Let $n$ be a non-zero natural number. We consider the polynomial $R(X) = X^{2n} + 1$. For $k$ in $\llbracket 1, 2n \rrbracket$, we denote $\varphi_k = \frac{\pi}{2n} + \frac{k\pi}{n}$ and $\omega_k = \mathrm{e}^{\mathrm{i}\varphi_k}$. Show that $$R(X) = \prod_{k=1}^{2n}(X - \omega_k)$$

The number of distinct real roots of the equation $x ^ { 5 } \left( x ^ { 3 } - x ^ { 2 } - x + 1 \right) + x \left( 3 x ^ { 3 } - 4 x ^ { 2 } - 2 x + 4 \right) - 1 = 0$ is $\_\_\_\_$.

Consider the polynomial
$$P = x^2 + 2(a-1)x - 8a - 8.$$
(1) Let $a$ be a rational number. If the value of $P$ is a rational number when $x = 1 - \sqrt{2}$, then $a =$ $\mathbf{K}$ and in this case the value of $P$ is $P =$ $\mathbf{LM}$.
(2) Let $x$ and $a$ be positive integers. We are to investigate $x$ and $a$ which are such that the value of $P$ is a prime number.
When we factorize $P$, we have
$$P = (x - \mathbf{N}\mathbf{N})(x + \mathbf{O}a + \mathbf{P}).$$
Hence $x$ must be $\mathbf{Q}$.
Furthermore, the smallest possible $a$ is $\mathbf{R}$, and in this case the value of $P$ is $P = \mathbf{ST}$.

Consider the polynomial
$$P = x^2 + 2(a-1)x - 8a - 8.$$
(1) Let $a$ be a rational number. If the value of $P$ is a rational number when $x = 1 - \sqrt{2}$, then $a =$ $\mathbf{K}$ and in this case the value of $P$ is $P =$ $\mathbf{LM}$.
(2) Let $x$ and $a$ be positive integers. We are to investigate $x$ and $a$ which are such that the value of $P$ is a prime number.
When we factorize $P$, we have
$$P = (x - \mathbf{N})(x + \mathbf{O}a + \mathbf{P}).$$
Hence $x$ must be $\mathbf{Q}$.
Furthermore, the smallest possible $a$ is $\mathbf{R}$, and in this case the value of $P$ is $P = \mathbf{ST}$.

Set $P = 10a^2 + 14ab - 21bc - 15ca$.
(1) Factorizing $P$, we obtain $$P = (\mathbf{A}a + \mathbf{B}b)(\mathbf{C}a - \mathbf{D}c).$$
(2) If $5a = \sqrt{6}$, $14b = \sqrt{2} + \sqrt{3} - \sqrt{6}$ and $15c = \sqrt{12} - \sqrt{8}$, then $$P = \frac{\mathbf{E}}{\mathbf{E} + \mathbf{F}} \frac{\mathbf{G}}{\mathbf{H}}$$ and hence the greatest integer less than $P$ is $\mathbf{I}$.

(a) Factorize the expression $x ^ { 2 } + 3 x - 10$.
(b) If $x ^ { 3 } + a x ^ { 2 } + b x + c = ( x - \alpha ) ( x - \beta ) ( x - \gamma )$ for all values of $x$, find $a , b , c$ in terms of $\alpha , \beta , \gamma$.
(c) Find a value of $b$ for which $x ^ { 3 } + b x + 2 = 0$ has exactly two distinct solutions.

The expression $3 x ^ { 3 } + 13 x ^ { 2 } + 8 x + a$, where $a$ is a constant, has ( $x + 2$ ) as a factor. Which one of the following is a complete factorisation of the expression?
A $( x + 2 ) ( x - 1 ) ( 3 x - 2 )$ B $( x + 2 ) ( x + 1 ) ( 3 x - 2 )$ C $( x + 2 ) ( x + 1 ) ( 3 x + 2 )$ D $( x + 2 ) ( x - 3 ) ( 3 x + 2 )$ E $( x + 2 ) ( x + 3 ) ( 3 x - 2 )$ F $( x + 2 ) ( x + 3 ) ( 3 x + 2 )$

Let $a$ and $b$ be integers,
$$P(x) = x^{3} + ax^{2} + bx - 2$$
It is known that the polynomial has exactly one real root.
If $\mathbf{P}(1) = 0$, what is the smallest value that the integer $a$ can take?
A) $-6$ B) $-5$ C) $-4$ D) $-3$ E) $-2$