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CAPITULO XIV

Miscelania sobre Fracciones

Ejercicio 140

Simplificar:

$\begin{matrix} \frac{12 x^{2} + 31 x + 20}{18 x^{2} + 21 x - 4} & = \frac{12 x^{2} + 16 x + 15 x + 20}{18 x^{2} - 3 x + 24 x - 4} \\ = \frac{4 x (3 x + 4) + 5 (3 x + 4)}{3 x (6 x - 1) + 4 (6 x - 1)} \\ = \frac{(4 x + 5) (3 x + 4)}{(3 x + 4) (6 x - 1)} \\ = \frac{4 x + 5}{6 x - 1} \end{matrix}$
$\begin{matrix} (\frac{1}{a} + \frac{2}{a^{2}} + \frac{1}{a^{3}}) \div (a + 2 - \frac{2 a + 1}{a}) & = \frac{1}{a} (1 + \frac{2}{a} + \frac{1}{a^{2}}) \div [\frac{a^{2} + 2 a - (2 a + 1)}{a}] \\ = \frac{1}{a} (\frac{a^{2} + 2 a + 1}{a^{2}}) \div (\frac{a^{2} + 2 a - 2 a - 1}{a}) \\ = \frac{1}{a} [\frac{{(a + 1)}^{2}}{a^{2}}] \div (\frac{a^{2} - 1}{a}) \\ = \frac{1}{a} [\frac{{(a + 1)}^{2}}{a^{2}}] \times [\frac{a}{(a + 1) (a - 1)}] \\ = \frac{a + 1}{a^{2} (a - 1)} \end{matrix}$
$\begin{matrix} \frac{x^{3} + 3 x^{2} + 9 x}{x^{5} - 27 x^{2}} & = \frac{x (x^{2} + 3 x + 9)}{x^{2} (x^{3} - 27)} \\ = \frac{x^{2} + 3 x + 9}{x (x - 3) (x^{2} + 3 x + 9)} \\ = \frac{1}{x (x - 3)} \end{matrix}$
$\begin{matrix} \frac{{(x + y)}^{2}}{y} - \frac{x {(x - y)}^{2}}{x y} & = \frac{1}{y} [{(x + y)}^{2} - \frac{x {(x - y)}^{2}}{x}] \\ = \frac{1}{y} [{(x + y)}^{2} - {(x - y)}^{2}] \\ = \frac{1}{y} {[(x + y) + (x - y)] [(x + y) - (x - y)]} \\ = \frac{1}{y} (x + y + x - y) (x + y - x + y) \\ = \frac{1}{y} (2 x) (2 y) \\ = 4 x \end{matrix}$
$\begin{matrix} \frac{a^{4} - 2 b^{3} + a^{2} b (b - 2)}{a^{4} - a^{2} b - 2 b^{2}} & = \frac{a^{4} - 2 b^{3} + a^{2} b^{2} - 2 a^{2} b}{a^{4} + a^{2} b - 2 a^{2} b - 2 b^{2}} \\ = \frac{a^{4} + a^{2} b^{2} - 2 a^{2} b - 2 b^{3}}{a^{2} (a^{2} + b) - 2 b (a^{2} + b)} \\ = \frac{a^{2} (a^{2} + b^{2}) - 2 b (a^{2} + b^{2})}{(a^{2} - 2 b) (a^{2} + b)} \\ = \frac{(a^{2} - 2 b) (a^{2} + b^{2})}{(a^{2} - 2 b) (a^{2} + b)} \\ = \frac{a^{2} + b^{2}}{a^{2} + b} \end{matrix}$
Multiplicar $a + \frac{1 + 5 a}{a^{2} - 5}$ por $a - \frac{a + 5}{a + 1}$
$\begin{matrix} (a + \frac{1 + 5 a}{a^{2} - 5}) (a - \frac{a + 5}{a + 1}) & = [\frac{a (a^{2} - 5) + 1 + 5 a}{a^{2} - 5}] [\frac{a (a + 1) - (a + 5)}{a + 1}] \\ = (\frac{a^{3} - 5 a + 1 + 5 a}{a^{2} - 5}) (\frac{a^{2} + a - a - 5}{a + 1}) \\ = (\frac{a^{3} + 1}{a^{2} - 5}) (\frac{a^{2} - 5}{a + 1}) \\ = \frac{(a + 1) (a^{2} - a + 1)}{a + 1} \\ = a^{2} - a + 1 \end{matrix}$
Dividir $x^{2} + 5 x - 4 - \frac{x^{3} - 29}{x - 5}$ entre $x + 34 + \frac{170 - x^{2}}{x - 5}$
$\begin{matrix} (x^{2} + 5 x - 4 - \frac{x^{3} - 29}{x - 5}) \div (x + 34 + \frac{170 - x^{2}}{x - 5}) & = [\frac{(x - 5) (x^{2} + 5 x - 4) - (x^{3} - 29)}{x - 5}] \div [\frac{(x - 5) (x + 34) + 170 - x^{2}}{x - 5}] \\ = (\frac{x^{3} + 5 x^{2} - 4 x - 5 x^{2} - 25 x + 20 - x^{3} + 29}{x - 5}) \div (\frac{x^{2} + 29 x - 170 + 170 - x^{2}}{x - 5}) \\ = (\frac{49 - 29 x}{x - 5}) \div (\frac{29 x}{x - 5}) \\ = (\frac{49 - 29 x}{x - 5}) (\frac{x - 5}{29 x}) \\ = \frac{49 - 29 x}{29 x} \end{matrix}$
Descomponer las expresiones siguientes en la suma o resta de tres fracciones simples irreducibles
$\begin{matrix} \frac{4 x^{2} - 5 x y + y^{2}}{3 x} & = \frac{4 x^{2}}{3 x} - \frac{5 x y}{3 x} + \frac{y^{2}}{3 x} \\ = \frac{4 x}{3} - \frac{5 y}{3} + \frac{y^{2}}{3 x} \end{matrix}$
$\begin{matrix} \frac{m - n - x}{m n x} & = \frac{m}{m n x} - \frac{n}{m n x} - \frac{x}{m n x} \\ = \frac{1}{n x} - \frac{1}{m x} - \frac{1}{m n} \end{matrix}$
Probar que $\frac{x^{3} - x y^{2}}{x - y} = x^{2} + x y$
$\begin{matrix} \frac{x^{3} - x y^{2}}{x - y} & = x^{2} + x y \\ \frac{x (x^{2} - y^{2})}{x - y} & = x (x + y) \\ \frac{x (x + y) (x - y)}{x - y} & = x (x + y) \\ x (x + y) & = x (x + y) \end{matrix}$
Probar que $x^{2} - 2 x + 1 - \frac{9 x - 3 x^{2}}{x - 3} = \frac{x^{3} - 1}{x - 1}$
$\begin{matrix} x^{2} - 2 x + 1 - \frac{9 x - 3 x^{2}}{x - 3} & = \frac{x^{3} - 1}{x - 1} \\ \frac{(x - 3) (x^{2} - 2 x + 1) - (9 x - 3 x^{2})}{x - 3} & = \frac{(x - 1) (x^{2} + x + 1)}{x - 1} \\ \frac{(x - 3) (x^{2} - 2 x + 1) + 3 x (x - 3)}{x - 3} & = x^{2} + x + 1 \\ \frac{(x - 3) [(x^{2} - 2 x + 1) + 3 x]}{x - 3} & = x^{2} + x + 1 \\ x^{2} - 2 x + 1 + 3 x & = x^{2} + x + 1 \\ x^{2} + x + 1 & = x^{2} + x + 1 \end{matrix}$
Probar que $\frac{a^{4} - 5 a^{2} + 4}{a^{3} + a^{2} - 4 a - 4} = a - 3 + \frac{2 + 4 a}{2 a + 1}$
$\begin{matrix} \frac{a^{4} - 5 a^{2} + 4}{a^{3} + a^{2} - 4 a - 4} & = a - 3 + \frac{2 + 4 a}{2 a + 1} \\ \frac{a^{4} - a^{2} - 4 a^{2} + 4}{a^{2} (a + 1) - 4 (a + 1)} & = \frac{(2 a + 1) (a - 3) + 2 + 4 a}{2 a + 1} \\ \frac{a^{2} (a^{2} - 1) - 4 (a^{2} - 1)}{(a^{2} - 4) (a + 1)} & = \frac{2 a^{2} + a - 6 a - 3 + 2 + 4 a}{2 a + 1} \\ \frac{(a^{2} - 4) (a^{2} - 1)}{(a^{2} - 4) (a + 1)} & = \frac{2 a^{2} - a - 1}{2 a + 1} \\ \frac{(a + 1) (a - 1)}{a + 1} & = \frac{2 a^{2} - 2 a + a - 1}{2 a + 1} \\ a - 1 & = \frac{2 a (a - 1) + (a - 1)}{2 a + 1} \\ a - 1 & = \frac{(2 a + 1) (a - 1)}{2 a + 1} \\ a - 1 & = a - 1 \end{matrix}$
Simplificar:
$\begin{matrix} \frac{1}{a - b} + \frac{1}{a + b} + \frac{2 a}{a^{2} - a b + b^{2}} & = \frac{(a + b) (a^{2} - a b + b^{2}) + (a - b) (a^{2} - a b + b^{2}) + 2 a (a + b) (a - b)}{(a - b) (a + b) (a^{2} - a b + b^{2})} \\ = \frac{a^{3} - a^{2} b + a b^{2} + a^{2} b - a b^{2} + b^{3} + a^{3} - a^{2} b + a b^{2} - a^{2} b + a b^{2} - b^{3} + 2 a (a^{2} - b^{2})}{(a - b) (a^{3} + b^{3})} \\ = \frac{2 a^{3} - 2 a^{2} b + 2 a b^{2} + 2 a (a^{2} - b^{2})}{(a - b) (a^{3} + b^{3})} \\ = \frac{2 a [a^{2} - a b + b^{2} + a^{2} - b^{2}]}{(a - b) (a^{3} + b^{3})} \\ = \frac{2 a [2 a^{2} - a b]}{(a - b) (a^{3} + b^{3})} \\ = \frac{2 a^{2} (2 a - b)}{(a - b) (a^{3} + b^{3})} \end{matrix}$
$\begin{matrix} (\frac{a^{2}}{1 - a^{2}} - \frac{a^{4}}{1 - a^{4}}) \times (1 - a + \frac{1 + a^{3}}{a^{2}}) & = [\frac{a^{2}}{1 - a^{2}} - \frac{a^{4}}{(1 - a^{2}) (1 + a^{2})}] \times [\frac{a^{2} (1 - a) + 1 + a^{3}}{a^{2}}] \\ = \frac{a^{2}}{1 - a^{2}} [1 - \frac{a^{2}}{1 + a^{2}}] \times [\frac{a^{2} - a^{3} + 1 + a^{3}}{a^{2}}] \\ = \frac{1}{1 - a^{2}} (\frac{1 + a^{2} - a^{2}}{1 + a^{2}}) \times (a^{2} + 1) \\ = \frac{1}{1 - a^{2}} \end{matrix}$
$\begin{matrix} (\frac{x^{2} - 9}{x^{2} - x - 12} \div \frac{x - 3}{x^{2} + 3 x}) \times \frac{a^{2} x^{2} - 16 a^{2}}{2 x^{2} + 7 x + 3} \times (\frac{2}{a^{2} x} + \frac{1}{a^{2} x^{2}}) & = [\frac{(x - 3) (x + 3)}{(x - 4) (x + 3)} \times \frac{x (x + 3)}{x - 3}] \times \frac{a^{2} (x^{2} - 16)}{2 x^{2} + x + 6 x + 3} \times \frac{1}{a^{2} x} (2 + \frac{1}{x}) \\ = \frac{1}{x - 4} \times x (x + 3) \times \frac{a^{2} (x - 4) (x + 4)}{x (2 x + 1) + 3 (2 x + 1)} \times \frac{1}{a^{2} x} (\frac{2 x + 1}{x}) \\ = (x + 3) \times \frac{x + 4}{(x + 3) (2 x + 1)} \times (\frac{2 x + 1}{x}) \\ = \frac{x + 4}{x} \end{matrix}$
$\begin{matrix} \frac{16 - 81 x^{2}}{72 x^{2} - 5 x - 12} & = \frac{(4 - 9 x) (4 + 9 x)}{72 x^{2} - 32 x + 27 x - 12} \\ = \frac{(4 - 9 x) (4 + 9 x)}{8 x (9 x + 4) - 3 (9 x + 4)} \\ = \frac{(4 - 9 x) (4 + 9 x)}{(8 x - 3) (9 x + 4)} \\ = \frac{4 - 9 x}{8 x - 3} \end{matrix}$
$\begin{matrix} (\frac{1}{x} - \frac{2}{x + 2} + \frac{3}{x + 3}) \div (\frac{x}{x + 2} + \frac{x}{x + 3} + \frac{6}{x^{2} + 5 x + 6}) & = [\frac{(x + 2) (x + 3) - 2 x (x + 3) + 3 x (x + 2)}{x (x + 2) (x + 3)}] \div (\frac{x}{x + 2} + \frac{x}{x + 3} + \frac{6}{(x + 2) (x + 3)}) \\ = [\frac{x^{2} + 5 x + 6 - 2 x^{2} - 6 x + 3 x^{2} + 6 x}{x (x + 2) (x + 3)}] \div [\frac{x (x + 3) + x (x + 2) + 6}{(x + 2) (x + 3)}] \\ = [\frac{2 x^{2} + 5 x + 6}{x (x + 2) (x + 3)}] \div [\frac{x^{2} + 3 x + x^{2} + 2 x + 6}{(x + 2) (x + 3)}] \\ = \frac{2 x^{2} + 5 x + 6}{x (x + 2) (x + 3)} \times \frac{(x + 2) (x + 3)}{2 x^{2} + 5 x + 6} \\ = \frac{1}{x} \end{matrix}$
$\begin{matrix} \frac{\frac{b}{a}}{1 - \frac{b^{2}}{a^{2}}} + \frac{1 + \frac{b}{a - b}}{2 - \frac{a - 3 b}{a - b}} & = \frac{\frac{b}{a}}{\frac{a^{2} - b^{2}}{a^{2}}} + \frac{\frac{a - b - b}{a - b}}{\frac{2 (a - b) - (a - 3 b)}{a - b}} \\ = \frac{a b}{a^{2} - b^{2}} + \frac{a}{2 a - 2 b - a + 3 b} \\ = \frac{a b}{(a + b) (a - b)} + \frac{a}{a + b} \\ = \frac{a}{a + b} [\frac{b}{a - b} + 1] \\ = \frac{a}{a + b} [\frac{b + a - b}{a - b}] \\ = \frac{a^{2}}{a^{2} - b^{2}} \end{matrix}$
$\begin{matrix} \frac{1}{3} (\frac{x^{2} - 36}{x} \div \frac{x}{x^{2} - 4}) \times \frac{1}{x - \frac{36}{x}} \times \frac{1}{x - \frac{4}{x}} & = \frac{1}{3} [\frac{(x + 6) (x - 6)}{x} \times \frac{(x - 2) (x + 2)}{x}] \times \frac{1}{\frac{x^{2} - 36}{x}} \times \frac{1}{\frac{x^{2} - 4}{x}} \\ = \frac{1}{3} \frac{(x + 6) (x - 6)}{x} \times \frac{(x - 2) (x + 2)}{x} \times \frac{x}{x^{2} - 36} \times \frac{x}{x^{2} - 4} \\ = \frac{1}{3} \end{matrix}$
$\begin{matrix} \frac{\frac{3 a}{{(a - 2 b)}^{2}} + \frac{5}{a - 5 b} + \frac{1}{a - 2 b}}{\frac{3 a^{2} - 14 a b + 10 b^{2}}{a^{2} - 4 a b + 4 b^{2}}} & = \frac{\frac{3 a (a - 5 b) + 5 {(a - 2 b)}^{2} + (a - 5 b) (a - 2 b)}{(a - 5 b) {(a - 2 b)}^{2}}}{\frac{3 a^{2} - 14 a b + 10 b^{2}}{{(a - 2 b)}^{2}}} \\ = \frac{3 a^{2} - 15 a b + 5 (a^{2} - 4 a b + 4 b^{2}) + a^{2} - 7 a b + 10 b^{2}}{(a - 5 b) (3 a^{2} - 14 a b + 10 b^{2})} \\ = \frac{3 a^{2} - 22 a b + 5 a^{2} - 20 a b + 20 b^{2} + a^{2} + 10 b^{2}}{(a - 5 b) (3 a^{2} - 14 a b + 10 b^{2})} \\ = \frac{9 a^{2} - 42 a b + 30 b^{2}}{(a - 5 b) (3 a^{2} - 14 a b + 10 b^{2})} \\ = \frac{3 (3 a^{2} - 14 a b + 10 b^{2})}{(a - 5 b) (3 a^{2} - 14 a b + 10 b^{2})} \\ = \frac{3}{a - 5 b} \end{matrix}$
$\begin{matrix} \frac{\frac{x + 1}{x - 1} - \frac{x - 1}{x + 1}}{\frac{x - 1}{x + 1} + \frac{x + 1}{x - 1}} \times \frac{x^{2} + 1}{2 a^{2} - 2 b} \div \frac{2 x}{a^{2} - b} & = \frac{\frac{{(x + 1)}^{2} - {(x - 1)}^{2}}{(x - 1) (x + 1)}}{\frac{{(x - 1)}^{2} + {(x + 1)}^{2}}{(x - 1) (x + 1)}} \times \frac{x^{2} + 1}{2 (a^{2} - b)} \times \frac{a^{2} - b}{2 x} \\ = \frac{{(x + 1)}^{2} - {(x - 1)}^{2}}{{(x - 1)}^{2} + {(x + 1)}^{2}} \times \frac{x^{2} + 1}{2} \times \frac{1}{2 x} \\ = \frac{[(x + 1) + (x - 1)] [(x + 1) - (x - 1)]}{x^{2} - 2 x + 1 + x^{2} + 2 x + 1} \times \frac{x^{2} + 1}{4 x} \\ = \frac{(x + 1 + x - 1) (x + 1 - x + 1)}{2 x^{2} + 2} \times \frac{x^{2} + 1}{4 x} \\ = \frac{2 (2 x)}{2 (x^{2} + 1)} \times \frac{x^{2} + 1}{4 x} \\ = \frac{1}{2} \end{matrix}$
$\begin{matrix} \frac{1}{3 x - 9} - \frac{1}{6 x + 12} - \frac{1}{2 {(x - 3)}^{2}} + \frac{1}{x - 6 + \frac{9}{x}} & = \frac{1}{3 (x - 3)} - \frac{1}{6 (x + 2)} - \frac{1}{2 {(x - 3)}^{2}} + \frac{1}{\frac{x (x - 6) + 9}{x}} \\ = \frac{1}{3 (x - 3)} - \frac{1}{6 (x + 2)} - \frac{1}{2 {(x - 3)}^{2}} + \frac{1}{\frac{x^{2} - 6 x + 9}{x}} \\ = \frac{1}{3 (x - 3)} - \frac{1}{6 (x + 2)} - \frac{1}{2 {(x - 3)}^{2}} + \frac{x}{x^{2} - 6 x + 9} \\ = \frac{1}{3 (x - 3)} - \frac{1}{6 (x + 2)} - \frac{1}{2 {(x - 3)}^{2}} + \frac{x}{{(x - 3)}^{2}} \\ = \frac{2 (x + 2) (x - 3) - {(x - 3)}^{2} - 3 (x + 2) + 6 x (x + 2)}{6 (x + 2) {(x - 3)}^{2}} \\ = \frac{2 (x^{2} - x - 6) - (x^{2} - 6 x + 9) - 3 x - 6 + 6 x^{2} + 12 x}{6 (x + 2) {(x - 3)}^{2}} \\ = \frac{2 x^{2} - 2 x - 12 - x^{2} + 6 x - 9 - 6 + 6 x^{2} + 9 x}{6 (x + 2) {(x - 3)}^{2}} \\ = \frac{7 x^{2} + 13 x - 27}{6 (x + 2) {(x - 3)}^{2}} \end{matrix}$
$\begin{matrix} \frac{a - b + \frac{a^{2} + b^{2}}{a + b}}{a + b - \frac{a^{2} - 2 b^{2}}{a - b}} \times \frac{b + \frac{b^{2}}{a}}{a - b} \times \frac{1}{1 + \frac{2 a - b}{b}} & = \frac{\frac{(a + b) (a - b) + a^{2} + b^{2}}{a + b}}{\frac{(a - b) (a + b) - (a^{2} - 2 b^{2})}{a - b}} \times \frac{\frac{a b + b^{2}}{a}}{a - b} \times \frac{1}{\frac{b + 2 a - b}{b}} \\ = \frac{\frac{a^{2} - b^{2} + a^{2} + b^{2}}{a + b}}{\frac{a^{2} - b^{2} - a^{2} + 2 b^{2}}{a - b}} \times \frac{\frac{b (a + b)}{a}}{a - b} \times \frac{b}{2 a} \\ = \frac{\frac{2 a^{2}}{a + b}}{\frac{b^{2}}{a - b}} \times \frac{b (a + b)}{a (a - b)} \times \frac{b}{2 a} \\ = \frac{2 a^{2} (a - b)}{b^{2} (a + b)} \times \frac{b (a + b)}{(a - b)} \times \frac{b}{2 a} \\ = 1 \end{matrix}$