This document contains examples of multiplying, expanding, and simplifying rational expressions. Some examples involve breaking rational expressions into sums and differences of fractions. Other examples use long division to write rational expressions in the form of a quotient plus a remainder over the divisor. The rational expressions involve variables and operations.

1. Multiplication and Division of Rational Expressions
Ex A. Simplify. Do not expand the results.
3 3.12x6 * 5 14 3
1. 10x * 2 3 2. 5x3 * 10x 4. 5x3 * 10x
5x 6x
5. 15x * 16 4 6. 10 * 183 7. 56x6* 635 8. 75x * 42 3
4 25x 9x4 5x 27 8x 49 25x
2x – 4 * 5x + 10 6 – 4x x – 2
9. 2x + 4 3x – 6 10. 3x – 2 * 2x + 4
9x – 12* 2 – x 4–x * x+4
11. 2x – 4 8 – 6x 12. –x – 4 x – 4
3x – 9 * 3 – x 42 – 6x 4 – 2x
13. 15x – 5 5 – 15x 14. –2x + 14 * –7x + 14
(x2 + x – 2 ) (x2 – 4 ) (x2 + 2x – 3 ) (x2 – 2x – 3)
15. 16. *
(x – 2)
*
(x2 – x) (x – 9)
2
(x2 – x – 2 )
(x2 – x – 2 ) (x2 + x ) (x2 + 5x – 6 ) (x2 – 5x + 6)
17. 18. 2 *
(x – 1)
2
*
(x2 + 2x + 1) (x + 5x + 6) (x2 – 5x – 6 )
(x2 – 3x – 4 ) (x2 – 3x + 2) 20. (– x + 6 – x ) (6 – x – x)
2 2
19. * *
(x + 5x + 6) (x2 – x – 12 )
2
(x2 – 1) (x2 – 2x – 8)

2. Multiplication and Division of Rational Expressions
Ex. A. Simplify. Do not expand the results.
(2x2 + x – 1 ) ÷ (4x2 – 1) (3x2 – 2x – 1) (x2 + x – 2 )
21. 22.
(1 – 2x) (2x – x )
2 (1 – 9x2) ÷ (x2 + 4x + 4)
(x2 – x + 2) ÷ (3x2 + 4x + 1) 24. (x + 1 – 6x ) ÷ (2x + x – 1 )
2 2
23. (3x2 – x – 2)
(–x – 3x2) (–x2 – 4) (x2 – 5x – 6)
(x3 – 4x) (x2 + 2) (–x3 + 9x ) (x2 + 6x + 9)
25. ÷ 26.
(–x + 4x – 4) (–x + 2)
2
(x2 + 3x) ÷ (–3x2 – 9x)
Ex. B. Multiply, expand and simplify the results.
27. x + 3 (x2 – 1) 28. x – 3 (x2 – 4) 29. 2x + 3 (x2 – 1)
x+1 x–2 1–x
30. 3 – 2x (x + 2)(x +1) 31. 3 – 2x (3x + 2)(1 – 2x)
x+2 2x – 1
32. ( x – 2 + x + 1 )( x – 3)(x + 3)
x–3 x+3
33. ( 2x – 1 – x + 2 ) ( 2x – 3)(x + 2)
x+2 2x – 3

3. Multiplication and Division of Rational Expressions
Ex B. Multiply, expand and simplify the results.
34. ( 4 – x – x – 1 )( x – 3)(2x + 3)
x – 3 2x + 3
35. ( 3 – x – 2x + 3 )(x – 3)(x + 2)
x+2 x–3
36. ( 3 – 4x – 1 – 2x )( x + 3)(x + 1)
x+1 x+3
37. ( 5x – 7 – 4 – 5x )(x – 3)(x + 5)
x+5 x–3
38. ( x2– 2 – 2 x + 1 ) ( x – 3)(x + 3)(x + 1)
x –9 x – 2x – 3
39. ( x2+ 3 – 2 2x + 1 ) ( x – 2)(x + 2)(x – 1)
x –4 x +x–2
40. ( x – 1 – x2 x + 1 – 3 ) ( x – 3)(x + 2)(x + 1)
x2 – x – 6 – 2x
41. ( x + 2 – x2 2x2x 1 3 ) ( x – 3)(x + 3)(x – 1)
+
x2 – 4x +3 + –

4. Multiplication and Division of Rational Expressions
Ex. C. Break up the following expressions as sums and
differences of fractions.
x –4 x2 – 4 x2 + 4x – 6
2
43. 44.
42. x 2 2x2
2x
12x3 – 9x2 + 6x 46. x – x – x
8
x6 4
47. x – x 2 – x
8 6 4
45. 3x x
Ex D. Use long division and write each rational expression in
R
the form of Q + D .
48. 2x++26 49. 3x––25 50. 4x– 1+3
x x x
51. 5x––34 52. 3x– x8
+ 53. –4x – 5
x 2 1–x
(x2 + x – 2 ) (2x2 + x – 3 ) (3x2 – 3x – 2 )
54. 55. 56.
(x – 1) (x – 2) (x + 2)
(–x2 + 4x – 3 ) (4x2 + 2 ) (5x2 – 1 )
57. 58. 59.
(x – 3) (x + 3) (x – 4)