Polynomial Roots: all problems

Let $f$ be a function for which $f\left(\frac{x}{3}\right) = x^2 + x + 1$. Find the sum of all values of $z$ for which $f(3z) = 7$.

Suppose that the roots of $x^3 + 3x^2 + 4x - 11 = 0$ are $a$, $b$, and $c$, and that the roots of $x^3 + rx^2 + sx + t = 0$ are $a + b$, $b + c$, and $c + a$. Find $t$.

For certain real numbers $a$, $b$, and $c$, the polynomial

has three distinct roots, and each root of $g(x)$ is also a root of the polynomial

What is $f(1)$?

Let $P(x)$ be a polynomial with rational coefficients such that when $P(x)$ is divided by the polynomial $x^2 + x + 1$, the remainder is $x + 2$, and when $P(x)$ is divided by the polynomial $x^2 + 1$, the remainder is $2x + 1$. There is a unique polynomial of least degree with these two properties. What is the sum of the squares of the coefficients of that polynomial?

Let $P(x)$ be the unique polynomial of minimal degree with the following properties:

• $P(x)$ has leading coefficient $1$,
• $1$ is a root of $P(x) - 1$,
• $2$ is a root of $P(x - 2)$,
• $3$ is a root of $P(3x)$, and
• $4$ is a root of $4P(x)$.

The roots of $P(x)$ are integers, with one exception. The root that is not an integer can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?

Polynomials $P(x)$ and $Q(x)$ each have degree $3$ and leading coefficient $1$, and their roots are all elements of $\{1, 2, 3, 4, 5\}$. The function $f(x) = \frac{P(x)}{Q(x)}$ has the property that there exist real numbers $a < b < c < d$ such that the set of all real numbers $x$ such that $f(x) \le 0$ consists of the closed interval $[a, b]$ together with the open interval $(c, d)$. How many functions $f(x)$ are possible?

Given the equations $x^2 + kx + 6 = 0$ and $x^2 - kx + 6 = 0$. If, when the roots of the equation are suitably listed, each root of the second equation is $5$ more than the corresponding root of the first equation, then $k$ equals:

What is the product of all the roots of the equation

Let $P(x)$ be a polynomial such that when $P(x)$ is divided by $x - 19$, the remainder is $99$, and when $P(x)$ is divided by $x - 99$, the remainder is $19$. What is the remainder when $P(x)$ is divided by $(x - 19)(x - 99)$?

The real root of the equation $8x^3 - 3x^2 - 3x - 1 = 0$ can be written in the form $\frac{\sqrt[3]{a} + \sqrt[3]{b} + 1}{c}$, where $a$, $b$, and $c$ are positive integers. Find $a + b + c$.

Let $x_1 < x_2 < x_3$ be the three real roots of the equation $\sqrt{2014}\,x^3 - 4029x^2 + 2 = 0$. Find $x_2(x_1 + x_3)$.

The graph of $y = x^6 - 10x^5 + 29x^4 - 4x^3 + ax^2$ lies above the line $y = bx + c$ except at three values of $x$, where the graph and the line intersect. What is the largest of these values?

The equation $2000x^6 + 100x^5 + 10x^3 + x - 2 = 0$ has exactly two real roots, one of which is $\frac{m + \sqrt{n}}{r}$, where $m$, $n$ and $r$ are integers, $m$ and $r$ are relatively prime, and $r > 0$. Find $m + n + r$.

Let $P(x)$ be a nonzero polynomial such that $(x - 1)P(x + 1) = (x + 2)P(x)$ for every real $x$, and $\left(P(2)\right)^2 = P(3)$. Then $P\left(\frac{7}{2}\right) = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let $P(x)$ be a quadratic polynomial with complex coefficients whose $x^2$ coefficient is $1$. Suppose the equation $P(P(x)) = 0$ has four distinct solutions, $x = 3, 4, a, b$. Find the sum of all possible values of $(a + b)^2$.

A quadratic polynomial with real coefficients and leading coefficient $1$ is called disrespectful if the equation $p(p(x)) = 0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\tilde{p}(x)$ for which the sum of the roots is maximized. What is $\tilde{p}(1)$?