For the polynomial $\frac{x^{3}+2 x+1}{5}-\frac{7}{2} x^{2}-x^{6},$ write:
$(i)$ the degree of the polynomial
$(ii)$ the coefficient of $x^{3}$
$(iii)$ the coefficient of $x^{6}$
$(iv)$ the constant term

(N/A) $(i)$ The degree of a polynomial is the highest power of the variable present in the expression. In the given polynomial $\frac{1}{5}x^{3} + \frac{2}{5}x + \frac{1}{5} - \frac{7}{2}x^{2} - x^{6}$,the highest power of $x$ is $6$. Thus,the degree is $6$.
$(ii)$ The term containing $x^{3}$ is $\frac{x^{3}}{5}$,which can be written as $\frac{1}{5}x^{3}$. Therefore,the coefficient of $x^{3}$ is $\frac{1}{5}$.
$(iii)$ The term containing $x^{6}$ is $-x^{6}$,which is $-1 \cdot x^{6}$. Therefore,the coefficient of $x^{6}$ is $-1$.
$(iv)$ The constant term is the term independent of $x$. In the expanded form $\frac{1}{5}x^{3} - \frac{7}{2}x^{2} + \frac{2}{5}x + \frac{1}{5}$,the constant term is $\frac{1}{5}$.