We consider the general case, without having information on the stability and multiplicity of the roots of $p$, and we seek to calculate $\sigma(p)$.
We construct the two polynomials $f$ and $g$ satisfying $f = p \wedge p_0$ and $p = fg$.
Show that $\sigma(g) = \pi(J(g))$.

We consider the general case, without having information on the stability and multiplicity of the roots of $p$, and we seek to calculate $\sigma(p)$.
We construct the two polynomials $f$ and $g$ satisfying $f = p \wedge p_0$ and $p = fg$.
Propose a method allowing us to construct a finite number (possibly zero) of polynomials $g_1, \ldots, g_\ell$, whose roots are stable and of multiplicity 1, such that $f = g_1 g_2 \cdots g_\ell$. Express $\sigma(p)$ using $n$, $\deg g$, $\pi(J(g))$, $\ell$, $\pi(J(g))$ as well as $\pi(J(g_1')), \ldots, \pi(J(g_\ell'))$.

If $a^4 + a^3 + a^2 + a + 1 = 0$, find the value of $a^{2m} + a^m + 1/a^m + 1/a^{2m}$ when $m$ is a multiple of 5, and find $a^{4m} + a^{3m} + a^{2m} + a^m$.

Let $s, sr, sr^2, sr^3$ be the roots of $x^4 + ax^3 + bx^2 + cx + d = 0$ (roots in geometric progression). Show that $c^2 = a^2 d$.

The number of integers $n$ for which the cubic equation $X ^ { 3 } - X + n = 0$ has 3 distinct integer solutions is:
(A) 0
(B) 1
(C) 2
(D) infinite.

Let $a _ { 0 } , a _ { 1 } , \cdots , a _ { 19 } \in \mathbb { R }$ and
$$P ( x ) = x ^ { 20 } + \sum _ { i = 0 } ^ { 19 } a _ { i } x ^ { i } , \quad x \in \mathbb { R }$$
If $P ( x ) = P ( - x )$ for all $x \in \mathbb { R }$, and
$$P ( k ) = k ^ { 2 } , \text{ for } k = 0,1,2 \cdots , 9$$
then find
$$\lim _ { x \rightarrow 0 } \frac { P ( x ) } { \sin ^ { 2 } x }$$

The number of different values of $a$ for which the equation $x ^ { 3 } - x + a = 0$ has two identical real roots is
(A) 0 .
(B) 1 .
(C) 2 .
(D) 3 .

Let $P(x)$ be an odd degree polynomial in $x$ with real coefficients. Show that the equation $P(P(x)) = 0$ has at least as many distinct real roots as the equation $P(x) = 0$.

Let $p$ and $q$ be two non-zero polynomials such that the degree of $p$ is less than or equal to the degree of $q$, and $p ( a ) q ( a ) = 0$ for $a = 0,1,2 , \ldots , 10$. Which of the following must be true?
(A) degree of $q \neq 10$
(B) degree of $p \neq 10$
(C) degree of $q \neq 5$
(D) degree of $p \neq 5$

(a) Let $n \geq 1$ be an integer. Prove that $X ^ { n } + Y ^ { n } + Z ^ { n }$ can be written as a polynomial with integer coefficients in the variables $\alpha = X + Y + Z$, $\beta = X Y + Y Z + Z X$ and $\gamma = X Y Z$.
(b) Let $G _ { n } = x ^ { n } \sin ( n A ) + y ^ { n } \sin ( n B ) + z ^ { n } \sin ( n C )$, where $x , y , z , A , B , C$ are real numbers such that $A + B + C$ is an integral multiple of $\pi$. Using (a) or otherwise, show that if $G _ { 1 } = G _ { 2 } = 0$, then $G _ { n } = 0$ for all positive integers $n$.

Suppose that the equations $x ^ { 2 } + b x + c a = 0$ and $x ^ { 2 } + c x + a b = 0$ have exactly one common non-zero root. Then
(A) $a + b + c = 0$.
(B) the two roots which are not common must necessarily be real.
(C) the two roots which are not common may not be real.
(D) the two roots which are not common are either both real or both not real.

Let $\alpha, \beta$ be the roots of the equation $x^2 - px + r = 0$ and $\frac{\alpha}{2}, 2\beta$ be the roots of the equation $x^2 - qx + r = 0$. Then the value of $r$ is
(A) $\frac{2}{9}(p-q)(2q-p)$
(B) $\frac{2}{9}(q-p)(2p-q)$
(C) $\frac{2}{9}(q-2p)(2q-p)$
(D) $\frac{2}{9}(2p-q)(2q-p)$

Let $p$ and $q$ be real numbers such that $p \neq 0 , p ^ { 3 } \neq q$ and $p ^ { 3 } \neq - q$. If $\alpha$ and $\beta$ are nonzero complex numbers satisfying $\alpha + \beta = - p$ and $\alpha ^ { 3 } + \beta ^ { 3 } = q$, then a quadratic equation having $\frac { \alpha } { \beta }$ and $\frac { \beta } { \alpha }$ as its roots is
A) $\left( p ^ { 3 } + q \right) x ^ { 2 } - \left( p ^ { 3 } + 2 q \right) x + \left( p ^ { 3 } + q \right) = 0$
B) $\left( p ^ { 3 } + q \right) x ^ { 2 } - \left( p ^ { 3 } - 2 q \right) x + \left( p ^ { 3 } + q \right) = 0$
C) $\left( \mathrm { p } ^ { 3 } - \mathrm { q } \right) \mathrm { x } ^ { 2 } - \left( 5 \mathrm { p } ^ { 3 } - 2 \mathrm { q } \right) \mathrm { x } + \left( \mathrm { p } ^ { 3 } - \mathrm { q } \right) = 0$
D) $\left( p ^ { 3 } - q \right) x ^ { 2 } - \left( 5 p ^ { 3 } + 2 q \right) x + \left( p ^ { 3 } - q \right) = 0$

A value of $b$ for which the equations $$\begin{aligned} & x ^ { 2 } + b x - 1 = 0 \\ & x ^ { 2 } + x + b = 0 \end{aligned}$$ have one root in common is
(A) $- \sqrt { 2 }$
(B) $- i \sqrt { 3 }$
(C) $i \sqrt { 5 }$
(D) $\sqrt { 2 }$

Let $f ( x ) = x ^ { 4 } + a x ^ { 3 } + b x ^ { 2 } + c$ be a polynomial with real coefficients such that $f ( 1 ) = - 9$. Suppose that $i \sqrt { 3 }$ is a root of the equation $4 x ^ { 3 } + 3 a x ^ { 2 } + 2 b x = 0$, where $i = \sqrt { - 1 }$. If $\alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 }$, and $\alpha _ { 4 }$ are all the roots of the equation $f ( x ) = 0$, then $\left| \alpha _ { 1 } \right| ^ { 2 } + \left| \alpha _ { 2 } \right| ^ { 2 } + \left| \alpha _ { 3 } \right| ^ { 2 } + \left| \alpha _ { 4 } \right| ^ { 2 }$ is equal to $\_\_\_\_$ .

If $p$ and $q$ are non-zero real numbers and $\alpha ^ { 3 } + \beta ^ { 3 } = - p , \alpha \beta = q$, then a quadratic equation whose roots are $\frac { \alpha ^ { 2 } } { \beta } , \frac { \beta ^ { 2 } } { \alpha }$ is :
(1) $p x ^ { 2 } - q x + p ^ { 2 } = 0$
(2) $q x ^ { 2 } + p x + q ^ { 2 } = 0$
(3) $p x ^ { 2 } + q x + p ^ { 2 } = 0$
(4) $q x ^ { 2 } - p x + q ^ { 2 } = 0$

Let $\alpha$ and $\beta$ be the roots of equation $px^2 + qx + r = 0$, $p \neq 0$. If $p$, $q$, $r$ are in A.P. and $\frac{1}{\alpha} + \frac{1}{\beta} = 4$, then the value of $|\alpha - \beta|$ is:
(1) $\frac{\sqrt{61}}{9}$
(2) $\frac{2\sqrt{17}}{9}$
(3) $\frac{\sqrt{34}}{9}$
(4) $\frac{2\sqrt{13}}{9}$

If $\lambda \in \mathrm { R }$ is such that the sum of the cubes of the roots of the equation, $x ^ { 2 } + ( 2 - \lambda ) x + ( 10 - \lambda ) = 0$ is minimum, then the magnitude of the difference of the roots of this equation is
(1) 20
(2) $2 \sqrt { 5 }$
(3) $2 \sqrt { 7 }$
(4) $4 \sqrt { 2 }$

The value of $\lambda$ such that sum of the squares of the roots of the quadratic equation, $x ^ { 2 } + ( 3 - \lambda ) x + 2 = \lambda$ has the least value is:
(1) 2
(2) $\frac { 4 } { 9 }$
(3) $\frac { 15 } { 8 }$
(4) 1

If $\lambda$ be the ratio of the roots of the quadratic equation in $x , 3 m ^ { 2 } x ^ { 2 } + m ( m - 4 ) x + 2 = 0$, then the least value of $m$ for which $\lambda + \frac { 1 } { \lambda } = 1$, is :
(1) $2 - \sqrt { 3 }$
(2) $- 2 + \sqrt { } \overline { 2 }$
(3) $4 - 2 \sqrt { 3 }$
(4) $4 - 3 \sqrt { 2 }$

If $\alpha$ and $\beta$ be the roots of the equation $x^2 - 2x + 2 = 0$, then the least value of $n$ for which $\frac{\alpha^n}{\beta} = 1$ is
(1) 5
(2) 4
(3) 2
(4) 3

Let $\alpha$ and $\beta$ be the roots of the equation $x ^ { 2 } - x - 1 = 0$. If $p _ { k } = ( \alpha ) ^ { k } + ( \beta ) ^ { k } , k \geq 1$, then which one of the following statements is not true?
(1) $p _ { 3 } = p _ { 5 } - p _ { 4 }$
(2) $p _ { 5 } = 11$
(3) $\left( p _ { 1 } + p _ { 2 } + p _ { 3 } + p _ { 4 } + p _ { 5 } \right) = 26$
(4) $p _ { 5 } = p _ { 2 } \cdot p _ { 3 }$

Let $\alpha$ and $\beta$ be the roots of the equation, $5x^{2} + 6x - 2 = 0$. If $S_{n} = \alpha^{n} + \beta^{n}, n = 1,2,3,\ldots$, then
(1) $6S_{6} + 5S_{5} = 2S_{4}$
(2) $5S_{6} + 6S_{5} + 2S_{4} = 0$
(3) $5S_{6} + 6S_{5} = 2S_{4}$
(4) $6S_{6} + 5S_{5} + 2S_{4} = 0$

If $\alpha$ and $\beta$ are the roots of the equation $x ^ { 2 } + p x + 2 = 0$ and $\frac { 1 } { \alpha }$ and $\frac { 1 } { \beta }$ are the roots of the equation $2 x ^ { 2 } + 2 q x + 1 = 0$, then $\left( \alpha - \frac { 1 } { \alpha } \right) \left( \beta - \frac { 1 } { \beta } \right) \left( \alpha + \frac { 1 } { \beta } \right) \left( \beta + \frac { 1 } { \alpha } \right)$ is equal to:
(1) $\frac { 9 } { 4 } \left( 9 + q ^ { 2 } \right)$
(2) $\frac { 9 } { 4 } \left( 9 - q ^ { 2 } \right)$
(3) $\frac { 9 } { 4 } \left( 9 + p ^ { 2 } \right)$
(4) $\frac { 9 } { 4 } \left( 9 - p ^ { 2 } \right)$

Let $\alpha$ and $\beta$ be the roots of $x ^ { 2 } - 3 x + p = 0$ and $\gamma$ and $\delta$ be the roots of $x ^ { 2 } - 6 x + q = 0$. If $\alpha , \beta , \gamma , \delta$ from a geometric progression. Then ratio $( 2 q + p ) : ( 2 q - p )$ is
(1) $3 : 1$
(2) $9 : 7$
(3) $5 : 3$
(4) $33 : 31$