Find a polynomial $p(x)$ that simultaneously has both the following properties.
(i) When $p(x)$ is divided by $x^{100}$ the remainder is the constant polynomial 1.
(ii) When $p(x)$ is divided by $(x-2)^{3}$ the remainder is the constant polynomial 2.

For a cubic function $f ( x )$ with leading coefficient 1 and a function $g ( x )$ that is continuous on the set of all real numbers, the following conditions are satisfied. (가) For all real numbers $x$, $f ( x ) g ( x ) = x ( x + 3 )$. (나) $g ( 0 ) = 1$ When $f ( 1 )$ is a natural number, what is the minimum value of $g ( 2 )$? [4 points]
(1) $\frac { 5 } { 13 }$
(2) $\frac { 5 } { 14 }$
(3) $\frac { 1 } { 3 }$
(4) $\frac { 5 } { 16 }$
(5) $\frac { 5 } { 17 }$

Let $P ( X ) = X ^ { 4 } + a _ { 3 } X ^ { 3 } + a _ { 2 } X ^ { 2 } + a _ { 1 } X + a _ { 0 }$ be a polynomial in $X$ with real coefficients. Assume that
$$P ( 0 ) = 1 , P ( 1 ) = 2 , P ( 2 ) = 3 , \text { and } P ( 3 ) = 4 .$$
Then, the value of $P ( 4 )$ is
(A) 5
(B) 24
(C) 29
(D) not determinable from the given data.

Let $p ( x )$ be a quadratic polynomial such that $p ( 0 ) = 1$. If $p ( x )$ leaves remainder 4 when divided by $x - 1$ and it leaves remainder 6 when divided by $x + 1$ then:
(1) $p ( - 2 ) = 19$
(2) $p ( 2 ) = 19$
(3) $p ( - 2 ) = 11$
(4) $p ( 2 ) = 11$

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) $\frac{8}{5}$

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

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

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

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 $P(x)$ 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