Ejercicio 104 – Solucionario Baldor

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DESCARGA

CAPITULO X

Descomposición Factorial

Ejercicio 104

Descomponer en 2 factores:

$\begin{matrix} 1 + {(x + y)}^{3} & = [1 + (x + y)] [1 - (x + y) + {(x + y)}^{2}] \\ = (x + y + 1) (1 - x - y + x^{2} + 2 x y + y^{2}) \end{matrix}$
$\begin{matrix} 1 - {(a + b)}^{3} & = [1 - (a + b)] [1 + (a + b) + {(a + b)}^{2}] \\ = (1 - a - b) (1 + a + b + a^{2} + 2 a b + b^{2}) \end{matrix}$
$\begin{matrix} 27 + {(m - n)}^{3} & = [3 + (m - n)] [3^{2} - 3 (m - n) + {(m - n)}^{2}] \\ = (3 + m - n) (9 - 3 m + 3 n + m^{2} - 2 m n + n^{2}) \end{matrix}$
$\begin{matrix} {(x - y)}^{3} - 8 & = [(x - y) - 2] [{(x - y)}^{2} + 2 (x - y) + 4] \\ = (x - y - 2) (x^{2} - 2 x y + y^{2} + 2 x - 2 y + 4) \end{matrix}$
$\begin{matrix} {(x + 2 y)}^{3} + 1 & = [(x + 2 y) + 1] [{(x + 2 y)}^{2} - (x + 2 y) + 1] \\ = (x + 2 y + 1) (x^{2} + 4 x y + 4 y^{2} - x - 2 y + 1) \end{matrix}$
$\begin{matrix} 1 - {(2 a - b)}^{3} & = [1 - (2 a - b)] [1 + (2 a - b) + {(2 a - b)}^{2}] \\ = (1 - 2 a + b) (1 + 2 a - b + 4 a^{2} - 4 a b + b^{2}) \end{matrix}$
$\begin{matrix} a^{3} + {(a + 1)}^{3} & = [a + (a + 1)] [a^{2} - a (a + 1) + {(a + 1)}^{2}] \\ = (2 a + 1) (a^{2} - a^{2} - a + a^{2} + 2 a + 1) \\ = (2 a + 1) (a^{2} - a + 1) \end{matrix}$
$\begin{matrix} 8 a^{3} - {(a - 1)}^{3} & = [2 a - (a - 1)] [{(2 a)}^{2} + 2 a (a - 1) + {(a - 1)}^{2}] \\ = (2 a - a + 1) (4 a^{2} + 2 a^{2} - 2 a + a^{2} - 2 a + 1) \\ = (a + 1) (7 a^{2} - 4 a + 1) \end{matrix}$
$\begin{matrix} 27 x^{3} - {(x - y)}^{3} & = [3 x - (x - y)] [{(3 x)}^{2} + 3 x (x - y) + {(x - y)}^{2}] \\ = (3 x - x + y) (9 x^{2} + 3 x^{2} - 3 x y + x^{2} - 2 x y + y^{2}) \\ = (2 x + y) (13 x^{2} - 5 x y + y^{2}) \end{matrix}$
$\begin{matrix} {(2 a - b)}^{3} - 27 & = [(2 a - b) - 3] [{(2 a - b)}^{2} + 3 (2 a - b) + 3^{2}] \\ = (2 a - b - 3) (4 a^{2} - 4 a b + b^{2} + 6 a - 3 b + 9) \end{matrix}$
$\begin{matrix} x^{6} - {(x + 2)}^{3} & = [x^{2} - (x + 2)] [{(x^{2})}^{2} + x^{2} (x + 2) + {(x + 2)}^{2}] \\ = (x^{2} - x - 2) (x^{4} + x^{4} + 2 x^{2} + x^{2} + 4 x + 4) \\ = (x^{2} - x - 2) (2 x^{4} + 3 x^{2} + 4 x + 4) \end{matrix}$
$\begin{matrix} {(a + 1)}^{3} + {(a - 3)}^{3} & = [(a + 1) + (a - 3)] [{(a + 1)}^{2} - (a + 1) (a - 3) + {(a - 3)}^{2}] \\ = (a + 1 + a - 3) [a^{2} + 2 a + 1 - (a^{2} - 2 a - 3) + a^{2} - 6 a + 9] \\ = (2 a - 2) [2 a^{2} - 4 a + 10 - a^{2} + 2 a + 3] \\ = 2 (a - 1) (a^{2} - 2 a + 13) \end{matrix}$
$\begin{matrix} {(x - 1)}^{3} - {(x + 2)}^{3} & = [(x - 1) - (x + 2)] [{(x - 1)}^{2} + (x - 1) (x + 2) + {(x + 2)}^{2}] \\ = [x - 1 - x - 2] [x^{2} - 2 x + 1 + x^{2} + x - 2 + x^{2} + 4 x + 4] \\ = - 3 (3 x^{2} + 3 x + 3) \\ = - 9 (x^{2} + x + 1) \end{matrix}$
$\begin{matrix} {(x - y)}^{3} - {(x + y)}^{3} & = [(x - y) - (x + y)] [{(x - y)}^{2} + (x - y) (x + y) + {(x + y)}^{2}] \\ = [x - y - x - y] [x^{2} - 2 x y + y^{2} + x^{2} - y^{2} + x^{2} + 2 x y + y^{2}] \\ = - 2 y (3 x^{2} + y^{2}) \end{matrix}$
$\begin{matrix} {(m - 2)}^{3} + {(m - 3)}^{3} & = [(m - 2) + (m - 3)] [{(m - 2)}^{2} - (m - 2) (m - 3) + {(m - 3)}^{2}] \\ = [m - 2 + m - 3] [m^{2} - 4 m + 4 - (m^{2} - 5 m + 6) + m^{2} - 6 m + 9] \\ = (2 m - 5) [2 m^{2} - 10 m + 13 - m^{2} + 5 m - 6] \\ = (2 m - 5) (m^{2} - 5 m + 7) \end{matrix}$
$\begin{matrix} {(2 x - y)}^{3} + {(3 x + y)}^{3} & = [(2 x - y) + (3 x + y)] [{(2 x - y)}^{2} - (2 x - y) (3 x + y) + {(3 x + y)}^{2}] \\ = [2 x - y + 3 x + y] [4 x^{2} - 4 x y + y^{2} - (6 x^{2} - x y - y^{2}) + 9 x^{2} + 6 x y + y^{2}] \\ = 5 x [13 x^{2} + 2 x y + 2 y^{2} - 6 x^{2} + x y + y^{2}] \\ = 5 x (7 x^{2} + 3 x y + 3 y^{2}) \end{matrix}$
$\begin{matrix} 8 {(a + b)}^{3} + {(a - b)}^{3} & = [2 (a + b) + (a - b)] {{[2 (a + b)]}^{2} - 2 (a + b) (a - b) + {(a - b)}^{2}} \\ = [2 a + 2 b + a - b] {4 {(a + b)}^{2} - 2 (a^{2} - b^{2}) + a^{2} - 2 a b + b^{2}} \\ = (3 a + b) {4 (a^{2} + 2 a b + b^{2}) - 2 a^{2} + 2 b^{2} + a^{2} - 2 a b + b^{2}} \\ = (3 a + b) {4 a^{2} + 8 a b + 4 b^{2} - a^{2} + 3 b^{2} - 2 a b} \\ = (3 a + b) (3 a^{2} + 6 a b + 7 b^{2}) \end{matrix}$
$\begin{matrix} 64 {(m + n)}^{3} - 125 & = [4 (m + n) - 5] {{[4 (m + n)]}^{2} + 20 (m + n) + 5^{2}} \\ = [4 m + 4 n - 5] {16 {(m + n)}^{2} + 20 m + 20 n + 25} \\ = (4 m + 4 n - 5) {16 (m^{2} + 2 m n + n^{2}) + 20 m + 20 n + 25} \\ = (4 m + 4 n - 5) (16 m^{2} + 32 m n + 16 n^{2} + 20 m + 20 n + 25) \end{matrix}$