When does the polynomial $1 + x + \cdots + x ^ { n }$ have $x - a$ as a factor? Here $n$ is a positive integer greater than 1000 and $a$ is a real number.
(A) if and only if $a = - 1$
(B) if and only if $a = - 1$ and $n$ is odd
(C) if and only if $a = - 1$ and $n$ is even
(D) We cannot decide unless $n$ is known.

The polynomial $x ^ { 10 } + x ^ { 5 } + 1$ is divisible by
(A) $x ^ { 2 } + x + 1$.
(B) $x ^ { 2 } - x + 1$.
(C) $x ^ { 2 } + 1$.
(D) $x ^ { 5 } - 1$.

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) $\frac{3 + \sqrt{-5}}{2}$

6. It is given that $x + 2$ is a factor of $x ^ { 3 } + 4 c x ^ { 2 } + x ( c + 1 ) ^ { 2 } - 6$.
The sum of the possible values of $c$ is
A - 10
B - 6
C 0
D 6
E 10

( 2 x + 1 )$ and $( x - 2 )$ are factors of $2 x ^ { 3 } + p x ^ { 2 } + q$.
What is the value of $2 p + q$ ?

$(2x+1)$ and $(x-2)$ are factors of $2x^3 + px^2 + q$
What is the value of $2p + q$?
A $-10$
B $-\frac{38}{5}$
C $-\frac{22}{3}$
D $\frac{22}{3}$
E $\frac{38}{5}$
F $10$

$$P(x) = 2x^{3} - (m+1)x^{2} - nx + 3m - 1$$
Given that the polynomial is completely divisible by $x^{2} - x$, what is $m - n$?
A) $\frac{-1}{3}$
B) $\frac{-1}{2}$
C) $\frac{3}{2}$
D) $2$
E) $3$

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 $a, b, c \in \mathbb{R}$ 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