Vieta's Formulas for Polynomials

Find the sum and product of the roots of the quadratic $2x^2 - 24x + 17$.

There are two values of $a$ for which the equation $4x^2 + ax + 8x + 9 = 0$ has only one solution for $x$. What is the sum of these values of $a$?

What is the sum of the reciprocals of the roots of the equation $\frac{2003}{2004}x + 1 + \frac{1}{x} = 0$?

What is the sum of all real numbers $x$ for which $|x^2 - 12x + 34| = 2$?

Let $a$ and $b$ be the roots of the equation $x^2 - mx + 2 = 0$. Suppose that $a + \frac{1}{b}$ and $b + \frac{1}{a}$ are the roots of the equation $x^2 - px + q = 0$. What is $q$?

The quadratic equation $x^2 + mx + n$ has roots twice those of $x^2 + px + m$, and none of $m$, $n$, and $p$ is zero. What is the value of $\frac{n}{p}$?

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 $a$ and $b$ are nonzero real numbers, and that the equation $x^2 + ax + b = 0$ has solutions $a$ and $b$. Then the pair $(a, b)$ is

Let $a$, $b$, and $c$ be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation $(x - a)(x - b) + (x - b)(x - c) = 0$?

The polynomial $x^3 - ax^2 + bx - 2010$ has three positive integer roots. What is the smallest possible value of $a$?

A rectangular box $\mathcal{P}$ has distinct edge lengths $a$, $b$, and $c$. The sum of the lengths of all $12$ edges of $\mathcal{P}$ is $13$, the sum of the areas of all $6$ faces of $\mathcal{P}$ is $\frac{11}{2}$, and the volume of $\mathcal{P}$ is $\frac{1}{2}$. What is the length of the longest interior diagonal connecting two vertices of $\mathcal{P}$?

For how many ordered pairs $(a, b)$ of integers does the polynomial $x^3 + ax^2 + bx + 6$ have $3$ distinct integer roots?

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 $f(1)$?

All the roots of the polynomial $z^6 - 10z^5 + Az^4 + Bz^3 + Cz^2 + Dz + 16$ are positive integers, possibly repeated. What is the value of $B$?

Let $a$, $b$, and $c$ be the roots of the polynomial $x^3 + kx + 1$. What is the sum

(Give your answer in terms of $k$.)

All the roots of polynomial $z^6 - 10z^5 + Az^4 + Bz^3 + Cz^2 + Dz + 16$ are positive integers, possibly repeated. What is the value of $B$?

In the polynomial $x^4 - 18x^3 + kx^2 + 200x - 1984 = 0$, the product of $2$ of its roots is $-32$. Find $k$.

The zeros of the function $f(x) = x^2 - ax + 2a$ are integers. What is the sum of the possible values of $a$?

Let $r$, $s$, and $t$ be the three roots of the equation

Find $(r + s)^3 + (s + t)^3 + (t + r)^3$.

Find the sum of the roots, real and non-real, of the equation $x^{2001} + \left(\frac12 - x\right)^{2001} = 0$, given that there are no multiple roots.

How many distinct ordered triples $(x, y, z)$ satisfy the following equations?

Consider the polynomials $P(x) = x^6 - x^5 - x^3 - x^2 - x$ and $Q(x) = x^4 - x^3 - x^2 - 1$. Given that $z_1$, $z_2$, $z_3$, and $z_4$ are the roots of $Q(x) = 0$, find $P(z_1) + P(z_2) + P(z_3) + P(z_4)$.

For integers $a$, $b$, $c$ and $d$, let $f(x) = x^2 + ax + b$ and $g(x) = x^2 + cx + d$. Find the number of ordered triples $(a, b, c)$ of integers with absolute values not exceeding $10$ for which there is an integer $d$ such that $g(f(2)) = g(f(4)) = 0$.

Let $P(x) = x^2 - 3x - 7$, and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$. David computes each of the three sums $P + Q$, $P + R$, and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If $Q(0) = 2$, then $R(0) = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.