A set of questions on reducing rational expressions are presented. The answers to the questions are at the bottom of the page and the solutions with detailed explanations to these questions are also 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.

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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) -1/2
   B) 1 / (x - 5)
   C) -2
   D) - 1 / (x - 5)
   E) 1/2
   
   
3. For all \( x \ne -4 \), which of the given expressions is equivalent to the expression \( \dfrac{16 - x^2}{x + 4} \) ?
   A) x - 4
   B) 16 - 1
   C) x + x
   D) - x - 4
   E) 4 - x
   
   
4. Simplify the following rational expression \( \dfrac{x + 2}{x^2 + 2x} \).
   A) 1 / 2x
   B) 1 / 2x for all x not equal to - 2
   C) 1 / x
   D) 1 / x , for all x not equal to - 2
   E) 1 / 2
   
   
5. For all \( x \ne 3\), which of the given expressions is equivalent to the expression \( \dfrac{3-x}{x^2 - x - 6} \) ?
   A) - 1 / (x + 2)
   B) 1 / (x - 2)
   C) -1 / (x - 2)
   D) 1 / (x - 3)
   E) -1 / (x - 3)
   
   
6. \( \dfrac{x^3 - x}{x^2 - 1} = \)
   A) x
   B) x , for all x not equal to 1
   C) x , for all x not equal to 1 or -1
   D) 1 / x
   E) x - 1
   
   
7. \( \dfrac{x^2 - 4}{x^2 + 4x - 12} = \)
   A) (x + 2) / (x + 6) , for all x
   B) (x + 2) / (x + 6) , for all x not equal to 2
   C) (x + 2) / (x + 6) , for all x not equal to - 2
   D) (x + 2) / (x + 6) , for all x not equal to 0
   E) 1 / 3
   
   
8. Simplify the rational expression \( \dfrac{x^2 + 1}{x^3 + x} \).
   A) 1 / x for all x not equal to 1
   B) x + 1 , for all x not equal to - 1
   C) 1 / 2x
   D) 1 / (x + 1) for all x not equal to 1
   E) 1 / x
   
   
9. \( \dfrac{x^2 + 2x - 3}{2x^2 + 3x - 5} = \)
   A) (x + 3) / (2x + 5) , for all x not equal to 1
   B) (x + 3) / (2x + 5) , for all x
   C) x + 3 , for all x not equal to 1
   D) 1 /(2x + 5) , for all x
   E) (x + 3) / (2x + 5) , for all x not equal to -3
   
   
10. For all \( x \ne 1 \), which of the following is equivalent to the rational expression \( \dfrac{x-1}{(x^2 - 1)(x + 3)} \) ?
    A) 1 / (x + 3)
    B) 1 / (x² + 4 x + 3)
    C) 1 / (x + 1)
    D) 1 / x
    E) 1 / (x - 1)

1. C
2. A
3. E
4. D
5. A
   
   
6. C
7. B
8. E
9. A
10. B
    More on Simplifying Rational Expressions are included.

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