Question

Which of the following statements about closure is false?

A Polynomials are closed under addition. When you add polynomials, the result will always be a polynomial.
B Polynomials are closed under subtraction. When you subtract polynomials, the result will always be a polynomial.
CC Polynomials are closed under division. When you divide polynomials, the result will always be a polynomial.
D Polynomials are closed under multiplication. When you multiply polynomials, the result will always be a polynomial.

Answer 1

Let us discuss each option briefly:

A)Polynomials are closed under addition. When you add polynomials, the result will always be a polynomial.

Example :
`(2x+1) + (4x+3) = 6x +4`

As seen in this example when we add two polynomials , the answer is also a polynomial.

This statement is true .

B) Polynomials are closed under subtraction. When you subtract polynomials, the result will always be a polynomial.

Example:
`(2x+1) - (x+3) = x-2`

As seen in example the subtraction of two polynomials is also a polynomial.

So this statement is true.

C) Polynomials are closed under division. When you divide polynomials, the result will always be a polynomial.

Example : suppose we want to divide x⁻⁴ by x²

`(x^(-4))/(x^(2))`

On dividing we will get: x⁻⁶

which is not a polynomial.

So the closure property does not hold true for division of polynomials.

D) Polynomials are closed under multiplication. When you multiply polynomials, the result will always be a polynomial.

Example: (4x+1)(2x+3) = 8x² +14x +3

The result here is also a polynomial.

So the closure property is true for multiplication in polynomials.

Answer is option C) division

Answer 2

C is false.

When you divide polynomials, the result is NOT always a polynomial.