Assume $f ( x )$ is a fifth-degree polynomial with real coefficients, and the remainder when $f ( x )$ is divided by $x ^ { n } - 1$ is $r _ { n } ( x )$ , where $n$ is a positive integer. Select the correct options.
(1) $r _ { 1 } ( x ) = f ( 1 )$
(2) $r _ { 2 } ( x )$ is a first-degree polynomial with real coefficients
(3) The remainder when $r _ { 4 } ( x )$ is divided by $x ^ { 2 } - 1$ equals $r _ { 2 } ( x )$
(4) $r _ { 5 } ( x ) = r _ { 6 } ( x )$
(5) If $f ( - x ) = - f ( x )$ , then $r _ { 3 } ( - x ) = - r _ { 3 } ( x )$

Let the polynomials $f(x) = x^3 + 2x^2 - 2x + k$ and $g(x) = x^2 + ax + 1$, where $k, a$ are real numbers. Given that $g(x)$ divides $f(x)$ and the equation $g(x) = 0$ has complex roots, select the option that is a root of the equation $f(x) = 0$.
(1) $-3$
(2) $0$
(3) $1$
(4) $\frac{1 + \sqrt{-3}}{2}$
(5) $\frac{3 + \sqrt{-5}}{2}$

Let $a , b , c$ be real numbers, and the polynomial $f ( x ) = a ( x - 1 ) ( x - 3 ) + b ( x - 1 ) ( x - 4 ) + c ( x - 3 ) ( x - 4 )$ simplifies to $f ( x ) = x ^ { 2 }$ . Regarding the magnitude relationship of $a , b , c$, select the correct option.
(1) $a > b > c$
(2) $a > c > b$
(3) $b > c > a$
(4) $c > a > b$
(5) $c > b > a$

A second-degree polynomial $P ( x )$ with leading coefficient 3 satisfies
$$P ( 1 ) - P ( 0 ) = 2$$
Given this, what is the value of $\mathbf { P } ( \mathbf { 2 } ) - \mathbf { P } ( \mathbf { 1 } )$?
A) 4
B) 5
C) 6
D) 7
E) 8

$$\frac { x ^ { 3 } - x ^ { 2 } y - x y ^ { 2 } + y ^ { 3 } } { 2 x ^ { 2 } - 4 x y + 2 y ^ { 2 } } = \frac { 1 } { 2 }$$
Given that, what is the sum $\mathbf { x } + \mathbf { y }$?
A) 1
B) 2
C) 4
D) $\frac { 3 } { 2 }$
E) $\frac { 4 } { 3 }$

$$P ( x ) = \left( x ^ { 2 } + 2 \right) ^ { 3 } + ( x - 3 ) ^ { 5 }$$
In this polynomial, what is the coefficient of the $x ^ { 4 }$ term?
A) - 9
B) - 3
C) 1
D) 11
E) 21

$$P ( x ) = x ^ { 2 } - 3 x + 2$$
Given that, when $P ( x - 1 ) + P ( 3 x - 3 )$ is divided by $x - 1$, which of the following is the quotient obtained?
A) $4 x - 10$
B) $4 x - 22$
C) $10 x - 16$
D) $10 x - 18$
E) $10 x - 22$

$$\frac { x ^ { 4 } + x ^ { 2 } y - x ^ { 2 } y ^ { 2 } - y ^ { 3 } } { x ^ { 3 } + x y - x ^ { 2 } y - y ^ { 2 } }$$
Which of the following is the simplified form of this expression?
A) $x$
B) $y$
C) $x y$
D) $x - y$
E) $x + y$

$\frac { x z - y z + x y - y ^ { 2 } } { x ^ { 2 } - x y + x z - y z }$\ Which of the following is the simplified form of this expression?\ A) $\frac { z - y } { x - z }$\ B) $\frac { y + z } { x + z }$\ C) $\frac { x + z } { y + z }$\ D) $\frac { x } { x + y }$\ E) $\frac { y - z } { x + y }$

It is known that a fourth-degree polynomial whose leading coefficient is 1 has roots that are all integers. Some parts of this polynomial's graph where it intersects the axes in the rectangular coordinate plane are given below.
Accordingly, what is the sum of the coefficients of this polynomial?
A) 72
B) 80
C) 84
D) 92
E) 96