Let $P(x)$ be a third-degree polynomial function such that $$P(-4) = P(-3) = P(5) = 0, \quad P(0) = 2$$ Given this, what is $P(1)$?
A) $\frac{7}{3}$
B) $\frac{8}{3}$
C) $\frac{7}{4}$
D) $\frac{9}{4}$
E) $\frac{8}{5}$

Real coefficient polynomials $P ( x ) , Q ( x )$ and $R ( x )$ are given. For the polynomial $\mathrm { P } ( \mathrm { x } )$ whose constant term is nonzero,
$$P ( x ) = Q ( x ) \cdot R ( x + 1 )$$
the equality is satisfied. If the constant term of P is twice the constant term of Q, what is the sum of the coefficients of R?
A) $\frac { 2 } { 3 }$
B) $\frac { 1 } { 4 }$
C) $\frac { 3 } { 4 }$
D) 1
E) 2

$$P ( x ) = x ^ { 11 } - 2 x ^ { 10 } + x - 2$$
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) $3 ^ { 11 } - 1$

$$P ( x ) = x ^ { 2 } - 3 x + 2$$
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) $10 x - 22$

Let $\mathrm { P } ( \mathrm { x } )$ be a second-degree polynomial and $\mathrm { Q } ( \mathrm { x } ) = \mathrm { k }$ be a constant polynomial such that
$$\begin{aligned} & P ( x ) + Q ( x ) = 2 x ^ { 2 } + 3 \\ & P ( Q ( x ) ) = 9 \end{aligned}$$
Accordingly, what is the sum of the values that k can take?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 2 } { 3 }$
D) $\frac { 1 } { 4 }$
E) $\frac { 3 } { 4 }$

The third-degree polynomial $\mathrm { P } ( \mathrm { x } )$ with leading coefficient 1 is divisible without remainder by $x ^ { 2 } + 4$. The remainder obtained from dividing the polynomial $P ( 2 x )$ by $2 x - 3$ is 52.
Accordingly, what is the value of $\mathbf { P } ( 2 )$?
A) 20
B) 22
C) 24
D) 26
E) 28

$$P ( x ) = x ^ { 3 } - m x + 1$$
The remainder when $P ( x - 1 )$ is divided by $x + 1$ equals the remainder when $P ( x + 1 )$ is divided by $x - 1$.
Accordingly, what is m?
A) 2
B) 4
C) 6
D) - 1
E) - 8

A third-degree polynomial $\mathrm { P } ( \mathrm { x } )$ with real coefficients and leading coefficient 1 satisfies the equalities
$$P ( 1 ) = P ( 3 ) = P ( 5 ) = 7$$
Accordingly, what is the value of $\mathbf { P } ( \mathbf { 0 } )$?
A) - 1
B) - 4
C) - 8
D) 4
E) 8

The polynomial $P ( x ) = ( x + 1 ) + ( x + 2 ) + \ldots + ( x + 9 )$
$$Q ( x ) = ( x + 1 ) + ( x + 2 ) + \ldots + ( x + 5 )$$
is divided by the polynomial.
What is the remainder obtained from this division?
A) 10 B) 12 C) 14 D) 16 E) 18

Let $x$, $y$ and $z$ be distinct prime numbers,
$$\begin{aligned} & x ( z - y ) = 18 \\ & y ( z - x ) = 40 \end{aligned}$$
the equalities are given.
Accordingly, what is the sum $\mathbf { x } + \mathbf { y } + \mathbf { z }$?
A) 17 B) 19 C) 21 D) 23 E) 25

Let $P(x)$ be a polynomial. A number $a$ satisfying the equation $P(a) = 0$ is called a root of this polynomial. For polynomials $P(x)$ and $R(x)$
$$\begin{aligned} &\mathrm{P}(\mathrm{x}) = \mathrm{x}^{2} - 1 \\ &\mathrm{R}(\mathrm{x}) = \mathrm{P}(\mathrm{P}(\mathrm{x})) \end{aligned}$$
the following equations are given.
Accordingly,
I. $-1$ II. $0$ III. $1$
which of these numbers are roots of the polynomial $\mathbf{R}(\mathbf{x})$?
A) Only I
B) Only II
C) Only III
D) I and III
E) II and III

$P(x)$ and $Q(x)$ are non-constant polynomials, and $R(x)$ is a first-degree polynomial, where
$$P(x) = Q(x) \cdot R(x)$$
the equality is satisfied.
Accordingly, I. The constant terms of polynomials $P(x)$ and $R(x)$ are the same. II. If the graph of $P(x)$ is a parabola, then the graph of $Q(x)$ is a line. III. Every root of polynomial $Q(x)$ is also a root of polynomial $R(x)$. Which of the following statements are always true?
A) Only II
B) Only III
C) I and II
D) I and III
E) II and III

For third-degree real-coefficient polynomials $P(x)$ and $R(x)$ whose highest degree terms have coefficient 1, the numbers 2 and 6 are common roots. When the polynomial $P(x) - R(x)$ is divided by $x - 1$, the remainder is 10.
Accordingly, what is the value of $P(0) - R(0)$?
A) 24
B) 27
C) 30
D) 33
E) 36

Let $a, b, c \in \mathbb{R}$ and $a \neq 0$. To factor the polynomial $ax^2 + bx + c$, we search for $m, n, r, s \in \mathbb{R}$ such that $a = m \cdot r$, $c = n \cdot s$, and $b = m \cdot s + n \cdot r$. If numbers satisfying these conditions can be found, the polynomial is factored as $ax^2 + bx + c = (mx + n)(rx + s)$.
Using the method described above, Sude wants to factor the polynomial $2x^2 + bx - 21$ where $b \in \mathbb{R}$. After finding the real numbers $m, n, r$, and $s$ that satisfy the given conditions, she notices that these numbers are each integers. Later, she confuses the places where she should write the numbers $n$ and $s$, and mistakenly factors the polynomial as $(mx + s)(rx + n)$ instead of $(mx + n)(rx + s)$, and finds the factors of the polynomial $2x^2 + x - 21$.
Accordingly, what is b?
A) 11 B) 12 C) 13 D) 14 E) 15

A third-degree polynomial $\mathrm{P}(\mathrm{x})$ with real coefficients and leading coefficient 3 is known to have exactly 2 different real roots.
If $\mathbf{P}(1) = \mathbf{P}(2) = \mathbf{0}$, then the value $\mathbf{P}(3)$ is
I. 6 II. 12 III. 18
Which of these numbers can it equal?
A) Only I B) Only II C) Only III D) I and II E) II and III