Section 3.1
Reducing Fractions
If $x$ and $y$ are real numbers, the quotient $\dfrac{x}{y}\;(y\neq0)$ is called a fraction. The number $x$ is called the numerator, and the number $y$ is called the denominator.
Both the numerator and denominator can also be algebraic expressions. Reducing fractions, whether it consists of numbers or algebraic expressions, consists of canceling like factors. To reduce a fraction, you must find a factor that appears in both the numerator and the denominator. Any such factor can be canceled. This is because of the property $$\frac{A\cdot C}{B\cdot C}=\frac{A}{B}\cdot\frac{C}{C}=\frac{A}{B}\cdot 1=\frac{A}{B}.$$
The factor of $x$, common to the numerator and the denominator in the initial fraction, can be canceled. In all cases, for now, assume the value of the denominator is non-zero.
Examples:
a. $$\dfrac{2(x-3)}{4(x+1)}=\dfrac{x-3}{2(x+1)}$$
b. \begin{align}
\frac{x^2-4x-12}{x^2-4}&=\frac{(x-6)(x+2)}{(x-2)(x+2)}\\
&=\frac{x-6}{x-2}
\end{align}
c. \begin{align}
\frac{3-x}{x^2-5x+6}&=\frac{-1(x-3)}{(x-3)(x-2)}\\
&=\frac{-1}{x-2}
\end{align}
d. \begin{align}
\frac{ax-3a-x^2+3x}{x^2+x-12}&=\frac{a(x-3)-x(x-3)}{(x+4)(x-3)}\\
&=\frac{(a-x)(x-3)}{(x+4)(x-3)}\\
&=\frac{a-x}{x+4}
\end{align}
Simplifying rational expressions introduction: Simplifying Rational Expressions
Exercises:
Reduce each of the following fractions.
\begin{array}{l l l}
\renewcommand{\arraystretch}{4}
1. \;\;\frac{4{{x}^{2}}y}{6x{{y}^{2}}}\;\;& 2. \;\;\frac{12{{a}^{4}}{{b}^{2}}}{8{{a}^{2}}b}\;\;& 3. \;\;\frac{144{{a}^{12}}{{b}^{20}}}{36{{a}^{14}}{{b}^{18}}}\;\; \\
4. \;\;\frac{6p-18}{-3}\;\;& 5. \;\;\frac{10{{K}^{3}}+25K}{15{{K}^{2}}}\;\;& 6. \;\;\frac{-2t}{4{{t}^{2}}-6t}\;\; \\
7. \;\;\frac{2x+6}{8x-2}\;\;& 8. \;\;\frac{5{{x}^{2}}-10x}{30{{x}^{3}}-15{{x}^{2}}}\;\;& 9. \;\;\frac{6x-9}{9x-6}\;\; \\
10. \;\;\frac{4{{x}^{2}}+xy}{6x{{y}^{2}}}\;\;& 11. \;\;\frac{12{{a}^{4}}+16{{a}^{2}}{{b}^{2}}}{8{{a}^{2}}b}\;\;& 12. \;\;\frac{144{{a}^{12}}{{b}^{20}}}{36{{a}^{14}}-24{{a}^{15}}{{b}^{18}}}\;\; \\
13. \;\;\frac{4{{x}^{2}}-12x}{6x-9}\;\;& 14. \;\;\frac{6x-18}{{{x}^{2}}-9}\;\;& 15. \;\;\frac{4{{x}^{2}}-12x}{{{x}^{2}}+x-12}\;\; \\
16. \;\;\frac{{{x}^{2}}-7x+12}{{{x}^{2}}-9}\;\;& 17. \;\;\frac{{{x}^{2}}+5x+4}{2{{x}^{2}}+22x+56}\;\;& 18. \;\;\frac{{{x}^{2}}-3x-28}{{{x}^{2}}-8x+7}\;\; \\
19. \;\;\frac{{{x}^{2}}+6x-16}{3{{x}^{2}}-192}\;\;& 20. \;\;\frac{3{{x}^{3}}-12{{x}^{2}}+9x}{6{{x}^{4}}-6{{x}^{2}}}\;\;& 21. \;\;\frac{\left( {{x}^{2}}-5x-6 \right)\left( {{x}^{2}}-2x+1 \right)}{\left( {{x}^{2}}-1 \right)\left( {{x}^{2}}-36 \right)}\;\; \\
22. \;\;\frac{5(x+3)-y(x+3)}{{{y}^{2}}-25}\;\;& 23. \;\;\frac{{{x}^{2}}y-x}{y-x{{y}^{2}}}\;\;& 24. \;\;\frac{x(x-7)-3(x-7)}{{{x}^{2}}+2x-15}\;\; \\
25. \;\;\frac{{{x}^{2}}-9x+xy-9y}{{{x}^{2}}-8x-9}\;\;& 26. \;\;\frac{7{{a}^{2}}{{x}^{3}}-343{{a}^{4}}x}{{{x}^{2}}-8ax+7{{a}^{2}}}\;\;& 27. \;\;\frac{{{(x+3)}^{2}}-{{K}^{2}}}{{{(x+3)}^{2}}-2K(x+3)+{{K}^{2}}}\;\;
\end{array}
\begin{array}{l l}
28. \;\;\frac{\left( {{x}^{3}}-x \right)\left( {{x}^{2}}-2x-8 \right)}{\left( {{x}^{2}}-3x-4 \right)\left( 2{{x}^{2}}+2x-4 \right)}\;\;& 29. \;\;\frac{6{{x}^{2}}-19x+3}{7x-2{{x}^{2}}-3}\;\; \\
30. \;\;\frac{{{x}^{3}}-3{{x}^{2}}-28x}{{{x}^{4}}-16{{x}^{2}}}\;\;& 31. \;\;\frac{5{{x}^{3}}+5{{x}^{2}}-10x}{25{{x}^{5}}-125{{x}^{3}}+100x}\;\; \\
32. \;\;\frac{3x(x-1)-2(x-1)}{9{{x}^{3}}-4x}\;\;& 33. \;\;\frac{{{x}^{3}}-3{{x}^{2}}+2x}{{{x}^{3}}+3{{x}^{2}}+2x}\;\; \\
34. \;\;\frac{12{{x}^{2}}-17x+6}{3{{x}^{2}}-5x+2}\;\;& 35. \;\;\frac{{{x}^{2}}-3x+2}{1-x}\;\; \\
36. \;\;\frac{{{x}^{2}}-4x+4}{4-{{x}^{2}}}\;\;& 37. \;\;\frac{{{x}^{4}}-13{{x}^{2}}+36}{(2-x)(3-x)}\;\; \\
38. \;\;\frac{{{x}^{2}}-2ax-24{{a}^{2}}}{{{x}^{2}}-5ax-6{{a}^{2}}}\;\;& 39. \;\;\frac{(1-x)(2-x)(3-x)}{\left( {{x}^{2}}-1 \right)\left( {{x}^{2}}-4 \right)}\;\; \\
40. \;\;\frac{ax+2ay-bx-2by}{{{x}^{2}}-4{{y}^{2}}}\;\;& 41. \;\;\frac{ax+2ay-bx-2by}{{{a}^{2}}-{{b}^{2}}}\;\; \\
42. \;\;\frac{3ax-2bx-3{{a}^{2}}+2ab}{4{{x}^{2}}-4{{a}^{2}}}\;\;& 43. \;\;\frac{3ax-2bx-3{{a}^{2}}+2ab}{9{{a}^{2}}-4{{b}^{2}}}\;\; \\
44. \;\;\frac{\left( {{x}^{3}}-3{{x}^{2}}-4x \right)\left( {{x}^{2}}+6x+9 \right)}{\left( {{x}^{2}}-x-12 \right)\left( {{x}^{2}}+4x+3 \right)}\;\;& 45. \;\;\frac{{{x}^{4}}-13{{x}^{2}}+36}{\left( {{x}^{2}}-x-12 \right)\left( {{x}^{2}}+x-12 \right)}\;\; \\
46. \;\;\frac{2{{x}^{2}}-3x-2}{4{{x}^{2}}-1}\;\;& 47. \;\;\frac{2-x-{{x}^{2}}}{{{x}^{2}}-1}\;\; \\
48. \;\;\frac{4{{x}^{2}}-14x+6}{4{{x}^{2}}+10x-6}\;\;& 49. \;\;\frac{6{{x}^{2}}+11x-10}{9{{x}^{2}}+6x-8}\;\; \\
50. \;\;\frac{2-x}{4x-8}\;\; 51. \;\;\frac{6y-12{{y}^{2}}}{9y}\;\;& 52. \;\;\frac{3{{x}^{2}}-15x+18}{{{x}^{2}}-x-6}\;\;\\
53. \;\;\frac{{{(x+6)}^{2}}(1-x)}{\left( 36-{{x}^{2}} \right)(x-1)}\;\;& 54. \;\;\frac{5{{a}^{2}}-35a-8}{2{{a}^{2}}-14a-16}\;\; \\ 55. \;\;\frac{24{{x}^{3}}+120{{x}^{2}}}{4{{x}^{3}}-100x}\;\;
\end{array}