The given quantified statement is:
$\forall p \in \P_3, \forall q \in \P_3, p-q \in \P_3.$
($\P_3$ stands for "third degree polynomial")
The question asks to state if the original is true or false. I negated the statement to be: $\exists p \in \P_3, \exists q \in \P_3, p-q \in \P_3.$ I found the negation to be false, and the original to be true. Can I get a confirmation? I am not able to think of any counter examples, making me believe that the original is true. Thanks!
On
The first statement is a false statement.
Strategy to find counter example: Well, to change the degree, make sure the leading coefficients of the two polynomials $p$ and $q$ are the same, hence the leading term will be eliminated.
Secondly, you negated the statement wrongly, the negation should be $$\exists p \in P_3, \exists q \in P_3, p-q \notin P_3.$$
The statement "$\exists p \in P_3, \exists q \in P_3, p-q \in P_3$" is true. Prove this by finding an example.
The original statement is false
Use $p=4x^3 + 2x^2 +6x -2$ and $q=4x^3$ for a simple example (there are plenty more)