Question

Find two polynomial expressions whose quotient, when simplified, is 1/x . ?

Recall the definition of a polynomial expression. Find two polynomial expressions whose quotient, when simplified, is 1/x . Use that division problem to determine whether polynomials are closed under division. Then describe how the other three operations—addition, subtraction, and multiplication—are different from division of polynomials.

Answer 1

See explanation...

Explanation:

Actually `1` and `x` are both polynomials, so `1/x` is already a quotient of polynomials.

If you prefer, you can multiply these polynomials by the same polynomial to get a more complicated quotient.

For example:

`(x+1)/(x^2+x) = (x+1)/(x(x+1)) = 1/x`

Note that `1/x` is not a polynomial, so polynomials are not closed under division.

Polynomials are closed under addition, subtraction and multiplication. Addition and multiplication are associative and commutative.

There is an identity under addition, namely the polynomial `0`.

There is an identity under multiplication, namely the polynomial `1`.

So they form a commutative ring but not a field.