Questions on Reducing Rational Expressions with Solutions

A set of questions on reducing rational expressions are presented. The answers are at the bottom of the page with detailed explanations included.

The math behind reducing rational expressions is similar to the math in reducing fractions: find an equivalent rational expression by dividing the numerator and denominator by their common factors.

1. For all \( x \ne 1 \), which of the following is equivalent to the rational expression:

   \[ \dfrac{x^2 + 5x - 6}{x - 1} \]

   A) \( x - 6 \)
   B) \( x - 1 \)
   C) \( x + 6 \)
   D) \( -x - 6 \)
   E) \( 6 - x \)

2. Which of the following is a simplified expression equal to:

   \[ \dfrac{5 - x}{2x - 10} \]

   for all \( x \ne 5 \)?

   A) \( -\frac{1}{2} \)
   B) \( \frac{1}{x - 5} \)
   C) \( -2 \)
   D) \( -\frac{1}{x - 5} \)
   E) \( \frac{1}{2} \)

3. For all \( x \ne -4 \), which of the given expressions is equivalent to:

   \[ \dfrac{16 - x^2}{x + 4} \]

   A) \( x - 4 \)
   B) \( 16 - 1 \)
   C) \( x + x \)
   D) \( -x - 4 \)
   E) \( 4 - x \)

4. Simplify the rational expression:

   \[ \dfrac{x + 2}{x^2 + 2x} \]

   A) \( \frac{1}{2x} \)
   B) \( \frac{1}{2x} \) for all \( x \neq -2 \)
   C) \( \frac{1}{x} \)
   D) \( \frac{1}{x} \), for all \( x \neq -2 \)
   E) \( \frac{1}{2} \)

5. For all \( x \ne 3 \), which expression is equivalent to:

   \[ \dfrac{3-x}{x^2 - x - 6} \]

   A) \( -\frac{1}{x + 2} \)
   B) \( \frac{1}{x - 2} \)
   C) \( -\frac{1}{x - 2} \)
   D) \( \frac{1}{x - 3} \)
   E) \( -\frac{1}{x - 3} \)

6. Simplify:

   \[ \dfrac{x^3 - x}{x^2 - 1} = \]

   A) \( x \)
   B) \( x \), for all \( x \neq 1 \)
   C) \( x \), for all \( x \neq 1 \) or \( -1 \)
   D) \( \frac{1}{x} \)
   E) \( x - 1 \)

7. Simplify:

   \[ \dfrac{x^2 - 4}{x^2 + 4x - 12} = \]

   A) \( \frac{x + 2}{x + 6} \), for all \( x \)
   B) \( \frac{x + 2}{x + 6} \), for all \( x \neq 2 \)
   C) \( \frac{x + 2}{x + 6} \), for all \( x \neq -2 \)
   D) \( \frac{x + 2}{x + 6} \), for all \( x \neq 0 \)
   E) \( \frac{1}{3} \)

8. Simplify the rational expression:

   \[ \dfrac{x^2 + 1}{x^3 + x} \]

   A) \( \frac{1}{x} \) for all \( x \neq 1 \)
   B) \( x + 1 \), for all \( x \neq -1 \)
   C) \( \frac{1}{2x} \)
   D) \( \frac{1}{x + 1} \) for all \( x \neq 1 \)
   E) \( \frac{1}{x} \)

9. Simplify:

   \[ \dfrac{x^2 + 2x - 3}{2x^2 + 3x - 5} = \]

   A) \( \frac{x + 3}{2x + 5} \), for all \( x \neq 1 \)
   B) \( \frac{x + 3}{2x + 5} \), for all \( x \)
   C) \( x + 3 \), for all \( x \neq 1 \)
   D) \( \frac{1}{2x + 5} \), for all \( x \)
   E) \( \frac{x + 3}{2x + 5} \), for all \( x \neq -3 \)

10. For all \( x \ne 1 \), which expression is equivalent to:

    \[ \dfrac{x-1}{(x^2 - 1)(x + 3)} \]

    A) \( \frac{1}{x + 3} \)
    B) \( \frac{1}{x^2 + 4x + 3} \)
    C) \( \frac{1}{x + 1} \)
    D) \( \frac{1}{x} \)
    E) \( \frac{1}{x - 1} \)