Ejercicio 117 – Solucionario Baldor

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CAPITULO XII

Mínimo Común Múltiplo

Ejercicio 117

Hallar el m.c.m de:

$\begin{matrix} 3 x + 3, 6 x - 6 \\ 3 x + 3 = 3 (x + 1) \\ 6 x - 6 = 6 (x - 1) = 2.3 (x - 1) \\ m . c . m = 2.3 (x - 1) (x + 1) \\ m . c . m = 6 (x - 1) (x + 1) \end{matrix}$
$\begin{matrix} 5 x + 10, 10 x^{2} - 40 \\ 5 x + 10 = 5 (x + 2) \\ 10 x^{2} - 40 = 10 (x^{2} - 4) = 2.5 (x + 2) (x - 2) \\ m . c . m = 2.5 (x + 2) (x - 2) \\ m . c . m = 10 (x + 2) (x - 2) \end{matrix}$
$\begin{matrix} x^{3} + 2 x^{2} y, x^{2} - 4 y^{2} \\ x^{3} + 2 x^{2} y = x^{2} (x + 2 y) \\ x^{2} - 4 y^{2} = (x - 2 y) (x + 2 y) \\ m . c . m = x^{2} (x + 2 y) (x - 2 y) \end{matrix}$
$\begin{matrix} 3 a^{2} x - 9 a^{2}, x^{2} - 6 x + 9 \\ 3 a^{2} x - 9 a^{2} = 3 a^{2} (x - 3) \\ x^{2} - 6 x + 9 = {(x - 3)}^{2} \\ m . c . m = 3 a^{2} {(x - 3)}^{2} \end{matrix}$
$\begin{matrix} 4 a^{2} - 9 b^{2}, 4 a^{2} - 12 a b + 9 b^{2} \\ 4 a^{2} - 9 b^{2} = (2 a - 3 b) (2 a + 3 b) \\ 4 a^{2} - 12 a b + 9 b^{2} = {(2 a - 3 b)}^{2} \\ m . c . m = {(2 a - 3 b)}^{2} (2 a + 3 b) \end{matrix}$
$\begin{matrix} a^{3} + a^{2} b, a^{3} + 2 a^{2} b + a b^{2} \\ a^{3} + a^{2} b = a^{2} (a + b) \\ a^{3} + 2 a^{2} b + a b^{2} = a (a^{2} + 2 a b + b^{2}) = a {(a + b)}^{2} \\ m . c . m = a^{2} {(a + b)}^{2} \end{matrix}$
$\begin{matrix} 3 a x + 12 a, 2 b x^{2} + 6 b x - 8 b \\ 3 a x + 12 a = 3 a (x + 4) \\ 2 b x^{2} + 6 b x - 8 b = 2 b (x^{2} + 3 x - 4) = 2 b (x + 4) (x - 1) \\ m . c . m = 2.3 a b (x + 4) (x - 1) \\ m . c . m = 6 a b (x + 4) (x - 1) \end{matrix}$
$\begin{matrix} x^{3} - 25 x, x^{2} + 2 x - 15 \\ x^{3} - 25 x = x (x^{2} - 25) = x (x - 5) (x + 5) \\ x^{2} + 2 x - 15 = (x + 5) (x - 3) \\ m . c . m = x (x - 5) (x + 5) (x - 3) \end{matrix}$
$\begin{matrix} {(x - 1)}^{2}, x^{2} - 1 \\ {(x - 1)}^{2} \\ x^{2} - 1 = (x + 1) (x - 1) \\ m . c . m = (x + 1) {(x - 1)}^{2} \end{matrix}$
$\begin{matrix} {(x + 1)}^{2}, x^{2} + 1 \\ {(x + 1)}^{2} \\ x^{2} + 1 \\ m . c . m = (x^{2} + 1) {(x + 1)}^{2} \end{matrix}$
$\begin{matrix} x^{3} + y^{3}, {(x + y)}^{3} \\ x^{3} + y^{3} = (x + y) (x^{2} - x y + y^{2}) \\ {(x + y)}^{3} \\ m . c . m = {(x + y)}^{3} (x^{2} - x y + y^{2}) \end{matrix}$
$\begin{matrix} x^{3} - y^{3}, {(x - y)}^{3} \\ x^{3} - y^{3} = (x - y) (x^{2} + x y + y^{2}) \\ {(x - y)}^{3} \\ m . c . m = {(x - y)}^{3} (x^{2} + x y + y^{2}) \end{matrix}$
$\begin{matrix} x^{2} + 3 x - 10, 4 x^{2} - 7 x - 2 \\ x^{2} + 3 x - 10 = (x + 5) (x - 2) \\ 4 x^{2} - 7 x - 2 = 4 x^{2} - 8 x + x - 2 = 4 x (x - 2) + (x - 2) = (4 x + 1) (x - 2) \\ m . c . m = (4 x + 1) (x - 2) (x + 5) \end{matrix}$
$\begin{matrix} a^{2} + a - 30, a^{2} + 3 a - 18 \\ a^{2} + a - 30 = (a + 6) (a - 5) \\ a^{2} + 3 a - 18 = (a + 6) (a - 3) \\ m . c . m = (a + 6) (a - 5) (a - 3) \end{matrix}$
$\begin{matrix} x^{3} - 9 x + 5 x^{2} - 45, x^{4} + 2 x^{3} - 15 x^{2} \\ x^{3} - 9 x + 5 x^{2} - 45 = x (x^{2} - 9) + 5 (x^{2} - 9) = (x^{2} - 9) (x + 5) = (x - 3) (x + 3) (x + 5) \\ x^{4} + 2 x^{3} - 15 x^{2} = x^{2} (x^{2} + 2 x - 15) = (x + 5) (x - 3) \\ m . c . m = (x - 3) (x + 3) (x + 5) \end{matrix}$
$\begin{matrix} x^{6} - 4 x^{3} - 32, a x^{4} + 2 a x^{3} + 4 a x^{2} \\ x^{6} - 4 x^{3} - 32 = x^{6} - 8 x^{3} + 4 x^{3} - 32 = x^{3} (x^{3} - 8) + 4 (x^{3} - 8) = (x^{3} - 8) (x^{3} + 4) = (x - 2) (x^{2} + 2 x + 4) (x^{3} + 4) \\ a x^{4} + 2 a x^{3} + 4 a x^{2} = a x^{2} (x^{2} + 2 x + 4) \\ m . c . m = a x^{2} (x - 2) (x^{2} + 2 x + 4) (x^{3} + 4) \end{matrix}$
$\begin{matrix} 8 {(x - y)}^{2}, 12 (x^{2} - y^{2}) \\ 8 {(x - y)}^{2} = 2^{3} {(x - y)}^{2} \\ 12 (x^{2} - y^{2}) = 2^{2} .3 (x + y) (x - y) \\ m . c . m = 2^{3} .3 (x + y) {(x - y)}^{2} \\ m . c . m = 24 (x + y) {(x - y)}^{2} \end{matrix}$
$\begin{matrix} 5 {(x + y)}^{2}, 10 (x^{2} + y^{2}) \\ 5 {(x + y)}^{2} \\ 10 (x^{2} + y^{2}) = 2.5 (x^{2} + y^{2}) \\ m . c . m = 2.5 (x^{2} + y^{2}) {(x + y)}^{2} \\ m . c . m = 10 (x^{2} + y^{2}) {(x + y)}^{2} \end{matrix}$
$\begin{matrix} 6 a {(m + n)}^{3}, 4 a^{2} b (m^{3} + n^{3}) \\ 6 a {(m + n)}^{3} = 2.3 a {(m + n)}^{3} \\ 4 a^{2} b (m^{3} + n^{3}) = 2^{2} a^{2} b (m^{3} + n^{3}) = 2^{2} a^{2} b (m + n) (m^{2} - m n + n^{2}) \\ m . c . m = 2^{2} .3 a^{2} b {(m + n)}^{3} (m^{2} - m n + n^{2}) \\ m . c . m = 12 a^{2} b {(m + n)}^{3} (m^{2} - m n + n^{2}) \end{matrix}$
$\begin{matrix} a x {(m - n)}^{3}, x^{3} (m^{3} - n^{3}) \\ a x {(m - n)}^{3} \\ x^{3} (m^{3} - n^{3}) = x^{3} (m - n) (m^{2} + m n + n^{2}) \\ m . c . m = a x^{3} {(m - n)}^{3} (m^{2} + m n + n^{2}) \end{matrix}$
$\begin{matrix} 2 a^{2} + 2 a, 3 a^{2} - 3 a, a^{4} - a^{2} \\ 2 a^{2} + 2 a = 2 a (a + 1) \\ 3 a^{2} - 3 a = 3 a (a - 1) \\ a^{4} - a^{2} = a^{2} (a^{2} - 1) = a^{2} (a - 1) (a + 1) \\ m . c . m = 2.3 a^{2} (a - 1) (a + 1) \\ m . c . m = 6 a^{2} (a - 1) (a + 1) \end{matrix}$
$\begin{matrix} x^{2} + 2 x, x^{3} - 2 x^{2}, x^{2} - 4 \\ x^{2} + 2 x = x (x + 2) \\ x^{3} - 2 x^{2} = x^{2} (x - 2) \\ x^{2} - 4 = (x - 2) (x + 2) \\ m . c . m = x^{2} (x - 2) (x + 2) \end{matrix}$
$\begin{matrix} x^{2} + x - 2, x^{2} - 4 x + 3, x^{2} - x - 6 \\ x^{2} + x - 2 = (x + 2) (x - 1) \\ x^{2} - 4 x + 3 = (x - 3) (x - 1) \\ x^{2} - x - 6 = (x - 3) (x + 2) \\ m . c . m = (x - 3) (x + 2) (x - 1) \end{matrix}$
$\begin{matrix} 6 a^{2} + 13 a + 6, 3 a^{2} + 14 a + 8, 4 + 12 a + 9 a^{2} \\ 6 a^{2} + 13 a + 6 = 6 a^{2} + 9 a + 4 a + 6 = 3 a (2 a + 3) + 2 (2 a + 3) = (2 a + 3) (3 a + 2) \\ 3 a^{2} + 14 a + 8 = 3 a^{2} + 12 a + 2 a + 8 = 3 a (a + 4) + 2 (a + 4) = (a + 4) (3 a + 2) \\ 4 + 12 a + 9 a^{2} = {(2 + 3 a)}^{2} = {(3 a + 2)}^{2} \\ m . c . m = {(3 a + 2)}^{2} (a + 4) (2 a + 3) \end{matrix}$
$\begin{matrix} 10 x^{2} + 10, 15 x + 15, 5 x^{2} - 5 \\ 10 x^{2} + 10 = 10 (x^{2} + 1) = 2.5 (x^{2} + 1) \\ 15 x + 15 = 15 (x + 1) = 3.5 (x + 1) \\ 5 x^{2} - 5 = 5 (x^{2} - 1) = 5 (x - 1) (x + 1) \\ m . c . m = 2.3.5 (x^{2} + 1) (x - 1) (x + 1) \\ m . c . m = 30 (x^{2} + 1) (x - 1) (x + 1) \end{matrix}$
$\begin{matrix} a x - 2 b x + a y - 2 b y, x^{2} + x y, x^{2} - x y \\ a x - 2 b x + a y - 2 b y = x (a - 2 b) + y (a - 2 b) = (a - 2 b) (x + y) \\ x^{2} + x y = x (x + y) \\ x^{2} - x y = x (x - y) \\ m . c . m = x (x + y) (x - y) (a - 2 b) \end{matrix}$
$\begin{matrix} 4 a^{2} b + 4 a b^{2}, 6 a - 6 b, 15 a^{2} - 15 b^{2} \\ 4 a^{2} b + 4 a b^{2} = 4 a b (a + b) = 2^{2} a b (a + b) \\ 6 a - 6 b = 6 (a - b) = 2.3 (a - b) \\ 15 a^{2} - 15 b^{2} = 15 (a^{2} - b^{2}) = 3.5 (a - b) (a + b) \\ m . c . m = 2^{2} .3.5 a b (a + b) (a - b) \\ m . c . m = 60 a b (a + b) (a - b) \end{matrix}$
$\begin{matrix} x^{2} - 25, x^{3} - 125, 2 x + 10 \\ x^{2} - 25 = (x - 5) (x + 5) \\ x^{3} - 125 = (x - 5) (x^{2} + 5 x + 25) \\ 2 x + 10 = 2 (x + 5) \\ m . c . m = 2 (x + 5) (x - 5) (x^{2} + 5 x + 25) \end{matrix}$
$\begin{matrix} a^{2} - 2 a b - 3 b^{2}, a^{3} b - 6 a^{2} b^{2} + 9 a b^{3}, a b^{2} + b^{3} \\ a^{2} - 2 a b - 3 b^{2} = a^{2} + a b - 3 a b - 3 b^{2} = a (a + b) - 3 b (a + b) = (a + b) (a - 3 b) \\ a^{3} b - 6 a^{2} b^{2} + 9 a b^{3} = a b (a^{2} - 6 a b + 9 b^{2}) = a b {(a - 3 b)}^{2} \\ a b^{2} + b^{3} = b^{2} (a + b) \\ m . c . m = a b^{2} (a + b) {(a - 3 b)}^{2} \end{matrix}$
$\begin{matrix} 2 m^{2} + 2 m n, 4 m n - 4 n^{2}, 6 m^{3} n - 6 m n^{3} \\ 2 m^{2} + 2 m n = 2 m (m + n) \\ 4 m n - 4 n^{2} = 4 n (m - n) = 2^{2} n (m - n) \\ 6 m^{3} n - 6 m n^{3} = 6 m n (m^{2} - n^{2}) = 2.3 m n (m - n) (m + n) \\ m . c . m = 2^{2} .3 m n (m - n) (m + n) \\ m . c . m = 12 m n (m - n) (m + n) \end{matrix}$
$\begin{matrix} 20 (x^{2} - y^{2}), 15 {(x - y)}^{2}, 12 {(x + y)}^{2} \\ 20 (x^{2} - y^{2}) = 2^{2} .5 (x - y) (x + y) \\ 15 {(x - y)}^{2} = 3.5 {(x - y)}^{2} \\ 12 {(x + y)}^{2} = 2^{2} .3 {(x + y)}^{2} \\ m . c . m = 2^{2} .3.5 {(x + y)}^{2} {(x - y)}^{2} \\ m . c . m = 60 {(x + y)}^{2} {(x - y)}^{2} \end{matrix}$
$\begin{matrix} a x^{2} + 5 a x - 14 a, x^{3} + 14 x^{2} + 49 x, x^{4} + 7 x^{3} - 18 x^{2} \\ a x^{2} + 5 a x - 14 a = a (x^{2} + 5 x - 14) = a (x + 7) (x - 2) \\ x^{3} + 14 x^{2} + 49 x = x (x^{2} + 14 x + 49) = x {(x + 7)}^{2} \\ x^{4} + 7 x^{3} - 18 x^{2} = x^{2} (x^{2} + 7 x - 18) = x^{2} (x + 9) (x - 2) \\ m . c . m = a x^{2} (x + 9) (x - 2) {(x + 7)}^{2} \end{matrix}$
$\begin{matrix} 2 x^{3} - 12 x^{2} + 18 x, 3 x^{4} - 27 x^{2}, 5 x^{3} + 30 x^{2} + 45 x \\ 2 x^{3} - 12 x^{2} + 18 x = 2 x (x^{2} - 6 x + 9) = 2 x {(x - 3)}^{2} \\ 3 x^{4} - 27 x^{2} = 3 x^{2} (x^{2} - 9) = 3 x^{2} (x + 3) (x - 3) \\ 5 x^{3} + 30 x^{2} + 45 x = 5 x (x^{2} + 6 x + 9) = 5 x {(x + 3)}^{2} \\ m . c . m = 2.3.5 x^{2} {(x + 3)}^{2} {(x - 3)}^{2} \\ m . c . m = 30 x^{2} {(x + 3)}^{2} {(x - 3)}^{2} \end{matrix}$
$\begin{matrix} 3 - 3 a^{2}, 6 + 6 a, 9 - 9 a, 12 + 12 a^{2} \\ 3 - 3 a^{2} = 3 (1 - a^{2}) = 3 (1 - a) (1 + a) \\ 6 + 6 a = 6 (1 + a) = 2.3 (1 + a) \\ 12 + 12 a^{2} = 12 (1 + a^{2}) = 2^{2} .3 (1 + a^{2}) \\ m . c . m = 2^{2} .3 (1 + a^{2}) (1 - a) (1 + a) \\ m . c . m = 12 (1 + a^{2}) (1 - a) (1 + a) \end{matrix}$
$\begin{matrix} 2 {(3 n - 2)}^{2}, 135 n^{3} - 40, 12 n - 8 \\ 2 {(3 n - 2)}^{2} \\ 135 n^{3} - 40 = 5 (27 n^{3} - 8) = 5 (3 n - 2) (9 n^{2} + 6 n + 4) \\ 12 n - 8 = 4 (3 n - 2) = 2^{2} (3 n - 2) \\ m . c . m = 2^{2} .5 {(3 n - 2)}^{2} (9 n^{2} + 6 n + 4) \\ m . c . m = 20 {(3 n - 2)}^{2} (9 n^{2} + 6 n + 4) \end{matrix}$
$\begin{matrix} 12 m n + 8 m - 3 n - 2, 48 m^{2} n - 3 n + 32 m^{2} - 2, 6 n^{2} - 5 n - 6 \\ 12 m n + 8 m - 3 n - 2 = 4 m (3 n + 2) - (3 n + 2) = (3 n + 2) (4 m - 1) \\ 48 m^{2} n - 3 n + 32 m^{2} - 2 = 3 n (16 m^{2} - 1) + 2 (16 m^{2} - 1) = (16 m^{2} - 1) (3 n + 2) = (4 m - 1) (4 m + 1) (3 n + 2) \\ 6 n^{2} - 5 n - 6 = 6 n^{2} - 9 n + 4 n - 6 = 3 n (2 n - 3) + 2 (2 n - 3) = (2 n - 3) (3 n + 2) \\ m . c . m = (4 m - 1) (4 m + 1) (3 n + 2) (2 n - 3) \end{matrix}$
$\begin{matrix} 18 x^{3} + 60 x^{2} + 50 x, 12 a x^{3} + 20 a x^{2}, 15 a^{2} x^{5} + 16 a^{2} x^{4} - 15 a^{2} x^{3} \\ 18 x^{3} + 60 x^{2} + 50 x = 2 x (9 x^{2} + 30 x + 25) = 2 x {(3 x + 5)}^{2} \\ 12 a x^{3} + 20 a x^{2} = 4 a x^{2} (3 x + 5) = 2^{2} a x^{2} (3 x + 5) \\ 15 a^{2} x^{5} + 16 a^{2} x^{4} - 15 a^{2} x^{3} = a^{2} x^{3} (15 x^{2} + 16 x - 15) = a^{2} x^{3} (15 x^{2} - 9 x + 25 x - 15) = a^{2} x^{3} [3 x (5 x - 3) + 5 (5 x - 3)] = a^{2} x^{3} (5 x - 3) (3 x + 5) \\ m . c . m = 2^{2} a^{2} x^{3} (5 x - 3) {(3 x + 5)}^{2} \\ m . c . m = 4 a^{2} x^{3} (5 x - 3) {(3 x + 5)}^{2} \end{matrix}$
$\begin{matrix} 16 - x^{4}, 16 + 8 x^{2} + x^{4}, 16 - 8 x^{2} + x^{4} \\ 16 - x^{4} = (4 + x^{2}) (4 - x^{2}) = (4 + x^{2}) (2 - x) (2 + x) \\ 16 + 8 x^{2} + x^{4} = {(4 + x^{2})}^{2} \\ 16 - 8 x^{2} + x^{4} = {(4 - x^{2})}^{2} = {[(2 - x) (2 + x)]}^{2} = {(2 - x)}^{2} {(2 + x)}^{2} \\ m . c . m = {(4 + x^{2})}^{2} {(2 - x)}^{2} {(2 + x)}^{2} \end{matrix}$
$\begin{matrix} 1 + a^{2}, {(1 + a)}^{2}, 1 + a^{3} \\ 1 + a^{2} \\ {(1 + a)}^{2} \\ 1 + a^{3} = (1 + a) (1 - a + a^{2}) \\ m . c . m = (1 + a^{2}) {(1 + a)}^{2} (1 - a + a^{2}) \end{matrix}$
$\begin{matrix} 8 n^{2} - 10 n - 3, 20 n^{2} + 13 n + 2, 10 n^{2} - 11 n - 6 \\ 8 n^{2} - 10 n - 3 = 8 n^{2} - 12 n + 2 n - 3 = 4 n (2 n - 3) + (2 n - 3) = (2 n - 3) (4 n + 1) \\ 20 n^{2} + 13 n + 2 = 20 n^{2} + 5 n + 8 n + 2 = 5 n (4 n + 1) + 2 (4 n + 1) = (4 n + 1) (5 n + 2) \\ 10 n^{2} - 11 n - 6 = 10 n^{2} - 15 n + 4 n - 6 = 5 n (2 n - 3) + 2 (2 n - 3) = (2 n - 3) (5 n + 2) \\ m . c . m = (2 n - 3) (4 n + 1) (5 n + 2) \end{matrix}$
$\begin{matrix} 6 a^{2} + a b - 2 b^{2}, 15 a^{2} + 22 a b + 8 b^{2}, 10 a^{2} + 3 a b - 4 b^{2} \\ 6 a^{2} + a b - 2 b^{2} = 6 a^{2} - 3 a b + 4 a b - 2 b^{2} = 3 a (2 a - b) + 2 b (2 a - b) = (2 a - b) (3 a + 2 b) \\ 15 a^{2} + 22 a b + 8 b^{2} = 15 a^{2} + 10 a b + 12 a b + 8 b^{2} = 5 a (3 a + 2 b) + 4 b (3 a + 2 b) = (3 a + 2 b) (5 a + 4 b) \\ 10 a^{2} + 3 a b - 4 b^{2} = 10 a^{2} - 5 a b + 8 a b - 4 b^{2} = 5 a (2 a - b) + 4 b (2 a - b) = (2 a - b) (5 a + 4 b) \\ m . c . m = (2 a - b) (3 a + 2 b) (5 a + 4 b) \end{matrix}$
$\begin{matrix} 12 x^{2} + 5 x y - 2 y^{2}, 15 x^{2} + 13 x y + 2 y^{2}, 20 x^{2} - x y - y^{2} \\ 12 x^{2} + 5 x y - 2 y^{2} = 12 x^{2} - 3 x y + 8 x y - 2 y^{2} = 3 x (4 x - y) + 2 y (4 x - y) = (4 x - y) (3 x + 2 y) \\ 15 x^{2} + 13 x y + 2 y^{2} = 15 x^{2} + 10 x y + 3 x y + 2 y^{2} = 5 x (3 x + 2 y) + y (3 x + 2 y) = (3 x + 2 y) (5 x + y) \\ 20 x^{2} - x y - y^{2} = 20 x^{2} - 5 x y + 4 x y - y^{2} = 5 x (4 x - y) + y (4 x - y) = (4 x - y) (5 x + y) \\ m . c . m = (4 x - y) (5 x + y) (3 x + 2 y) \end{matrix}$
$\begin{matrix} 6 b^{2} x^{2} + 6 b^{2} x^{3}, 3 a^{2} x - 3 a^{2} x^{2}, 1 - x^{4} \\ 6 b^{2} x^{2} + 6 b^{2} x^{3} = 6 b^{2} x^{2} (1 + x) = 2.3 b^{2} x^{2} (1 + x) \\ 3 a^{2} x - 3 a^{2} x^{2} = 3 a^{2} x (1 - x) \\ 1 - x^{4} = (1 + x^{2}) (1 - x^{2}) = (1 + x^{2}) (1 + x) (1 - x) \\ m . c . m = 2.3 a^{2} b^{2} x^{2} (1 + x^{2}) (1 + x) (1 - x) \\ m . c . m = 6 a^{2} b^{2} x^{2} (1 + x^{2}) (1 + x) (1 - x) \end{matrix}$
$\begin{matrix} x^{4} + 8 x - 4 x^{3} - 32, a^{2} x^{4} - 2 a^{2} x^{3} - 8 a^{2} x^{2}, 2 x^{4} - 4 x^{3} + 8 x^{2} \\ x^{4} + 8 x - 4 x^{3} - 32 = x^{3} (x - 4) + 8 (x - 4) = (x - 4) (x^{3} + 8) = (x - 4) (x + 2) (x^{2} - 2 x + 4) \\ a^{2} x^{4} - 2 a^{2} x^{3} - 8 a^{2} x^{2} = a^{2} x^{2} (x^{2} - 2 x - 8) = a^{2} x^{2} (x - 4) (x + 2) \\ 2 x^{4} - 4 x^{3} + 8 x^{2} = 2 x^{2} (x^{2} - 2 x + 4) \\ m . c . m = 2 a^{2} x^{2} (x - 4) (x + 2) (x^{2} - 2 x + 4) \end{matrix}$
$\begin{matrix} x^{3} - 9 x + x^{2} - 9, x^{4} - 10 x^{2} + 9, x^{2} + 4 x + 3, x^{2} - 4 x + 3 \\ x^{3} - 9 x + x^{2} - 9 = x (x^{2} - 9) + (x^{2} - 9) = (x^{2} - 9) (x + 1) = (x - 3) (x + 3) (x + 1) \\ x^{4} - 10 x^{2} + 9 = x^{4} - x^{2} - 9 x^{2} + 9 = x^{2} (x^{2} - 1) - 9 (x^{2} - 1) = (x^{2} - 1) (x^{2} - 9) = (x + 1) (x - 1) (x - 3) (x + 3) \\ x^{2} - 4 x + 3 = (x - 3) (x - 1) \\ m . c . m = (x + 1) (x - 1) (x - 3) (x + 3) \end{matrix}$
$\begin{matrix} 1 - a^{3}, 1 - a, 1 - a^{2}, 1 - 2 a + a^{2} \\ 1 - a^{3} = (1 - a) (1 + a + a^{2}) \\ 1 - a \\ 1 - a^{2} = (1 + a) (1 - a) \\ 1 - 2 a + a^{2} = {(1 - a)}^{2} \\ m . c . m = (1 + a) {(1 - a)}^{2} (1 + a + a^{2}) \end{matrix}$
$\begin{matrix} a^{2} b - a b^{2}, a^{4} b^{2} - a^{2} b^{4}, a {(a b - b^{2})}^{2}, b {(a^{2} + a b)}^{2} \\ a^{2} b - a b^{2} = a b (a - b) \\ a^{4} b^{2} - a^{2} b^{4} = a^{2} b^{2} (a^{2} - b^{2}) = a^{2} b^{2} (a + b) (a - b) \\ a {(a b - b^{2})}^{2} = a {[b (a - b)]}^{2} = a b^{2} {(a - b)}^{2} \\ b {(a^{2} + a b)}^{2} = b {[a (a + b)]}^{2} = b a^{2} {(a + b)}^{2} \\ m . c . m = a^{2} b^{2} {(a + b)}^{2} {(a - b)}^{2} \end{matrix}$
$\begin{matrix} m^{3} - 27 n^{3}, m^{2} - 9 n^{2}, m^{2} - 6 m n + 9 n^{2}, m^{2} + 3 m n + 9 n^{2} \\ m^{3} - 27 n^{3} = (m - 3 n) (m^{2} + 3 m n + 9 n^{2}) \\ m^{2} - 6 m n + 9 n^{2} = {(m - 3 n)}^{2} \\ m^{2} + 3 m n + 9 n^{2} \\ m . c . m = {(m - 3 n)}^{2} (m^{2} + 3 m n + 9 n^{2}) \end{matrix}$