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

A third-degree polynomial $P ( x )$ with real coefficients has roots $- 3$, $- 1$, and $2$.\ Given that $P ( 0 ) = 12$, what is the coefficient of the $x ^ { 2 }$ term?\ A) - 4\ B) - 3\ C) - 2\ D) 1\ E) 2

Let $a$ and $b$ be integers such that $$\begin{aligned}& P ( x ) = x ^ { 3 } - a x ^ { 2 } - ( b + 2 ) x + 4 b \\& Q ( x ) = x ^ { 2 } - 2 a x + b\end{aligned}$$ For the polynomials

• $\mathrm{P} ( - 4 ) = 0$
• $\mathrm{Q} ( - 4 ) \neq 0$

it is known that.\ If the roots of polynomial $\mathbf{Q} ( \mathbf{x} )$ are also roots of polynomial $\mathbf{P} ( \mathbf{x} )$, what is the difference $b - a$?\ A) 8\ B) 9\ C) 11\ D) 13\ E) 14

How many second-degree polynomials have coefficients from the set $\{ 0,1,2 , \ldots , 9 \}$ and have one root equal to $\frac { - 2 } { 3 }$?\ A) 5\ B) 7\ C) 8\ D) 10\ E) 11

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

Let $a$ and $b$ be integers,
$$P(x) = x^{3} + ax^{2} + bx - 2$$
It is known that the polynomial has exactly one real root.
If $\mathbf{P}(1) = 0$, what is the smallest value that the integer $a$ can take?
A) $-6$ B) $-5$ C) $-4$ D) $-3$ E) $-2$

Let $P(x)$ and $Q(x)$ be polynomials with real coefficients such that $P(x) + Q(x)$ is a second-degree polynomial and
$$\begin{aligned} & P(x) \cdot Q(x) = -4 \cdot (x-1)^{4} \cdot (x-2)^{2} \\ & P(3) = -16 \end{aligned}$$
are satisfied. Accordingly, what is the value of $Q(4)$?
A) 12 B) 24 C) 36 D) 48 E) 54