Question

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.
Please help, thanks so much!!

Answer 1

Answer:

Step-by-step explanation:

A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

Actually 1 and x both are polynomials. Therefore 1/x is already a quotient of polynomials.

Two polynomial expressions whose quotient, when simplified, is 1/x

x+1/x^2+x

Take the common of the denominator

x+1/x(x+1)

x+1 will be cancelled out by x+1

1/x

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 additive identity which is 0.

There is a multiplicative identity which is 1....

Answer 2

Answer:

Polynomials are not closed under division. Many examples can prove that they are not. Any division of polynomials that leaves a variable to the first or higher power in the denominator is not a polynomial because it has a variable with an exponent that is not a positive integer.

When we add or subtract two polynomials, we’re combining like terms, or terms with the same power of the same variable. Therefore, the exponents and variables don’t change. Only the coefficients of each term might change. This means that the result must be a polynomial, and therefore polynomials are closed under subtraction.

The product of polynomials can have the same original variables as the factors but with higher integer exponents. These products will also be polynomials.

Step-by-step explanation:

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