Question

. Find two polynomial expressions whose quotient, when simplified, is 1/x . Use that division problem to determine whether polynomials are closed under division.

Answer 1

The two polynomials are:

(x + 1) and (x² + x)

Explanation:

A polynomial is simply an expression which consists of variables & coefficients involving only the operations of addition, subtraction, multiplication, and non - negative integer exponents of variables.

Now, 1 and x are both polynomials. Thus; 1/x is already a quotient of a polynomial.

Now, to get two polynomial expressions whose quotient, when simplified, is 1/x, we will just multiply the numerator and denominator by the same polynomial to get more quotients.

So,

Let's multiply both numerator and denominator by (x + 1) to get;

(x + 1)/(x(x + 1))

This gives; (x + 1)/(x² + x)

Now, 1 and x are both polynomials but the expression "1/x" is not a polynomial but a quotient and thus polynomials are not closed under division.