Let $x^{3} + ax^{2} + bx + 8 = 0$ be a cubic equation with integer coefficients. Suppose both $r$ and $-r$ are roots of this equation, where $r > 0$ is a real number. List all possible pairs of values $(a, b)$.

The polynomial $p(x) = 10x^{400} + ax^{399} + bx^{398} + 3x + 15$, where $a, b$ are real constants, is given to be divisible by $x^{2}-1$.
(i) If you can, find the values of $a$ and $b$. Write your answers as $a =$ $\_\_\_\_$, $b =$ $\_\_\_\_$. If it is not possible to decide, state so.
(ii) If you can, find the sum of reciprocals of all 400 (complex) roots of $p(x)$. Write your answer as sum $=$ $\_\_\_\_$. If it is not possible to decide, state so.

[14 points] We want to find a nonzero polynomial $p ( x )$ with integer coefficients having the following property.
$$\text { Letting } q ( x ) : = \frac { p ( x ) } { x ( 1 - x ) } , \quad q ( x ) = q \left( \frac { 1 } { 1 - x } \right) \text { for all } x \notin \{ 0,1 \}$$
(i) Find one such polynomial with the smallest possible degree.
(ii) Find one such polynomial with the largest possible degree OR show that the degree of such polynomials is unbounded.

A function $f(x) = x^{3} + ax^{2} + bx + 4$ satisfies the following condition for two integers $a$ and $b$. What is the maximum value of $f(1)$? [4 points] For all real numbers $\alpha$, the limit $\lim_{x \rightarrow \alpha} \frac{f(2x+1)}{f(x)}$ exists.

Let $n \in \mathbf{N}^{*}$. Let $(z_{1}, z_{2}, \ldots, z_{n}) \in \mathbf{C}^{n}$. We define the two polynomials $P(X)$ and $Q(X)$ in $\mathbf{C}[X]$ by: $$P(X) = \prod_{k=1}^{n}(X - z_{k}) \quad \text{and} \quad Q(X) = \prod_{(k,l) \in \llbracket 1;n \rrbracket^{2}}(X - z_{k} - z_{l})$$
We assume $n = 2$ and $P \in \mathbf{R}_{2}[X]$. If the coefficients of $Q$ are strictly positive, is $P$ then a Hurwitz polynomial?

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.

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) $\sqrt { 2 }$

The number of pairs $a , b$ of real numbers, such that whenever $\alpha$ is a root of the equation $x ^ { 2 } + a x + b = 0 , \quad \alpha ^ { 2 } - 2$ is also a root of this equation, is :
(1) 6
(2) 8
(3) 4
(4) 2

Let $\alpha , \beta$ be the roots of the equation $x ^ { 2 } - 4 \lambda x + 5 = 0$ and $\alpha , \gamma$ be the roots of the equation $x ^ { 2 } - ( 3 \sqrt { 2 } + 2 \sqrt { 3 } ) x + 7 + 3 \lambda \sqrt { 3 } = 0$. If $\beta + \gamma = 3 \sqrt { 2 }$, then $( \alpha + 2 \beta + \gamma ) ^ { 2 }$ is equal to

Let $a \in R$ and let $\alpha , \beta$ be the roots of the equation $x ^ { 2 } + 60 ^ { \frac { 1 } { 4 } } x + a = 0$. If $\alpha ^ { 4 } + \beta ^ { 4 } = - 30$, then the product of all possible values of $a$ is $\_\_\_\_$.

Let $\lambda \neq 0$ be a real number. Let $\alpha , \beta$ be the roots of the equation $14 x ^ { 2 } - 31 x + 3 \lambda = 0$ and $\alpha , \gamma$ be the roots of the equation $35 x ^ { 2 } - 53 x + 4 \lambda = 0$. Then $\frac { 3 \alpha } { \beta }$ and $\frac { 4 \alpha } { \gamma }$ are the roots of the equation :
(1) $7 x ^ { 2 } + 245 x - 250 = 0$
(2) $7 x ^ { 2 } - 245 x + 250 = 0$
(3) $49 x ^ { 2 } - 245 x + 250 = 0$
(4) $49 x ^ { 2 } + 245 x + 250 = 0$

Q61. If 2 and 6 are the roots of the equation $a x ^ { 2 } + b x + 1 = 0$, then the quadratic equation, whose roots are $\frac { 1 } { 2 a + b }$ and $\frac { 1 } { 6 a + b }$, is :
(1) $2 x ^ { 2 } + 11 x + 12 = 0$
(2) $x ^ { 2 } + 8 x + 12 = 0$
(3) $4 x ^ { 2 } + 14 x + 12 = 0$
(4) $x ^ { 2 } + 10 x + 16 = 0$

The leading coefficient is 1, and the fourth-degree polynomial $\mathbf { P } ( \mathbf { x } )$ with real coefficients has roots $-i$ and $2i$. What is $\mathbf { P } ( \mathbf { 0 } )$?
A) 2
B) 4
C) 6
D) 7
E) 8

$$\begin{aligned} & P ( x ) = x ^ { 2 } - 2 x + m \\ & Q ( x ) = x ^ { 2 } + 3 x + n \end{aligned}$$
polynomials are given. These two polynomials have a common root and the roots of the polynomial $P(x)$ are equal, so what is the sum $m + n$?
A) $-5$
B) $-3$
C) 2
D) 4
E) 5

Let b and c be non-zero real numbers such that the roots of the equation
$$x ^ { 2 } + b x + c = 0$$
are b and c. Accordingly, what is the product $b \cdot c$?
A) $- 6$
B) $- 5$
C) $- 4$
D) $- 3$
E) $- 2$

The sum of the roots of the equation $x ^ { 2 } - a x + 1 = 0$, which has two real roots, is a root of the equation $$x ^ { 2 } + 6 x + a = 0$$ Accordingly, what is a?\ A) - 3\ B) - 4\ C) - 5\ D) - 6\ E) - 7

A 4th degree polynomial $P ( x )$ with real coefficients and leading coefficient 1 satisfies
$$P ( x ) = P ( - x )$$
for every real number $x$.
$$P ( 2 ) = P ( 3 ) = 0$$
Given that, what is $\mathbf { P ( 1 ) }$?
A) 12 B) 18 C) 24 D) 30 E) 36