Let $n$ be a non-zero natural number. Let $P$ be in $\mathbb{C}_{2n}[X]$, and for all $\lambda$ in $\mathbb{C}$, let $Q_\lambda$ be the quotient of $P_\lambda = P(\lambda X) - P(\lambda)$ by $X-1$: $$Q_\lambda(X) = \frac{P(\lambda X) - P(\lambda)}{X - 1} \in \mathbb{C}_{2n-1}[X]$$ Show that, for all $\lambda$ in $\mathbb{C}$, $Q_\lambda(1) = \lambda P'(\lambda)$.
$$P ( x ) = x ^ { 2 } - 3 x + 2$$ Given that, when $P ( x - 1 ) + P ( 3 x - 3 )$ is divided by $x - 1$, which of the following is the quotient obtained? A) $4 x - 10$ B) $4 x - 22$ C) $10 x - 16$ D) $10 x - 18$ E) $10 x - 22$
The third-degree polynomial $\mathrm { P } ( \mathrm { x } )$ with leading coefficient 1 is divisible without remainder by $x ^ { 2 } + 4$. The remainder obtained from dividing the polynomial $P ( 2 x )$ by $2 x - 3$ is 52. Accordingly, what is the value of $\mathbf { P } ( 2 )$? A) 20 B) 22 C) 24 D) 26 E) 28
$$P ( x ) = x ^ { 3 } - m x + 1$$ The remainder when $P ( x - 1 )$ is divided by $x + 1$ equals the remainder when $P ( x + 1 )$ is divided by $x - 1$. Accordingly, what is m? A) 2 B) 4 C) 6 D) - 1 E) - 8
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