Exponentials, Polynomials & Complex Roots

Practice Questions

Level 1: Easy (Foundational)

1. Determine $2^{-3}$ as an exact fraction.
   A) $\tfrac{1}{6}$
   B) $\tfrac{1}{8}$
   C) $-8$
   D) $-6$
2. Factor completely: $x^2 - 9$.
   A) $(x-3)(x+3)$
   B) $(x-3)^2$
   C) $(x+3)^2$
   D) $(x-9)(x+1)$
3. Determine the value of $\log_{10}(1000)$. (SPR)

Level 2: Medium (Standard)

1. If $p(x) = x^3 - 7x + 6$ and $p(1) = 0$, which factor of $p$ must exist?
   A) $x-1$
   B) $x+1$
   C) $x-6$
   D) $x+6$
2. Solve $x^2 + 2x + 5 = 0$ over $\mathbb{C}$. Which pair are the roots?
   A) $-1 \pm 2i$
   B) $1 \pm 2i$
   C) $-1 \pm i$
   D) $2 \pm i$
3. A population $P$ satisfies $P(t) = 500 \cdot 2^{t/3}$ where $t$ is in years. Determine the exact population when $t=6$. (SPR)
4. The graph of $y=2^{x}$ is transformed to $y = 2^{x-3}$. Which best describes the change?
   A) Shift left 3
   B) Shift right 3
   C) Shift up 3
   D) Shift down 3

Level 3: Hard (Advanced)

1. $p(x) = x^3 + ax^2 - x - 3$ has a factor $(x-1)$. Determine the exact value of $a$.
   A) $1$
   B) $2$
   C) $3$
   D) $-1$
2. If $\log_2 x + \log_2 (x-2) = 3$, determine the exact positive value of $x$.
   A) $1+\sqrt{5}$
   B) $1+\sqrt{7}$
   C) $1+\sqrt{9}$
   D) $4$
3. Determine the exact modulus $|z|$ of the complex number $z = 3 - 4i$. (SPR)