Question

Which of the following equations demonstrate that the set of polynomials is not closed under the certain operations?

Answer 1

Do you have the equations? Your question is missing the equations.

Answer 2

Multiplication: (x^2-5x+3)(x-5)=x^3-10x^2+28x-15
Division: (x^2-5x+3) / (x-5)=x+ 3/x-5
Division: (x^2+x)/(x+1)=x
Subtraction: (3x^4 +x^3)-(-2x^4+x^3)=5x

Answer 3

The best answer is "Division: (x^2-5x+3) / (x-5)=x+ 3/x-5"
Note that \((x+3)/(x-5)\) is not a polynomial.

Basically, a set is "closed" under an operation if you do that operation with members of the set and you still get a result that is still in the set.

A polynomial is an algebraic expression with variables and terms; the only operations in a polynomial should be addition, subtraction, multiplication. All variables should have nonnegative integer exponents, i.e. {0,1,2,3,...}. So things such as \(x^2 + 3x + 4\) and \(1 (=x^0)\) and \(3x^2y+5y\) are polynomials but \[ \frac{x^2+3}{4-x}\] is not a polynomial because there is division.

So you can see in that example, doing division got you something that is not a polynomial