Unit 1 test: Algebra

Let $r$ and $s$ be the roots of $x^2 - 8x + 5 = 0$. What is $\dfrac{1}{r} + \dfrac{1}{s}$?

For what positive value of $k$ does $x^2 - kx + 16 = 0$ have exactly one real solution?

Positive numbers $x, y, z$ satisfy $xy + xz = 14$, $xy + yz = 18$ and $xz + yz = 20$. What is $x + y + z$?

How many ordered pairs of positive integers $(x, y)$ satisfy $xy + x + y = 34$?

A function satisfies $f(x) + 2f(-x) = 3x + 3$ for all real $x$. What is $f(5)$?

A linear function $f(x) = ax + b$ with real coefficients satisfies $f(f(f(x))) = 8x + 21$ for all $x$. What is $f(10)$?

Solve $\log_3 x + \log_9 x + \log_{27} x = \dfrac{11}{2}$.

A real number $x$ satisfies $4^x + 4^{-x} = 23$. What is $2^x + 2^{-x}$?

What is the minimum value of $9x + \dfrac{4}{x}$ for $x > 0$?

Positive numbers $a$ and $b$ satisfy $a + b = 1$. What is the smallest possible value of $\left(1 + \dfrac{1}{a}\right)\left(1 + \dfrac{1}{b}\right)$?

What is the sum of the infinite geometric series $6 - 2 + \dfrac{2}{3} - \dfrac{2}{9} + \cdots$?

What is $\dfrac{1}{\sqrt{1} + \sqrt{2}} + \dfrac{1}{\sqrt{2} + \sqrt{3}} + \dots + \dfrac{1}{\sqrt{24} + \sqrt{25}}$?

Let $r, s, t$ be the roots of $P(x) = x^3 - 2x^2 - 5x + 3$. What is $(1 - r)(1 - s)(1 - t)$?

A polynomial $P(x)$ leaves remainder $4$ when divided by $x - 1$ and remainder $10$ when divided by $x - 3$. What is the remainder when $P(x)$ is divided by $(x - 1)(x - 3)$?

Let $z = \cos\dfrac{\pi}{3} + i\sin\dfrac{\pi}{3}$. What is $(3 - z)(3 - z^2)(3 - z^3)(3 - z^4)(3 - z^5)$?