$ and $2x + 4$ the result is a rational expression.

Therefore, dividing binomials operation is not closed for polynomials.

Subtract the two polynomials $x + 1$ and $2x + 4.$

$\begin{aligned}{\text{Subtract}}&= 2x + 4 - \left( {x + 1}\right)\\&=x + 3\\\end{aligned}$

The result after subtracting is a polynomial. Therefore, subtracting binomials is a closed for polynomials.

Multiplying the two polynomials $x + 1$ and $2x + 4.$

$\begin{aligned}{\text{Multiplying}}&= \left( {2x + 4} \right)\times \left( {x + 1} \right)\\&= 2{x^2} + 2x + 4x + 4\\&= 2{x^2} + 6x + 4\\\end{aligned}$

The result after subtracting is a polynomial. Therefore, multiplying binomials is a closed for polynomials.

The operation that is not closed for polynomial is $\boxed{{\text{Dividing binomials}}}.$ Option (B) is correct.

Option (A) is not correct as the adding binomials operation is closed for polynomials.

Option (B) is correct as the dividing binomials operation is not closed for polynomials.

Option (C) is not correct as the adding binomials operation is closed for polynomials.

Option (D) is not correct as the adding binomials operation is closed for polynomials.

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Answer details:

Grade: High School

Subject: Mathematics

Chapter: Polynomials

Keywords: polynomials, function, dividing, subtracting, adding, multiplying, binomials, closed for binomials.