Let us discuss each option briefly:
A)Polynomials are closed under addition. When you add polynomials, the result will always be a polynomial.
Example : [tex] (2x+1) + (4x+3) = 6x +4 [/tex]
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: [tex] (2x+1) - (x+3) = x-2 [/tex]
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
[tex] \frac{x^(-4)}{x^(2)} [/tex]
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
