Consider the polynomial $$p(x) = x^6 + 10x^5 + 11x^4 + 12x^3 + 13x^2 - 12x - 11.$$ Find the remainder when $p(x)$ is divided by $x+1$. [1 point]

For a natural number $n$, let $a _ { n }$ be the remainder when the polynomial $2 x ^ { 2 } - 3 x + 1$ is divided by $x - n$. Find the value of $\sum _ { n = 1 } ^ { 7 } \left( a _ { n } - n ^ { 2 } + n \right)$. [3 points]

12. A polynomial $p ( x )$ has the property that $p ( 1 ) = 2$.
Which one of the following can be deduced from this?
A $\quad p ( x ) = ( x - 1 ) q ( x ) + 2$ for some polynomial $q ( x )$.
B $\quad p ( x ) = ( x + 1 ) q ( x ) + 2$ for some polynomial $q ( x )$.
C $\quad p ( x ) = ( x - 1 ) q ( x ) - 2$ for some polynomial $q ( x )$.
D $\quad p ( x ) = ( x + 1 ) q ( x ) - 2$ for some polynomial $q ( x )$.
E $\quad p ( x ) = ( x - 2 ) q ( x ) + 1$ for some polynomial $q ( x )$. F $\quad p ( x ) = ( x + 2 ) q ( x ) + 1$ for some polynomial $q ( x )$. G $\quad p ( x ) = ( x - 2 ) q ( x ) - 1$ for some polynomial $q ( x )$. H $\quad p ( x ) = ( x + 2 ) q ( x ) - 1$ for some polynomial $q ( x )$.

When $\left( 3 x ^ { 2 } + 8 x - 3 \right)$ is multiplied by $( p x - 1 )$ and the resulting product is divided by $( x + 1 )$, the remainder is 24 .
What is the value of $p$ ?
A - 4
B 2
C 4
D $\frac { 8 } { 7 }$
E $\frac { 11 } { 4 }$

The function f is defined by $\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + c$.
$a , b$ and $c$ take the values 1,2 and 3 with no two of them being equal and not necessarily in this order.
The remainder when $\mathrm { f } ( x )$ is divided by ( $x + 2$ ) is $R$.
The remainder when $\mathrm { f } ( x )$ is divided by ( $x + 3$ ) is $S$.
What is the largest possible value of $R - S$ ?
A - 26
B 5
C 7
D 17
E 29

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