How to determine degree of polynomials?

Question

Let $A$ an integral domain and $f, g \in A[X]$ such that $\partial (f+g)=5$ and $\partial(f-g)=2$. Determine $\partial (fg)$; $\partial(f^{2}-g^{2})$ and $\partial(f^{2}+g^{2})$.

P.S.: $\partial$ denote the degree of polynomial

Answer 1

The degree of $f$ and the degree of $g$ must be $5$. But the three terms of greater degree of $f$ and $g$ have the same coefficients. Note that since the degree of $f+g$ is not equal to the degree of $f-g$, $1+1\neq0$ in the ring $A$.

Now, it is clear that the degree of $fg$ is $5+5=10$. Since $f^2-g^2=(f+g)(f-g)$, the degree of $f^2-g^2$ is $5+2=7$. Since the degree of $f^2$ and $g^2$ is $10$, the three terms of greater degree has the same coefficients. Thus, the degree of the sum, $f^2+g^2$ is also $10$.