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CAPITULO XIII

Reducción de fracciones

Reducir una fracción a expresión entera o mixta

Ejercicio 123

Reducir a expresión entera o mixta:

$\frac{6 a^{3} - 10 a^{2}}{2 a}$
$\begin{matrix} \frac{6 a^{3} - 10 a^{2}}{2 a} & = \frac{a^{3}}{2 a} - \frac{a^{2}}{2 a} \\ = 3 a^{2} - 5 a \end{matrix}$
$\frac{9 x^{2} y - 6 x^{2} y^{2} + 3 x y^{3}}{3 x y}$
$\begin{matrix} \frac{9 x^{2} y - 6 x^{2} y^{2} + 3 x y^{3}}{3 x y} & = \frac{x^{2} y}{3 x y} - \frac{x^{2} y^{2}}{3 x y} + \frac{3 x y^{3}}{3 x y} \\ = 3 x - 2 x y + y^{2} \end{matrix}$
$\frac{x^{2} + 3}{x}$
$\begin{matrix} \frac{x^{2} + 3}{x} & = \frac{x^{2}}{x} + \frac{3}{x} \\ = x + \frac{3}{x} \end{matrix}$
$\frac{10 a^{2} + 15 a - 2}{5 a}$

$2 a + 3 - \frac{2}{5 a}$
$\frac{9 x^{3} - 6 x^{2} + 3 x - 5}{3 x}$

$3 x^{2} - 2 x + 1 - \frac{5}{3 x}$
$\frac{x^{2} - 5 x - 16}{x + 2}$

$x - 7 - \frac{2}{x + 2}$
$\frac{12 x^{2} - 6 x - 2}{4 x - 1}$

$\begin{matrix} 3 x - \frac{3}{4} - \frac{\frac{11}{4}}{4 x - 1} & = 3 x - \frac{3}{4} - \frac{11}{4 (4 x - 1)} \\ = 3 x - \frac{3 (4 x - 1) + 11}{4 (4 x - 1)} \\ = 3 x - \frac{12 x - 3 + 11}{4 (4 x - 1)} \\ = 3 x - \frac{12 x + 8}{4 (4 x - 1)} \\ = 3 x - \frac{4 (3 x + 2)}{4 (4 x - 1)} \\ = 3 x - \frac{3 x + 2}{4 x - 1} \end{matrix}$
$\frac{a^{3} + 3 b^{3}}{a + 2 b}$

$a^{2} - 2 a b + 4 b^{2} - \frac{5 b^{3}}{a + 2 b}$
$\frac{x^{3} - x^{2} - 6 x + 1}{x^{2} - 3}$

$x - 1 - \frac{3 x + 2}{x^{2} - 3}$
$\frac{3 x^{3} + 4 x^{2} y + 2 x y^{2} - 6 y^{3}}{3 x - 2 y}$

$x^{2} + 2 x y + 2 y^{2} - \frac{2 y^{3}}{3 x - 2 y}$
$\frac{2 x^{3} - 7 x^{2} + 6 x - 8}{2 x^{2} - x + 1}$

$x - 3 + \frac{2 x - 5}{2 x^{2} - x + 1}$
$\frac{2 a^{4} - 3 a^{3} + a^{2}}{a^{2} - a + 1}$

$2 a^{2} - a - 2 - \frac{a - 2}{a^{2} - a + 1}$
$\frac{x^{4} - 4 x^{2} - 3 x}{x^{2} - 2}$

$x^{2} - 2 - \frac{3 x + 4}{x^{2} - 2}$
$\frac{10 n^{3} - 18 n^{2} - 5 n + 3}{2 n^{2} - 3 n + 1}$

$\begin{matrix} 5 n - \frac{3}{2} + \frac{- \frac{29}{2} n + \frac{9}{2}}{2 n^{2} - 3 n + 1} & = 5 n - \frac{3}{2} + \frac{\frac{9 - 29 n}{2}}{2 n^{2} - 3 n + 1} \\ = 5 n - \frac{3}{2} + \frac{- 29 n + 9}{2 (2 n^{2} - 3 n + 1)} \\ = 5 n - \frac{1}{2} (3 - \frac{9 - 29 n}{2 n^{2} - 3 n + 1}) \\ = 5 n - \frac{1}{2} (\frac{6 n^{2} - 9 n + 3 - 9 + 29 n}{2 n^{2} - 3 n + 1}) \\ = 5 n - \frac{1}{2} (\frac{6 n^{2} + 20 n - 6}{2 n^{2} - 3 n + 1}) \\ = 5 n - \frac{1}{2} [\frac{2 (3 n^{2} + 10 n - 3)}{2 n^{2} - 3 n + 1}] \\ = 5 n - \frac{3 n^{2} + 10 n - 3}{2 n^{2} - 3 n + 1} \end{matrix}$
$\frac{8 x^{4}}{4 x^{2} + 5 x + 6}$

$\begin{matrix} 2 x^{2} - \frac{5}{2} x + \frac{1}{8} + \frac{\frac{115}{8} x - \frac{3}{4}}{4 x^{2} + 5 x + 6} & = 2 x^{2} - \frac{5}{2} x + \frac{1}{8} + \frac{\frac{115 x - 6}{8}}{4 x^{2} + 5 x + 6} \\ = 2 x^{2} - \frac{5}{2} x + \frac{1}{8} + \frac{115 x - 6}{8 (4 x^{2} + 5 x + 6)} \\ = 2 x^{2} - \frac{1}{2} [5 x - \frac{1}{4} - \frac{115 x - 6}{4 (4 x^{2} + 5 x + 6)}] \\ = 2 x^{2} - \frac{1}{2} [\frac{20 x (4 x^{2} + 5 x + 6) - (4 x^{2} + 5 x + 6) - 115 x + 6}{4 (4 x^{2} + 5 x + 6)}] \\ = 2 x^{2} - \frac{1}{2} [\frac{80 x^{3} + 100 x^{2} + 120 x - 4 x^{2} - 5 x - 6 - 115 x + 6}{4 (4 x^{2} + 5 x + 6)}] \\ = 2 x^{2} - \frac{1}{2} [\frac{80 x^{3} + 96 x^{2}}{4 (4 x^{2} + 5 x + 6)}] \\ = 2 x^{2} - \frac{1}{2} [\frac{8 (10 x^{3} + 12 x^{2})}{4 (4 x^{2} + 5 x + 6)}] \\ = 2 x^{2} - \frac{10 x^{3} + 12 x^{2}}{4 x^{2} + 5 x + 6} \end{matrix}$
$\frac{6 m^{5} + 3 m^{4} n}{3 m^{3} - m n^{2} + n^{3}}$

$\begin{matrix} 2 m^{2} + m n + \frac{2}{3} n^{2} + \frac{- m^{2} n^{3} - \frac{1}{3} m n^{4} - \frac{2}{3} n^{5}}{3 m^{3} - m n^{2} + n^{3}} & = 2 m^{2} + m n + \frac{2}{3} n^{2} + \frac{\frac{- 3 m^{2} n^{3} - m n^{4} - 2 n^{5}}{3}}{3 m^{3} - m n^{2} + n^{3}} \\ = 2 m^{2} + m n + \frac{2}{3} n^{2} + \frac{- 3 m^{2} n^{3} - m n^{4} - 2 n^{5}}{3 (3 m^{3} - m n^{2} + n^{3})} \\ = 2 m^{2} + m n + \frac{1}{3} [2 n^{2} + \frac{- 3 m^{2} n^{3} - m n^{4} - 2 n^{5}}{3 m^{3} - m n^{2} + n^{3}}] \\ = 2 m^{2} + m n + \frac{1}{3} [\frac{2 n^{2} (3 m^{3} - m n^{2} + n^{3}) - 3 m^{2} n^{3} - m n^{4} - 2 n^{5}}{3 m^{3} - m n^{2} + n^{3}}] \\ = 2 m^{2} + m n + \frac{1}{3} [\frac{6 m^{3} n^{2} - 2 m n^{4} + 2 n^{5} - 3 m^{2} n^{3} - m n^{4} - 2 n^{5}}{3 m^{3} - m n^{2} + n^{3}}] \\ = 2 m^{2} + m n + \frac{1}{3} [\frac{6 m^{3} n^{2} - 3 m^{2} n^{3} - 3 m n^{4}}{3 m^{3} - m n^{2} + n^{3}}] \\ = 2 m^{2} + m n + \frac{1}{3} [\frac{3 (2 m^{3} n^{2} - m^{2} n^{3} - m n^{4})}{3 m^{3} - m n^{2} + n^{3}}] \\ = 2 m^{2} + m n + \frac{2 m^{3} n^{2} - m^{2} n^{3} - m n^{4}}{3 m^{3} - m n^{2} + n^{3}} \end{matrix}$