SOLUTION: A student states that polynomials are closed under division. Which of the following expressions are counterexamples to the statement? Choose all answers that are correct. _

SOLUTION: A student states that polynomials are closed under division. Which of the following expressions are counterexamples to the statement? Choose all answers that are correct. __

Question 934535: A student states that polynomials are closed under division. Which of the following expressions are counterexamples to the statement?
Choose all answers that are correct.
__ x^3 � x^6
__ x^6 � 3x^2
__ (x^4 � 121) � (x � 121)
__ (x^2 � 6x + 9) � (x � 3)
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
if polynomials are closed under division, then the answer must also be a polynomial.

a polynomial must have all variables raised to a non-negative integer.

x^3/x^6 = x^-3 which is not a polynomial because the exponent of the variable is negative.

(x^6) / (3x^2) = (x^4)/3 which is a polynomial because the exponent of the variable is a non-negative integer.

(x^4 - 121) / (x - 121) results in x^3 + 121x^2 + 121^2x + 121^3 + (121^4-121)/(x-121) which is not a polynomial because the last term has the variable being raised to a negative exponent because 1/(x-121) = (x-121)^-1.

(x^2 - 6x + 9) / (x-3) results in (x-3) which is a polynomial because the exponent of the variable is a non-negative integer.

looks like the first one and the third one are counter examples because the result of those operations is not a polynomial.

here's some examples of expressions that are polynomial and expressions that are not polynomial.

http://www.mathwarehouse.com/algebra/polynomial/polynomial-equation.php

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