Comparte esto 👍👍

DESCARGA

CAPITULO XIII

Fracciones algebraicas

Simplificación de fracciones cuyos términos sean polinomios

Ejercicio 119

Simplificar o reducir a su más simple expresión:

$\begin{matrix} \frac{3 a b}{2 a^{2} x + 2 a^{3}} & = \frac{3 a b}{2 a^{2} (x + a)} \\ = \frac{3 b}{2 a (x + a)} \end{matrix}$
$\begin{matrix} \frac{x y}{3 x^{2} y - 3 x y^{2}} & = \frac{x y}{3 x y (x - y)} \\ = \frac{1}{3 (x - y)} \end{matrix}$
$\begin{matrix} \frac{2 a x + 4 b x}{3 a y + 6 b y} & = \frac{2 x (a + 2 b)}{3 y (a + 2 b)} \\ = \frac{2 x}{3 y} \end{matrix}$
$\begin{matrix} \frac{x^{2} - 2 x - 3}{x - 3} & = \frac{(x - 3) (x + 1)}{x - 3} \\ = x + 1 \end{matrix}$
$\begin{matrix} \frac{10 a^{2} b^{3} c}{80 (a^{3} - a^{2} b)} & = \frac{10 a^{2} b^{3} c}{80 a^{2} (a - b)} \\ = \frac{b^{3} c}{8 (a - b)} \end{matrix}$
$\begin{matrix} \frac{x^{2} - 4}{5 a x + 10 a} & = \frac{(x - 2) (x + 2)}{5 a (x + 2)} \\ = \frac{x - 2}{5 a} \end{matrix}$
$\begin{matrix} \frac{3 x^{2} - 4 x - 15}{x^{2} - 5 x + 6} & = \frac{3 x^{2} - 9 x + 5 x - 15}{(x - 3) (x - 2)} \\ = \frac{3 x (x - 3) + 5 (x - 3)}{(x - 3) (x - 2)} \\ = \frac{(x - 3) (3 x + 5)}{(x - 3) (x - 2)} \\ = \frac{3 x + 5}{x - 2} \end{matrix}$
$\begin{matrix} \frac{15 a^{2} b n - 45 a^{2} b m}{10 a^{2} b^{2} n - 30 a^{2} b^{2} m} & = \frac{a^{2} b (n - 3 m)}{a^{2} b^{2} (n - 3 m)} \\ = \frac{3}{2 b} \end{matrix}$
$\begin{matrix} \frac{x^{2} - y^{2}}{x^{2} + 2 x y + y^{2}} & = \frac{(x + y) (x - y)}{{(x + y)}^{2}} \\ = \frac{x - y}{x + y} \end{matrix}$
$\begin{matrix} \frac{3 x^{2} y + 15 x y}{x^{2} - 25} & = \frac{3 x y (x + 5)}{(x - 5) (x + 5)} \\ = \frac{3 x y}{x - 5} \end{matrix}$
$\begin{matrix} \frac{a^{2} - 4 a b + 4 b^{2}}{a^{3} - 8 b^{3}} & = \frac{{(a - 2 b)}^{2}}{(a - 2 b) (a^{2} + 2 a b + 4 b^{2})} \\ = \frac{a - 2 b}{a^{2} + 2 a b + 4 b^{2}} \end{matrix}$
$\begin{matrix} \frac{x^{3} + 4 x^{2} - 21 x}{x^{3} - 9 x} & = \frac{x (x^{2} + 4 x - 21)}{x (x^{2} - 9)} \\ = \frac{x^{2} + 4 x - 21}{x^{2} - 9} \\ = \frac{(x + 7) (x - 3)}{(x - 3) (x + 3)} \\ = \frac{x + 7}{x + 3} \end{matrix}$
$\begin{matrix} \frac{6 x^{2} + 5 x - 6}{15 x^{2} - 7 x - 2} & = \frac{6 x^{2} + 9 x - 4 x - 6}{15 x^{2} - 10 x + 3 x - 2} \\ = \frac{3 x (2 x + 3) - 2 (2 x + 3)}{5 x (3 x - 2) + (3 x - 2)} \\ = \frac{(3 x - 2) (2 x + 3)}{(3 x - 2) (5 x + 1)} \\ = \frac{2 x + 3}{5 x + 1} \end{matrix}$
$\begin{matrix} \frac{a^{3} + 1}{a^{4} - a^{3} + a - 1} & = \frac{(a + 1) (a^{2} - a + 1)}{a^{3} (a - 1) + (a - 1)} \\ = \frac{(a + 1) (a^{2} - a + 1)}{(a - 1) (a^{3} + 1)} \\ = \frac{1}{a - 1} \end{matrix}$
$\begin{matrix} \frac{2 a x + a y - 4 b x - 2 b y}{a x - 4 a - 2 b x + 8 b} & = \frac{a (2 x + y) - 2 b (2 x + y)}{a (x - 4) - 2 b (x - 4)} \\ = \frac{(a - 2 b) (2 x + y)}{(a - 2 b) (x - 4)} \\ = \frac{2 x + y}{x - 4} \end{matrix}$
$\begin{matrix} \frac{a^{2} - a b - 6 b^{2}}{a^{3} x - 6 a^{2} b x + 9 a b^{2} x} & = \frac{(a - 3 b) (a + 2 b)}{a x (a^{2} - 6 a b + 9 b^{2})} \\ = \frac{(a - 3 b) (a + 2 b)}{a x {(a - 3 b)}^{2}} \\ = \frac{a + 2 b}{a x (a - 3 b)} \end{matrix}$
$\begin{matrix} \frac{m^{2} + n^{2}}{m^{4} - n^{4}} & = \frac{m^{2} + n^{2}}{(m^{2} - n^{2}) (m^{2} + n^{2})} \\ = \frac{1}{m^{2} - n^{2}} \end{matrix}$
$\begin{matrix} \frac{x^{3} + y^{3}}{{(x + y)}^{3}} & = \frac{(x + y) (x^{2} - x y + y^{2})}{{(x + y)}^{}} \\ = \frac{x^{2} - x y + y^{2}}{{(x + y)}^{2}} \end{matrix}$
$\begin{matrix} \frac{{(m - n)}^{2}}{m^{2} - n^{2}} & = \frac{{(m - n)}^{2}}{(m + n) (m - n)} \\ = \frac{m - n}{m + n} \end{matrix}$
$\begin{matrix} \frac{{(a - x)}^{3}}{a^{3} - x^{3}} & = \frac{{(a - x)}^{}}{(a - x) (a^{2} + a x + x^{2})} \\ = \frac{{(a - x)}^{2}}{a^{2} + a x + x^{2}} \end{matrix}$
$\begin{matrix} \frac{a^{2} - a - 20}{a^{2} - 7 a + 10} & = \frac{(a - 5) (a + 4)}{(a - 5) (a - 2)} \\ = \frac{a + 4}{a - 2} \end{matrix}$
$\begin{matrix} \frac{{(1 - a^{2})}^{2}}{a^{2} + 2 a + 1} & = \frac{{[(1 - a) (1 + a)]}^{2}}{{(a + 1)}^{2}} \\ = \frac{{(1 - a)}^{2} {(1 + a)}^{2}}{{(1 + a)}^{2}} \\ = {(1 - a)}^{2} \end{matrix}$
$\begin{matrix} \frac{a^{4} b^{2} - a^{2} b^{4}}{a^{4} - b^{4}} & = \frac{a^{2} b^{2} (a^{2} - b^{2})}{(a^{2} + b^{2}) (a^{2} - b^{2})} \\ = \frac{a^{2} b^{2}}{a^{2} + b^{2}} \end{matrix}$
$\begin{matrix} \frac{x^{2} - y^{2}}{x^{3} - y^{3}} & = \frac{(x + y) (x - y)}{(x - y) (x^{2} + x y + y^{2})} \\ = \frac{x + y}{x^{2} + x y + y^{2}} \end{matrix}$
$\begin{matrix} \frac{24 a^{3} b + 8 a^{2} b^{2}}{36 a^{4} + 24 a^{3} b + 4 a^{2} b^{2}} & = \frac{a^{2} b (3 a + b)}{4 a^{2} (9 a^{2} + 6 a b + b^{2})} \\ = \frac{2 b (3 a + b)}{{(3 a + b)}^{2}} \\ = \frac{2 b}{3 a + b} \end{matrix}$
$\begin{matrix} \frac{n^{3} - n}{n^{2} - 5 n - 6} & = \frac{n (n^{2} - 1)}{(n - 6) (n + 1)} \\ = \frac{n (n - 1) (n + 1)}{(n - 6) (n + 1)} \\ = \frac{n (n - 1)}{n - 6} \end{matrix}$
$\begin{matrix} \frac{8 n^{3} + 1}{8 n^{3} - 4 n^{2} + 2 n} & = \frac{(2 n + 1) (4 n^{2} - 2 n + 1)}{2 n (4 n^{2} - 2 n + 1)} \\ = \frac{2 n + 1}{2 n} \end{matrix}$
$\begin{matrix} \frac{a^{2} - {(b - c)}^{2}}{{(a + b)}^{2} - c^{2}} & = \frac{[a - (b - c)] [a + (b - c)]}{[(a + b) + c] [(a + b) - c]} \\ = \frac{(a - b + c) (a + b - c)}{(a + b + c) (a + b - c)} \\ = \frac{a - b + c}{a + b + c} \end{matrix}$
$\begin{matrix} \frac{{(a + b)}^{2} - {(c - d)}^{2}}{{(a + c)}^{2} - {(b - d)}^{2}} & = \frac{[(a + b) + (c - d)] [(a + b) - (c - d)]}{[(a + c) + (b - d)] [(a + c) - (b - d)]} \\ = \frac{(a + b + c - d) (a + b - c + d)}{(a + c + b - d) (a + c - b + d)} \\ = \frac{a + b - c + d}{a - b + c + d} \end{matrix}$
$\begin{matrix} \frac{3 x^{3} + 9 x^{2}}{x^{2} + 6 x + 9} & = \frac{3 x^{2} (x + 3)}{{(x + 3)}^{2}} \\ = \frac{3 x^{2}}{x + 3} \end{matrix}$
$\begin{matrix} \frac{10 a^{2} (a^{3} + b^{3})}{6 a^{4} - 6 a^{3} b + 6 a^{2} b^{2}} & = \frac{a^{2} (a + b) (a^{2} - a b + b^{2})}{a^{2} (a^{2} - a b + b^{2})} \\ = \frac{5 (a + b)}{3} \end{matrix}$
$\begin{matrix} \frac{a (4 a^{2} - 8 a b)}{x (3 a^{2} - 6 a b)} & = \frac{a .4 a (a - 2 b)}{x .3 a (a - 2 b)} \\ = \frac{4 a}{3 x} \end{matrix}$
$\begin{matrix} \frac{x^{3} - 6 x^{2}}{x^{2} - 12 x + 36} & = \frac{x^{2} (x - 6)}{{(x - 6)}^{2}} \\ = \frac{x^{2}}{x - 6} \end{matrix}$
$\begin{matrix} \frac{{(x - 4 y)}^{2}}{x^{5} - 64 x^{2} y^{3}} & = \frac{{(x - 4 y)}^{2}}{x^{2} (x^{3} - 64 y^{3})} \\ = \frac{{(x - 4 y)}^{2}}{x^{2} (x - 4 y) (x^{2} + 4 x y + 16 y^{2})} \\ = \frac{x - 4 y}{x^{2} (x^{2} + 4 x y + 16 y^{2})} \end{matrix}$
$\begin{matrix} \frac{x^{3} - 3 x y^{2}}{x^{4} - 6 x^{2} y^{2} + 9 y^{4}} & = \frac{x (x^{2} - 3 y^{2})}{{(x^{2} - 3 y^{2})}^{2}} \\ = \frac{x}{x^{2} - 3 y^{2}} \end{matrix}$
$\begin{matrix} \frac{m^{3} n + 3 m^{2} n + 9 m n}{m^{3} - 27} & = \frac{m n (m^{2} + 3 m + 9)}{(m - 3) (m^{2} + 3 m + 9)} \\ = \frac{m n}{m - 3} \end{matrix}$
$\begin{matrix} \frac{x^{4} - 8 x^{2} + 15}{x^{4} - 9} & = \frac{(x^{2} - 5) (x^{2} - 3)}{(x^{2} - 3) (x^{2} + 3)} \\ = \frac{x^{2} - 5}{x^{2} + 3} \end{matrix}$
$\begin{matrix} \frac{a^{4} + 6 a^{2} - 7}{a^{4} + 8 a^{2} - 9} & = \frac{(a^{2} + 7) (a^{2} - 1)}{(a^{2} + 9) (a^{2} - 1)} \\ = \frac{a^{2} + 7}{a^{2} + 9} \end{matrix}$
$\begin{matrix} \frac{3 x^{2} + 19 x + 20}{6 x^{2} + 17 x + 12} & = \frac{3 x^{2} + 15 x + 4 x + 20}{6 x^{2} + 9 x + 8 x + 12} \\ = \frac{3 x (x + 5) + 4 (x + 5)}{3 x (2 x + 3) + 4 (2 x + 3)} \\ = \frac{(x + 5) (3 x + 4)}{(3 x + 4) (2 x + 3)} \\ = \frac{x + 5}{2 x + 3} \end{matrix}$
$\begin{matrix} \frac{4 a^{4} - 15 a^{2} - 4}{a^{2} - 8 a - 20} & = \frac{4 a^{4} - 16 a^{2} + a^{2} - 4}{(a - 10) (a + 2)} \\ = \frac{4 a^{2} (a^{2} - 4) + (a^{2} - 4)}{(a - 10) (a + 2)} \\ = \frac{(a^{2} - 4) (4 a^{2} + 1)}{(a - 10) (a + 2)} \\ = \frac{(a + 2) (a - 2) (4 a^{2} + 1)}{(a - 10) (a + 2)} \\ = \frac{(a - 2) (4 a^{2} + 1)}{a - 10} \end{matrix}$
$\begin{matrix} \frac{125 a + a^{4}}{2 a^{3} + 20 a^{2} + 50 a} & = \frac{a (125 + a^{3})}{2 a (a^{2} + 10 a + 25)} \\ = \frac{(a + 5) (a^{2} - 5 a + 25)}{2 {(a + 5)}^{2}} \\ = \frac{a^{2} - 5 a + 25}{2 (a + 5)} \end{matrix}$
$\begin{matrix} \frac{a^{2} n^{2} - 36 a^{2}}{a n^{2} + a n - 30 a} & = \frac{a^{2} (n^{2} - 36)}{a (n^{2} + n - 30)} \\ = \frac{a (n + 6) (n - 6)}{(n + 6) (n - 5)} \\ = \frac{a (n - 6)}{n - 5} \end{matrix}$
$\begin{matrix} \frac{3 m^{2} + 5 m n - 8 n^{2}}{m^{3} - n^{3}} & = \frac{3 m^{2} - 3 m n + 8 m n - 8 n^{2}}{(m - n) (m^{2} + m n + n^{2})} \\ = \frac{3 m (m - n) + 8 n (m - n)}{(m - n) (m^{2} + m n + n^{2})} \\ = \frac{(m - n) (3 m + 8 n)}{(m - n) (m^{2} + m n + n^{2})} \\ = \frac{3 m + 8 n}{m^{2} + m n + n^{2}} \end{matrix}$
$\begin{matrix} \frac{15 a^{3} b - 18 a^{2} b}{20 a^{2} b^{2} - 24 a b^{2}} & = \frac{3 a^{2} b (5 a - 6)}{4 a b^{2} (5 a - 6)} \\ = \frac{3 a}{4 b} \end{matrix}$
$\begin{matrix} \frac{9 x^{2} - 24 x + 16}{9 x^{4} - 16 x^{2}} & = \frac{{(3 x - 4)}^{2}}{x^{2} (9 x^{2} - 16)} \\ = \frac{{(3 x - 4)}^{2}}{x^{2} (3 x + 4) (3 x - 4)} \\ = \frac{3 x - 4}{x^{2} (3 x + 4)} \end{matrix}$
$\begin{matrix} \frac{16 a^{2} x - 25 x}{12 a^{3} - 7 a^{2} - 10 a} & = \frac{x (16 a^{2} - 25)}{a (12 a^{2} - 7 a - 10)} \\ = \frac{x (4 x - 5) (4 x + 5)}{a (12 a^{2} + 8 a - 15 a - 10)} \\ = \frac{x (4 x - 5) (4 x + 5)}{a [4 a (3 a + 2) - 5 (3 a + 2)]} \\ = \frac{x (4 x - 5) (4 x + 5)}{a (4 x - 5) (3 a + 2)} \\ = \frac{x (4 x + 5)}{a (3 a + 2)} \end{matrix}$
$\begin{matrix} \frac{8 x^{4} - x y^{3}}{4 x^{4} - 4 x^{3} y + x^{2} y^{2}} & = \frac{x (8 x^{3} - y^{3})}{x^{2} (4 x^{2} - 4 x y + y^{2})} \\ = \frac{(2 x - y) (4 x^{2} + 2 x y + y^{2})}{x {(2 x - y)}^{2}} \\ = \frac{4 x^{2} + 2 x y + y^{2}}{x (2 x - y)} \end{matrix}$
$\begin{matrix} \frac{3 a n - 4 a - 6 b n + 8 b}{6 n^{2} - 5 n - 4} & = \frac{a (3 n - 4) - 2 b (3 n - 4)}{6 n^{2} + 3 n - 8 n - 4} \\ = \frac{(3 n - 4) (a - 2 b)}{3 n (2 n + 1) - 4 (2 n + 1)} \\ = \frac{(3 n - 4) (a - 2 b)}{(3 n - 4) (2 n + 1)} \\ = \frac{a - 2 b}{2 n + 1} \end{matrix}$
$\begin{matrix} \frac{x^{4} - 49 x^{2}}{x^{3} + 2 x^{2} - 63 x} & = \frac{x^{2} (x^{2} - 49)}{x (x^{2} + 2 x - 63)} \\ = \frac{x (x + 7) (x - 7)}{(x + 9) (x - 7)} \\ = \frac{x (x + 7)}{x + 9} \end{matrix}$
$\begin{matrix} \frac{x^{4} + x - x^{3} y - y}{x^{3} - x - x^{2} y + y} & = \frac{x (x^{3} + 1) - y (x^{3} + 1)}{x (x^{2} - 1) - y (x^{2} - 1)} \\ = \frac{(x - y) (x^{3} + 1)}{(x - y) (x^{2} - 1)} \\ = \frac{(x + 1) (x^{2} - x + 1)}{(x + 1) (x - 1)} \\ = \frac{x^{2} - x + 1}{x - 1} \end{matrix}$
$\begin{matrix} \frac{2 x^{3} + 6 x^{2} - x - 3}{x^{3} + 3 x^{2} + x + 3} & = \frac{2 x^{2} (x + 3) - (x + 3)}{x^{2} (x + 3) + (x + 3)} \\ = \frac{(x + 3) (2 x^{2} - 1)}{(x + 3) (x^{2} + 1)} \\ = \frac{2 x^{2} - 1}{x^{2} + 1} \end{matrix}$
$\begin{matrix} \frac{a^{3} m - 4 a m + a^{3} n - 4 a n}{a^{4} - 4 a^{3} - 12 a^{2}} & = \frac{a m (a^{2} - 4) + a n (a^{2} - 4)}{a^{2} (a^{2} - 4 a - 12)} \\ = \frac{(a^{2} - 4) (a m + a n)}{a^{2} (a - 6) (a + 2)} \\ = \frac{a (a - 2) (a + 2) (m + n)}{a^{2} (a - 6) (a + 2)} \\ = \frac{(a - 2) (m + n)}{a (a - 6)} \end{matrix}$
$\begin{matrix} \frac{4 a^{2} - {(x - 3)}^{2}}{{(2 a + x)}^{2} - 9} & = \frac{[2 a - (x - 3)] [2 a + (x - 3)]}{[(2 a + x) - 3] [(2 a + x) + 3]} \\ = \frac{(2 a - x + 3) (2 a + x - 3)}{(2 a + x - 3) (2 a + x + 3)} \\ = \frac{2 a - x + 3}{2 a + x + 3} \end{matrix}$
$\begin{matrix} \frac{m - a m + n - a n}{1 - 3 a + 3 a^{2} - a^{3}} & = \frac{m (1 - a) + n (1 - a)}{(1 - a^{3}) - 3 a (1 - a)} \\ = \frac{(1 - a) (m + n)}{(1 - a) (1 + a + a^{2}) - 3 a (1 - a)} \\ = \frac{(1 - a) (m + n)}{(1 - a) [(1 + a + a^{2}) - 3 a]} \\ = \frac{m + n}{1 - 2 a + a^{2}} \\ = \frac{m + n}{{(1 - a)}^{2}} \end{matrix}$
$\begin{matrix} \frac{6 x^{2} + 3}{42 x^{5} - 9 x^{3} - 15 x} & = \frac{3 (2 x^{2} + 1)}{3 x (14 x^{4} - 3 x^{2} - 5)} \\ = \frac{2 x^{2} + 1}{x (14 x^{4} + 7 x^{2} - 10 x^{2} - 5)} \\ = \frac{2 x^{2} + 1}{x [7 x^{2} (2 x^{2} + 1) - 5 (2 x^{2} + 1)]} \\ = \frac{2 x^{2} + 1}{x (2 x^{2} + 1) (7 x^{2} - 5)} \\ = \frac{1}{x (7 x^{2} - 5)} \end{matrix}$
$\begin{matrix} \frac{a^{2} - a^{3} - 1 + a}{a^{2} + 1 - a^{3} - a} & = \frac{a^{2} (1 - a) - (1 - a)}{(a^{2} + 1) - a (a^{2} + 1)} \\ = \frac{(1 - a) (a^{2} - 1)}{(a^{2} + 1) (1 - a)} \\ = \frac{a^{2} - 1}{a^{2} + 1} \end{matrix}$
$\begin{matrix} \frac{8 x^{3} + 12 x^{2} y + 6 x y^{2} + y^{3}}{6 x^{2} + x y - y^{2}} & = \frac{(8 x^{3} + y^{3}) + 6 x y (2 x + y)}{6 x^{2} + 3 x y - 2 x y - y^{2}} \\ = \frac{(2 x + y) (4 x^{2} - 2 x y + y^{2}) + 6 x y (2 x + y)}{3 x (2 x + y) - y (2 x + y)} \\ = \frac{(2 x + y) [(4 x^{2} - 2 x y + y^{2}) + 6 x y]}{(2 x + y) (3 x - y)} \\ = \frac{4 x^{2} - 2 x y + y^{2} + 6 x y}{3 x - y} \\ = \frac{4 x^{2} + 4 x y + y^{2}}{3 x - y} \\ = \frac{{(2 x + y)}^{2}}{3 x - y} \end{matrix}$
$\begin{matrix} \frac{8 n^{3} - 125}{25 - 20 n + 4 n^{2}} & = \frac{(2 n - 5) (4 n^{2} + 10 n + 25)}{4 n^{2} - 20 n + 25} \\ = \frac{(2 n - 5) (4 n^{2} + 10 n + 25)}{{(2 n - 5)}^{2}} \\ = \frac{4 n^{2} + 10 n + 25}{2 n - 5} \end{matrix}$
$\begin{matrix} \frac{6 - x - x^{2}}{15 + 2 x - x^{2}} & = \frac{- (x^{2} + x - 6)}{- (x^{2} - 2 x - 15)} \\ = \frac{x^{2} + x - 6}{x^{2} - 2 x - 15} \\ = \frac{(x + 3) (x - 2)}{(x - 5) (x + 3)} \\ = \frac{x - 2}{x - 5} \end{matrix}$
$\begin{matrix} \frac{3 + 2 x - 8 x^{2}}{4 + 5 x - 6 x^{2}} & = \frac{- (8 x^{2} - 2 x - 3)}{- (6 x^{2} - 5 x - 4)} \\ = \frac{8 x^{2} - 2 x - 3}{6 x^{2} - 5 x - 4} \\ = \frac{8 x^{2} + 4 x - 6 x - 3}{6 x^{2} + 3 x - 8 x - 4} \\ = \frac{4 x (2 x + 1) - 3 (2 x + 1)}{3 x (2 x + 1) - 4 (2 x + 1)} \\ = \frac{(2 x + 1) (4 x - 3)}{(2 x + 1) (3 x - 4)} \\ = \frac{4 x - 3}{3 x - 4} \end{matrix}$
$\begin{matrix} \frac{m^{2} n^{2} + 3 m n - 10}{4 - 4 m n + m^{2} n^{2}} & = \frac{(m n + 5) (m n - 2)}{m^{2} n^{2} - 4 m n + 4} \\ = \frac{(m n + 5) (m n - 2)}{{(m n - 2)}^{2}} \\ = \frac{m n + 5}{m n - 2} \end{matrix}$
$\begin{matrix} \frac{x^{3} + x^{2} y - 4 b^{2} x - 4 b^{2} y}{4 b^{2} - 4 b x + x^{2}} & = \frac{x^{2} (x + y) - 4 b^{2} (x + y)}{x^{2} - 4 b x + 4 b^{2}} \\ = \frac{(x + y) (x^{2} - 4 b^{2})}{{(x - 2 b)}^{2}} \\ = \frac{(x + y) (x + 2 b) (x - 2 b)}{{(x - 2 b)}^{2}} \\ = \frac{(x + y) (x + 2 b)}{x - 2 b} \end{matrix}$
$\begin{matrix} \frac{x^{6} + x^{3} - 2}{x^{4} - x^{3} y - x + y} & = \frac{(x^{3} + 2) (x^{3} - 1)}{x^{3} (x - y) - (x - y)} \\ = \frac{(x^{3} + 2) (x^{3} - 1)}{(x - y) (x^{3} - 1)} \\ = \frac{x^{3} + 2}{x - y} \end{matrix}$
$\begin{matrix} \frac{(x^{2} - x - 2) (x^{2} - 9)}{(x^{2} - 2 x - 3) (x^{2} + x - 6)} & = \frac{(x - 2) (x + 1) (x - 3) (x + 3)}{(x - 3) (x + 1) (x + 3) (x - 2)} \\ = 1 \end{matrix}$
$\begin{matrix} \frac{(a^{2} - 4 a + 4) (4 a^{2} - 4 a + 1)}{(a^{2} + a - 6) (2 a^{2} - 5 a + 2)} & = \frac{{(a - 2)}^{2} {(2 a - 1)}^{2}}{(a + 3) (a - 2) (2 a^{2} - 4 a - a + 2)} \\ = \frac{(a - 2) {(2 a - 1)}^{2}}{(a + 3) [2 a (a - 2) - (a - 2)]} \\ = \frac{(a - 2) {(2 a - 1)}^{2}}{(a + 3) (a - 2) (2 a - 1)} \\ = \frac{2 a - 1}{a + 3} \end{matrix}$
$\begin{matrix} \frac{(x^{3} - 3 x) (x^{3} - 1)}{(x^{4} + x^{3} + x^{2}) (x^{2} - 1)} & = \frac{x (x^{2} - 3) (x - 1) (x^{2} + x + 1)}{x^{2} (x^{2} + x + 1) (x + 1) (x - 1)} \\ = \frac{x^{2} - 3}{x (x + 1)} \end{matrix}$
$\begin{matrix} \frac{(4 n^{2} + 4 n - 3) (n^{2} + 7 n - 30)}{(2 n^{2} - 7 n + 3) (4 n^{2} + 12 n + 9)} & = \frac{(4 n^{2} - 2 n + 6 n - 3) (n + 10) (n - 3)}{(2 n^{2} - n - 6 n + 3) {(2 n + 3)}^{2}} \\ = \frac{[2 n (2 n - 1) + 3 (2 n - 1)] (n + 10) (n - 3)}{[n (2 n - 1) - 3 (2 n - 1)] {(2 n + 3)}^{2}} \\ = \frac{(2 n - 1) (2 n + 3) (n + 10) (n - 3)}{(2 n - 1) (n - 3) {(2 n + 3)}^{2}} \\ = \frac{n + 10}{2 n + 3} \end{matrix}$
$\begin{matrix} \frac{(x^{6} - y^{6}) (x + y)}{(x^{3} - y^{3}) (x^{3} + x^{2} y + x y^{2} + y^{3})} & = \frac{(x^{3} - y^{3}) (x^{3} + y^{3}) (x + y)}{(x^{3} - y^{3}) [x^{2} (x + y) + y^{2} (x + y)]} \\ = \frac{(x^{3} + y^{3}) (x + y)}{(x^{2} + y^{2}) (x + y)} \\ = \frac{x^{3} + y^{3}}{x^{2} + y^{2}} \end{matrix}$
$\begin{matrix} \frac{x^{3} + 3 x^{2} - 4}{x^{3} + x^{2} - 8 x - 12} & = \frac{x^{3} + 2 x^{2} + x^{2} - 4}{x^{3} - 4 x + x^{2} - 4 x - 12} \\ = \frac{(x^{3} + 2 x^{2}) + (x^{2} - 4)}{(x^{3} - 4 x) + (x^{2} - 4 x - 12)} \\ = \frac{x^{2} (x + 2) + (x + 2) (x - 2)}{x (x^{2} - 4) + (x - 6) (x + 2)} \\ = \frac{(x + 2) [x^{2} + (x - 2)]}{x (x + 2) (x - 2) + (x - 6) (x + 2)} \\ = \frac{(x + 2) (x^{2} + x - 2)}{(x + 2) [x (x - 2) + (x - 6)]} \\ = \frac{(x + 2) (x - 1)}{x^{2} - 2 x + x - 6} \\ = \frac{(x + 2) (x - 1)}{x^{2} - x - 6} \\ = \frac{(x + 2) (x - 1)}{(x - 3) (x + 2)} \\ = \frac{x - 1}{x - 3} \end{matrix}$
$\begin{matrix} \frac{x^{3} - x^{2} - 8 x + 12}{x^{4} - 2 x^{3} - 7 x^{2} + 20 x - 12} & = \frac{(x^{3} - 4 x) - (x^{2} + 4 x - 12)}{(x^{4} - 2 x^{3}) - (7 x^{2} - 20 x + 12)} \\ = \frac{x (x^{2} - 4) - (x + 6) (x - 2)}{x^{3} (x - 2) - (7 x^{2} - 14 x - 6 x + 12)} \\ = \frac{x (x - 2) (x + 2) - (x + 6) (x - 2)}{x^{3} (x - 2) - [7 x (x - 2) - 6 (x - 2)]} \\ = \frac{(x - 2) [x (x + 2) - (x + 6)]}{x^{3} (x - 2) - (x - 2) (7 x - 6)} \\ = \frac{(x - 2) (x^{2} + 2 x - x - 6)}{(x - 2) [x^{3} - (7 x - 6)]} \\ = \frac{x^{2} + x - 6}{x^{3} - 7 x + 6} \\ = \frac{(x + 3) (x - 2)}{x^{3} - x - 6 x + 6} \\ = \frac{(x + 3) (x - 2)}{x (x^{2} - 1) - 6 (x - 1)} \\ = \frac{(x + 3) (x - 2)}{x (x - 1) (x + 1) - 6 (x - 1)} \\ = \frac{(x + 3) (x - 2)}{(x - 1) [x (x + 1) - 6]} \\ = \frac{(x + 3) (x - 2)}{(x - 1) (x^{2} + x - 6)} \\ = \frac{1}{x - 1} \end{matrix}$
$\begin{matrix} \frac{x^{4} - 7 x^{2} - 2 x + 8}{x^{4} - 2 x^{3} - 9 x^{2} + 10 x + 24} & = \frac{x^{4} - x - 7 x^{2} - x + 8}{x^{4} - 9 x^{2} - 2 x^{3} + 10 x + 24} \\ = \frac{(x^{4} - x) - (7 x^{2} + x - 8)}{(x^{4} - 9 x^{2}) - (2 x^{3} - 10 x - 24)} \\ = \frac{x (x^{3} - 1) - (7 x^{2} - 7 x + 8 x - 8)}{x^{2} (x^{2} - 9) - 2 (x^{3} - 5 x - 12) \Rightarrow a p l i c o e l m é t o d o d e l a s r a i c e s} \\ = \frac{x (x - 1) (x^{2} + x + 1) - [7 x (x - 1) + 8 (x - 1)]}{x^{2} (x - 3) (x + 3) - 2 (x - 3) (x^{2} + 3 x + 4)} \\ = \frac{x (x - 1) (x^{2} + x + 1) - (x - 1) (7 x + 8)}{(x - 3) [x^{2} (x + 3) - 2 (x^{2} + 3 x + 4)]} \\ = \frac{(x - 1) [x (x^{2} + x + 1) - (7 x + 8)]}{(x - 3) [x^{3} + 3 x^{2} - 2 x^{2} - 6 x - 8]} \\ = \frac{(x - 1) [x^{3} + x^{2} + x - 7 x - 8]}{(x - 3) [x^{3} + x^{2} - 6 x - 8]} \\ = \frac{(x - 1) (x^{3} + x^{2} - 6 x - 8)}{(x - 3) (x + 2) (x^{2} - x - 4)} \\ = \frac{(x - 1) [(x^{3} - 4 x) + (x^{2} - 2 x - 8)]}{(x - 3) (x + 2) (x^{2} - x - 4)} \\ = \frac{(x - 1) [x (x^{2} - 4) + (x - 4) (x + 2)]}{(x - 3) (x + 2) (x^{2} - x - 4)} \\ = \frac{(x - 1) [x (x + 2) (x - 2) + (x - 4) (x + 2)]}{(x - 3) (x + 2) (x^{2} - x - 4)} \\ = \frac{(x - 1) (x + 2) [x (x - 2) + (x - 4)]}{(x - 3) (x + 2) (x^{2} - x - 4)} \\ = \frac{(x - 1) (x^{2} - 2 x + x - 4)}{(x - 3) (x^{2} - x - 4)} \\ = \frac{(x - 1) (x^{2} - x - 4)}{(x - 3) (x^{2} - x - 4)} \\ = \frac{x - 1}{x - 3} \end{matrix}$
$\begin{matrix} \frac{a^{5} - a^{3} - a^{2} + 1}{a^{5} - 2 a^{4} - 6 a^{3} + 8 a^{2} + 5 a - 6} & = \frac{(a^{5} - a^{3}) - (a^{2} - 1)}{(a^{5} + 5 a - 6) - (2 a^{4} + 6 a^{3} - 8 a^{2})} \\ = \frac{a^{3} (a^{2} - 1) - (a^{2} - 1)}{(a - 1) (a^{4} + a^{3} + a^{2} + a + 6) - 2 a^{2} (a^{2} + 3 a - 4)} \\ = \frac{(a^{2} - 1) (a^{3} - 1)}{(a - 1) (a^{4} + a^{3} + a^{2} + a + 6) - 2 a^{2} (a + 4) (a - 1)} \\ = \frac{(a - 1) (a + 1) (a - 1) (a^{2} + a + 1)}{(a - 1) [(a^{4} + a^{3} + a^{2} + a + 6) - 2 a^{2} (a + 4)]} \\ = \frac{(a + 1) (a - 1) (a^{2} + a + 1)}{a^{4} + a^{3} + a^{2} + a + 6 - 2 a^{3} - 8 a^{2}} \\ = \frac{(a + 1) (a - 1) (a^{2} + a + 1)}{a^{4} - a^{3} - 7 a^{2} + a + 6} \\ = \frac{(a + 1) (a - 1) (a^{2} + a + 1)}{(a^{4} - a^{3}) - (7 a^{2} - a - 6)} \\ = \frac{(a + 1) (a - 1) (a^{2} + a + 1)}{a^{3} (a - 1) - (7 a^{2} - 7 a + 6 a - 6)} \\ = \frac{(a + 1) (a - 1) (a^{2} + a + 1)}{a^{3} (a - 1) - [7 a (a - 1) + 6 (a - 1)]} \\ = \frac{(a + 1) (a - 1) (a^{2} + a + 1)}{a^{3} (a - 1) - (a - 1) (7 a + 6)} \\ = \frac{(a + 1) (a - 1) (a^{2} + a + 1)}{(a - 1) [a^{3} - (7 a + 6)]} \\ = \frac{(a + 1) (a^{2} + a + 1)}{a^{3} - 7 a - 6} \\ = \frac{(a + 1) (a^{2} + a + 1)}{a^{3} - a - 6 a - 6} \\ = \frac{(a + 1) (a^{2} + a + 1)}{a (a^{2} - 1) - 6 (a + 1)} \\ = \frac{(a + 1) (a^{2} + a + 1)}{a (a + 1) (a - 1) - 6 (a + 1)} \\ = \frac{(a + 1) (a^{2} + a + 1)}{(a + 1) [a (a - 1) - 6]} \\ = \frac{a^{2} + a + 1}{a^{3} - a - 6} \\ = \frac{a^{2} + a + 1}{(a - 3) (a + 2)} \end{matrix}$