You can prove that a statement isn't true by finding a single example that contradicts the statement, which is called a counterexample. Show that the set of polynomials is not closed under division by finding a counterexample of division of a polynomial by a polynomial that does not result in a polynomial.

allbleachix 2022-09-12 Answered
You can prove that a statement isn't true by finding a single example that contradicts the statement, which is called a counterexample. Show that the set of polynomials is not closed under division by finding a counterexample of division of a polynomial by a polynomial that does not result in a polynomial.
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Answers (1)

Gabriela Werner
Answered 2022-09-13 Author has 11 answers
The expressions x and x 2 are polynomials. When the expressions are divided, the result is 1 x or x 1 . This quotient is not a polynomial since the exponent of x is a negative number. This shows that dividing two polynomials does not necessarily result to a polynomial. Hence, the set of polynomials is not closed under the division operation.
Result:
Possible solution: x ÷ x 2
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