Let $f ( x ) = 7 x ^ { 32 } + 5 x ^ { 22 } + 3 x ^ { 12 } + x ^ { 2 }$. (i) Find the remainder when $f ( x )$ is divided by $x ^ { 2 } + 1$. (ii) Find the remainder when $x f ( x )$ is divided by $x ^ { 2 } + 1$. In each case your answer should be a polynomial of the form $a x + b$, where $a$ and $b$ are constants.

Let $f ( x )$ be a real polynomial function of degree 3 satisfying the condition that the remainder when $( x + 1 ) f ( x )$ is divided by $x ^ { 3 } + 2$ is $x + 2$. If $f ( 0 ) = 4$, what is the value of $f ( 2 )$?
(1) 8
(2) 10
(3) 15
(4) 18
(5) 20

Let $a$ and $b$ be real numbers (where $a > 0$). If the polynomial $ax^{2} + (2a+b)x - 12$ divided by $x^{2} + (2-a)x - 2a$ gives a remainder of 6, then the ordered pair $(a, b) = $ (14--1), 14--2).

It is known that $f(x), g(x), h(x)$ are all real-coefficient cubic polynomials, and their remainders when divided by $x^{2} - 2x + 3$ are $x + 1$, $x - 3$, and $-2$ respectively. If $xf(x) + ag(x) + bh(x)$ is divisible by $x^{2} - 2x + 3$, where $a, b$ are real numbers, then $a =$ (14－1)(14－2), $b =$ (14－3).

$$P ( x ) = x ^ { 11 } - 2 x ^ { 10 } + x - 2$$
What is the remainder when this polynomial is divided by $x ^ { 2 } - 5 x + 6$?
A) $3 ^ { 10 } + 1$
B) $3 ^ { 10 } - 1$
C) $3 ^ { 11 } + 1$
D) $3 ^ { 11 } - 1$