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.
14. If $\mathrm { f } ( \mathrm { x } ) \left| \begin{array} { c c c } 1 & x & x + 1 \\ 2 x & x ( x - 1 ) & ( x + 1 ) x \\ 3 x ( x - 1 ) & x ( x - 1 ) ( x - 2 ) & ( x + 1 ) x ( x - 1 ) \end{array} \right|$ then $\mathrm { f } ( 100 )$ is equal to : (A) 0 (B) 1 (C) 100 (D) $\quad - 100$
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