If the polynomials $ax^3 + 4x^2 + 3x - 4$ and $x^3 - 4x + a$ leave the same remainder when each is divided by $x - 3$,find the value of $a$.

Give an example of a polynomial,which is:
$(i)$ monomial of degree $1$
$(ii)$ binomial of degree $20$
$(iii)$ trinomial of degree $2$

Divide $p(x)=21+10 x+x^{2}$ by $g(x)=2+x$ and find the quotient and the remainder.

Classify the following polynomial as a polynomial in one variable,two variables,etc.:
$x^{2}-2xy+y^{2}+1$

On dividing $p(x) = x^{3} + 2x^{2} - 5ax - 7$ by $(x + 1)$,the remainder is $R_{1}$ and on dividing $q(x) = x^{3} + ax^{2} - 12x + 6$ by $(x - 2)$,the remainder is $R_{2}$. If $2R_{1} + R_{2} = 6$,then find the value of $a$.

On dividing $p(x) = 2x^3 - 3x^2 + ax - 3a + 9$ by $(x + 1)$,if the remainder is $16$,then find the value of $a$. Then,find the remainder on dividing $p(x)$ by $(x + 2)$.