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DESCARGA

CAPITULO X

Descomposición Factorial

Ejercicio 91

Factorar o descomponer en dos factores:

$\begin{matrix} a^{2} + a b + a x + b x & = a (a + b) + x (a + b) \\ = (a + b) (a + x) \end{matrix}$
$\begin{matrix} a m - b m + a n - b n & = m (a - b) + n (a - b) \\ = (a - b) (m + n) \end{matrix}$
$\begin{matrix} a x - 2 b x - 2 a y + 4 b y & = x (a - 2 b) - 2 y (a - 2 b) \\ = (a - 2 b) (x - 2 y) \end{matrix}$
$\begin{matrix} a^{2} x^{2} - 3 b x^{2} + a^{2} y^{2} - 3 b y^{2} & = x^{2} (a^{2} - 3 b) + y^{2} (a^{2} - 3 b) \\ = (a^{2} - 3 b) (x^{2} + y^{2}) \end{matrix}$
$\begin{matrix} 3 m - 2 n - 2 n x^{4} + 3 m x^{4} & = 3 m + 3 m x^{4} - 2 n - 2 n x^{4} \\ = 3 m (1 + x^{4}) - 2 n (1 + x^{4}) \\ = (1 + x^{4}) (3 m - 2 n) \end{matrix}$
$\begin{matrix} x^{2} - a^{2} + x - a^{2} x & = x^{2} + x - a^{2} x - a^{2} \\ = x (x + 1) - a^{2} (x + 1) \\ = (x + 1) (x - a^{2}) \end{matrix}$
$\begin{matrix} 4 a^{3} - 1 - a^{2} + 4 a & = 4 a + 4 a^{3} - 1 - a^{2} \\ = 4 a (1 + a^{2}) - (1 + a^{2}) \\ = (1 + a^{2}) (4 a - 1) \end{matrix}$
$\begin{matrix} x + x^{2} - x y^{2} - y^{2} & = x + x^{2} - y^{2} - x y^{2} \\ = x (1 + x) - y^{2} (1 + x) \\ = (1 + x) (x - y^{2}) \end{matrix}$
$\begin{matrix} 3 a b x^{2} - 2 y^{2} - 2 x^{2} + 3 a b y^{2} & = 3 a b x^{2} + 3 a b y^{2} - 2 x^{2} - 2 y^{2} \\ = 3 a b (x^{2} + y^{2}) - 2 (x^{2} + y^{2}) \\ = (x^{2} + y^{2}) (3 a b - 2) \end{matrix}$
$\begin{matrix} 3 a - b^{2} + 2 b^{2} x - 6 a x & = 3 a - 6 a x - b^{2} + 2 b^{2} x \\ = 3 a (1 - 2 x) - b^{2} (1 - 2 x) \\ = (1 - 2 x) (3 a - b^{2}) \end{matrix}$
$\begin{matrix} 4 a^{3} x - 4 a^{2} b + 3 b m - 3 a m x & = 4 a^{3} x - 4 a^{2} b - 3 a m x + 3 b m \\ = 4 a^{2} (a x - b) - 3 m (a x - b) \\ = (a x - b) (4 a^{2} - 3 m) \end{matrix}$
$\begin{matrix} 6 a x + 3 a + 1 + 2 x & = 3 a + 6 a x + (1 + 2 x) \\ = 3 a (1 + 2 x) + (1 + 2 x) \\ = (1 + 2 x) (3 a + 1) \end{matrix}$
$\begin{matrix} 3 x^{3} - 9 a x^{2} - x + 3 a & = 3 x^{2} (x - 3 a) - (x - 3 a) \\ = (x - 3 a) (3 x^{2} - 1) \end{matrix}$
$\begin{matrix} 2 a^{2} x - 5 a^{2} y + 15 b y - 6 b x & = 2 a^{2} x - 6 b x - 5 a^{2} y + 15 b y \\ = 2 x (a^{2} - 3 b) - 5 y (a^{2} - 3 b) \\ = (a^{2} - 3 b) (2 x - 5 y) \end{matrix}$
$\begin{matrix} 2 x^{2} y + 2 x z^{2} + y^{2} z^{2} + x y^{3} & = 2 x^{2} y + 2 x z^{2} + x y^{3} + y^{2} z^{2} \\ = 2 x (x y + z^{2}) + y^{2} (x y + z^{2}) \\ = (x y + z^{2}) (2 x + y^{2}) \end{matrix}$
$\begin{matrix} 6 m - 9 n + 21 n x - 14 m x & = 6 m - 9 n - 14 m x + 21 n x \\ = 3 (2 m - 3 n) - 7 x (2 m - 3 n) \\ = (2 m - 3 n) (3 - 7 x) \end{matrix}$
$\begin{matrix} n^{2} x - 5 a^{2} y^{2} - n^{2} y^{2} + 5 a^{2} x & = n^{2} x - n^{2} y^{2} + 5 a^{2} x - 5 a^{2} y^{2} \\ = n^{2} (x - y^{2}) + 5 a^{2} (x - y^{2}) \\ = (x - y^{2}) (n^{2} + 5 a^{2}) \end{matrix}$
$\begin{matrix} 1 + a + 3 a b + 3 b & = 1 + a + 3 b + 3 a b \\ = (1 + a) + 3 b (1 + a) \\ = (1 + a) (1 + 3 b) \end{matrix}$
$\begin{matrix} 4 a m^{3} - 12 a m n - m^{2} + 3 n & = 4 a m (m^{2} - 3 n) - (m^{2} - 3 n) \\ = (m^{2} - 3 n) (4 a m - 1) \end{matrix}$
$\begin{matrix} 20 a x - 5 b x - 2 b y + 8 a y & = 20 a x - 5 b x + 8 a y - 2 b y \\ = 5 x (4 a - b) + 2 y (4 a - b) \\ = (4 a - b) (5 x + 2 y) \end{matrix}$
$\begin{matrix} 3 - x^{2} + 2 a b x^{2} - 6 a b & = 3 - x^{2} - 6 a b + 2 a b x^{2} \\ = (3 - x^{2}) - 2 a b (3 - x^{2}) \\ = (3 - x^{2}) (1 - 2 a b) \end{matrix}$
$\begin{matrix} a^{3} + a^{2} + a + 1 & = a^{2} (a + 1) + (a + 1) \\ = (a + 1) (a^{2} + 1) \end{matrix}$
$\begin{matrix} 3 a^{2} - 7 b^{2} x + 3 a x - 7 a b^{2} & = 3 a x + 3 a^{2} - 7 b^{2} x - 7 a b^{2} \\ = 3 a (x + a) - 7 b^{2} (x + a) \\ = (x + a) (3 a - 7 b^{2}) \end{matrix}$
$\begin{matrix} 2 a m - 2 a n + 2 a - m + n - 1 & = 2 a (m - n + 1) - (m - n + 1) \\ = (m - n + 1) (2 a - 1) \end{matrix}$
$\begin{matrix} 3 a x - 2 b y - 2 b x - 6 a + 3 a y + 4 b & = 3 a x + 3 a y - 6 a - 2 b x - 2 b y + 4 b \\ = 3 a (x + y - 2) - 2 b (x + y - 2) \\ = (x + y - 2) (3 a - 2 b) \end{matrix}$
$\begin{matrix} a^{3} + a + a^{2} + 1 + x^{2} + a^{2} x^{2} & = a^{3} + a + a^{2} + 1 + a^{2} x^{2} + x^{2} \\ = a (a^{2} + 1) + (a^{2} + 1) + x^{2} (a^{2} + 1) \\ = (a^{2} + 1) (a + 1 + x^{2}) \end{matrix}$
$\begin{matrix} 3 a^{3} - 3 a^{2} b + 9 a b^{2} - a^{2} + a b - 3 b^{2} & = 3 a (a^{2} - a b + 3 b^{2}) - (a^{2} - a b + 3 b^{2}) \\ = (a^{2} - a b + 3 b^{2}) (3 a - 1) \end{matrix}$
$\begin{matrix} 2 x^{3} - n x^{2} + 2 x z^{2} - n z^{2} - 3 n y^{2} + 6 x y^{2} & = 2 x^{3} + 6 x y^{2} + 2 x z^{2} - n x^{2} - 3 n y^{2} - n z^{2} \\ = 2 x (x^{2} + 3 y^{2} + z^{2}) - n (x^{2} + 3 y^{2} + z^{2}) \\ = (x^{2} + 3 y^{2} + z^{2}) (2 x - n) \end{matrix}$
$\begin{matrix} 3 x^{3} + 2 a x y + 2 a y^{2} - 3 x y^{2} - 2 a x^{2} - 3 x^{2} y & = 2 a x y + 2 a y^{2} - 2 a x^{2} - 3 x^{2} y - 3 x y^{2} + 3 x^{3} \\ = 2 a (x y + y^{2} - x^{2}) - 3 x (x y + y^{2} - x^{2}) \\ = (x y + y^{2} - x^{2}) (2 a - 3 x) \end{matrix}$
$\begin{matrix} a^{2} b^{3} - n^{4} + a^{2} b^{3} x^{2} - n^{4} x^{2} - 3 a^{2} b^{3} x + 3 n^{4} x & = a^{2} b^{3} - 3 a^{2} b^{3} x + a^{2} b^{3} x^{2} - n^{4} + 3 n^{4} x - n^{4} x^{2} \\ = a^{2} b^{3} (1 - 3 x + x^{2}) - n^{4} (1 - 3 x + x^{2}) \\ = (1 - 3 x + x^{2}) (a^{2} b^{3} - n^{4}) \end{matrix}$