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DESCARGA

CAPITULO XIII

Simplificación de fracciones

Reducir una expresión mixta a fraccionaria

Ejercicio 124

Reducir a fracción:

$\begin{matrix} a + \frac{4 a}{a + 2} & = \frac{a (a + 2) + 4 a}{a + 2} \\ = \frac{a^{2} + 2 a + 4 a}{a + 2} \\ = \frac{a^{2} + 6 a}{a + 2} \\ = \frac{a (a + 6)}{a + 2} \end{matrix}$
$\begin{matrix} m - n - \frac{n^{2}}{m} & = \frac{(m - n) m - n^{2}}{m} \\ = \frac{m^{2} - m n - n^{2}}{m} \end{matrix}$
$\begin{matrix} x + 5 - \frac{3}{x - 2} & = \frac{x (x - 2) + 5 (x - 2) - 3}{x - 2} \\ = \frac{x^{2} - 2 x + 5 x - 10 - 3}{x - 2} \\ = \frac{x^{2} + 3 x - 13}{x - 2} \end{matrix}$
$\begin{matrix} a + \frac{a b}{a + b} & = \frac{a (a + b) + a b}{a + b} \\ = \frac{a [(a + b) + b]}{a + b} \\ = \frac{a [a + b + b]}{a + b} \\ = \frac{a (a + 2 b)}{a + b} \end{matrix}$
$\begin{matrix} \frac{1 - a^{2}}{a} + a - 3 & = \frac{1 - a^{2} + a^{2} - 3 a}{a} \\ = \frac{1 - 3 a}{a} \end{matrix}$
$\begin{matrix} 1 - \frac{a + x}{a - x} & = \frac{a - x - a - x}{a - x} \\ = \frac{- 2 x}{a - x} \\ = \frac{2 x}{x - a} \end{matrix}$
$\begin{matrix} \frac{2 a + x}{a + x} - 1 & = \frac{2 a + x - a - x}{a + x} \\ = \frac{a}{a + x} \end{matrix}$
$\begin{matrix} x + 2 - \frac{3}{x - 1} & = \frac{x (x - 1) + 2 (x - 1) - 3}{x - 1} \\ = \frac{x^{2} - x + 2 x - 2 - 3}{x - 1} \\ = \frac{x^{2} + x - 5}{x - 1} \end{matrix}$
$\begin{matrix} x^{2} - 3 x - \frac{x^{2} - 6 x}{x + 2} & = \frac{(x^{2} - 3 x) (x + 2) - (x^{2} - 6 x)}{x + 2} \\ = \frac{x^{3} - 3 x^{2} + 2 x^{2} - 6 x - x^{2} + 6 x}{x + 2} \\ = \frac{x^{3} - 2 x^{2}}{x + 2} \\ = \frac{x^{2} (x - 2)}{x + 2} \end{matrix}$
$\begin{matrix} x + y + \frac{x^{2} - y^{2}}{x - y} & = x + y + \frac{(x + y) (x - y)}{x - y} \\ = x + y + x + y \\ = 2 x + 2 y \\ = 2 (x + y) \end{matrix}$
$\begin{matrix} \frac{3 m n}{m - n} + m - 2 n & = \frac{3 m n + m (m - n) - 2 n (m - n)}{m - n} \\ = \frac{3 m n + m^{2} - m n - 2 m n + 2 n^{2}}{m - n} \\ = \frac{m^{2} + 2 n^{2}}{m - n} \end{matrix}$
$\begin{matrix} 2 a - 3 x - \frac{5 a x - 6 x^{2}}{a + 2 x} & = 2 a - 3 x - \frac{x (5 a - 6 x)}{a + 2 x} \\ = \frac{2 a (a + 2 x) - 3 x (a + 2 x) - x (5 a - 6 x)}{a + 2 x} \\ = \frac{2 a^{2} + 4 a x - 3 a x - 6 x^{2} - 5 a x + 6 x^{2}}{a + 2 x} \\ = \frac{2 a^{2} - 4 a x}{a + 2 x} \\ = \frac{2 a (a - 2 x)}{a + 2 x} \end{matrix}$
$\begin{matrix} m^{2} - 2 m + 4 - \frac{m^{3}}{m + 2} & = \frac{m^{2} (m + 2) - 2 m (m + 2) + 4 (m + 2) - m^{3}}{m + 2} \\ = \frac{m^{3} + 2 m^{2} - 2 m^{2} - 4 m + 4 m + 8 - m^{3}}{m + 2} \\ = \frac{8}{m + 2} \end{matrix}$
$\begin{matrix} x^{2} - 5 x - \frac{3 x (x + 2)}{x - 2} & = \frac{x^{2} (x - 2) - 5 x (x - 2) - 3 x (x + 2)}{x - 2} \\ = \frac{x^{3} - 2 x^{2} - 5 x^{2} + 10 x - 3 x^{2} - 6 x}{x - 2} \\ = \frac{x^{3} - 10 x^{2} + 4 x}{x - 2} \\ = \frac{x (x^{2} - 10 x + 4)}{x - 2} \end{matrix}$
$\begin{matrix} a^{2} + 3 a b - b^{2} + \frac{7 a b^{2} - b^{3}}{2 a - b} & = \frac{(a^{2} + 3 a b - b^{2}) (2 a - b) + 7 a b^{2} - b^{3}}{2 a - b} \\ = \frac{2 a^{3} + 6 a^{2} b - 2 a b^{2} - a^{2} b - 3 a b^{2} + b^{3} + 7 a b^{2} - b^{3}}{2 a - b} \\ = \frac{2 a^{3} + 5 a^{2} b + 2 a b^{2}}{2 a - b} \\ = \frac{a (2 a^{2} + 5 a b + 2 b^{2})}{2 a - b} \end{matrix}$
$\begin{matrix} \frac{x^{3} + 2}{x^{2} - x + 1} - (x + 1) & = \frac{x^{3} + 2 - (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1} \\ = \frac{x^{3} + 2 - (x^{3} + 1)}{x^{2} - x + 1} \\ = \frac{x^{3} + 2 - x^{3} - 1}{x^{2} - x + 1} \\ = \frac{1}{x^{2} - x + 1} \end{matrix}$
$\begin{matrix} x + 3 - \frac{x^{3} - 2 x^{2} + 1}{x^{2} - 4 x + 3} & = \frac{(x + 3) (x^{2} - 4 x + 3) - (x^{3} - 2 x^{2} + 1)}{x^{2} - 4 x + 3} \\ = \frac{x^{3} - 4 x^{2} + 3 x + 3 x^{2} - 12 x + 9 - x^{3} + 2 x^{2} - 1}{(x - 3) (x - 1)} \\ = \frac{x^{2} - 9 x + 8}{(x - 3) (x - 1)} \\ = \frac{(x - 8) (x - 1)}{(x - 3) (x - 1)} \\ = \frac{x - 8}{x - 3} \end{matrix}$
$\begin{matrix} 3 a + \frac{3 a^{2} b + 3 a b^{2}}{a^{2} - b^{2}} & = 3 a + \frac{3 a b (a + b)}{(a + b) (a - b)} \\ = 3 a + \frac{3 a b}{a - b} \\ = \frac{3 a (a - b) + 3 a b}{a - b} \\ = \frac{3 a^{2} - 3 a b + 3 a b}{a - b} \\ = \frac{3 a^{2}}{a - b} \end{matrix}$
$\begin{matrix} x - 3 - \frac{x^{3} - 27}{x^{2} - 6 x + 9} & = x - 3 - \frac{(x - 3) (x^{2} + 3 x + 9)}{{(x - 3)}^{2}} \\ = x - 3 - \frac{x^{2} + 3 x + 9}{x - 3} \\ = \frac{{(x - 3)}^{2} - (x^{2} + 3 x + 9)}{x - 3} \\ = \frac{x^{2} - 6 x + 9 - x^{2} - 3 x - 9}{x - 3} \\ = \frac{- 9 x}{x - 3} \\ = \frac{9 x}{3 - x} \end{matrix}$
$\begin{matrix} a^{2} - 3 a + 5 + \frac{2 a^{3} - 11 a + 9}{a^{2} + a - 2} & = a^{2} - 3 a + 5 + \frac{2 a^{3} - 2 a - 9 a + 9}{(a + 2) (a - 1)} \\ = a^{2} - 3 a + 5 + \frac{2 a (a^{2} - 1) - 9 (a - 1)}{(a + 2) (a - 1)} \\ = a^{2} - 3 a + 5 + \frac{2 a (a - 1) (a + 1) - 9 (a - 1)}{(a + 2) (a - 1)} \\ = a^{2} - 3 a + 5 + \frac{(a - 1) [2 a (a + 1) - 9]}{(a + 2) (a - 1)} \\ = a^{2} - 3 a + 5 + \frac{2 a^{2} + 2 a - 9}{a + 2} \\ = \frac{(a^{2} - 3 a + 5) (a + 2) + 2 a^{2} + 2 a - 9}{a + 2} \\ = \frac{a^{3} - 3 a^{2} + 5 a + 2 a^{2} - 6 a + 10 + 2 a^{2} + 2 a - 9}{a + 2} \\ = \frac{a^{3} + a^{2} + a + 1}{a + 2} \\ = \frac{a^{2} (a + 1) + (a + 1)}{a + 2} \\ = \frac{(a^{2} + 1) (a + 1)}{a + 2} \end{matrix}$