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CAPITULO XI

Máximo común divisor de polinomios por descomposición en factores

Ejercicio 112

Hallar, por descomposición en factores, el m.c.d. de:

$2 a^{2} + 2 a b, 4 a^{2} - 4 a b$
$\begin{matrix} 2 a^{2} + 2 a b = 2 a (a + b) \\ 4 a^{2} - 4 a b = 2^{2} a (a - b) \\ m . c . d = 2 a \end{matrix}$
$6 x^{3} y - 6 x^{2} y, 9 x^{3} y^{2} + 18 x^{2} y^{2}$
$\begin{matrix} 6 x^{3} y - 6 x^{2} y = 2.3 x^{2} y (x - 1) \\ 9 x^{3} y^{2} + 18 x^{2} y^{2} = 3^{2} x^{2} y^{2} (x + 2) \\ m . c . d = 3 x^{2} y \end{matrix}$
$12 a^{2} b^{3}, 4 a^{3} b^{2} - 8 a^{2} b^{3}$
$\begin{matrix} 12 a^{2} b^{3} = 3.4 a^{2} b^{3} \\ 4 a^{3} b^{2} - 8 a^{2} b^{3} = 4 a^{2} b^{2} (a - 2 b) \\ m . c . d = 4 a^{2} b^{2} \end{matrix}$
$a b + b, a^{2} + a$
$\begin{matrix} a b + b = b (a + 1) \\ a^{2} + a = a (a + 1) \\ m . c . d = a + 1 \end{matrix}$
$x^{2} - x, x^{3} - x^{2}$
$\begin{matrix} x^{2} - x = x (x - 1) \\ x^{3} - x^{2} = x^{2} (x - 1) \\ m . c . d = x - 1 \end{matrix}$
$30 a x^{2} - 15 x^{3}, 10 a x y^{2} - 20 x^{2} y^{2}$
$\begin{matrix} 30 a x^{2} - 15 x^{3} = 5.3 x^{2} (2 a - x) \\ 10 a x y^{2} - 20 x^{2} y^{2} = 5.2 x y^{2} (a - 2 x) \\ m . c . d = 5 x \end{matrix}$
$18 a^{2} x^{3} y^{4}, 6 a^{2} x^{2} y^{4} - 18 a^{2} x y^{4}$
$\begin{matrix} 18 a^{2} x^{3} y^{4} = 3.6 a^{2} x^{3} y^{4} \\ 6 a^{2} x^{2} y^{4} - 18 a^{2} x y^{4} = 6 a^{2} x y^{4} (x - 3) \\ m . c . d = 6 a^{2} x y^{4} \end{matrix}$
$5 a^{2} - 15 a, a^{3} - 3 a^{2}$
$\begin{matrix} 5 a^{2} - 15 a = 5 a (a - 3) \\ a^{3} - 3 a^{2} = a^{2} (a - 3) \\ m . c . d = a (a - 3) \end{matrix}$
$3 x^{3} + 15 x^{2}, a x^{2} + 5 a x$
$\begin{matrix} 3 x^{3} + 15 x^{2} = 3 x^{2} (x + 5) \\ a x^{2} + 5 a x = a x (x + 5) \\ m . c . d = x (x + 5) \end{matrix}$
$a^{2} - b^{2}, a^{2} - 2 a b + b^{2}$
$\begin{matrix} a^{2} - b^{2} = (a - b) (a + b) \\ a^{2} - 2 a b + b^{2} = {(a - b)}^{2} \\ m . c . d = a - b \end{matrix}$
$m^{3} + n^{3}, 3 a m + 3 a n$
$\begin{matrix} m^{3} + n^{3} = (m + n) (m^{2} - m n + n^{2}) \\ 3 a m + 3 a n = 3 a (m + n) \\ m . c . d = m + n \end{matrix}$
$x^{2} - 4, x^{3} - 8$
$\begin{matrix} x^{2} - 4 = (x - 2) (x + 2) \\ x^{3} - 8 = (x - 2) (x^{2} + 2 x + 4) \\ m . c . d = x - 2 \end{matrix}$
$2 a x^{2} + 4 a x, x^{3} - x^{2} - 6 x$
$\begin{matrix} 2 a x^{2} + 4 a x = 2 a x (x + 2) \\ x^{3} - x^{2} - 6 x = x (x^{2} - x - 6) = x (x - 3) (x + 2) \\ m . c . d = x (x + 2) \end{matrix}$
$9 x^{2} - 1, 9 x^{2} - 6 x + 1$
$\begin{matrix} 9 x^{2} - 1 = (3 x - 1) (3 x + 1) \\ 9 x^{2} - 6 x + 1 = {(3 x - 1)}^{2} \\ m . c . d = 3 x - 1 \end{matrix}$
$4 a^{2} + 4 a b + b^{2}, 2 a^{2} - 2 a b + a b - b^{2}$
$\begin{matrix} 4 a^{2} + 4 a b + b^{2} = {(2 a + b)}^{2} \\ 2 a^{2} - 2 a b + a b - b^{2} = 2 a (a - b) + b (a - b) = (a - b) (2 a + b) \\ m . c . d = 2 a + b \end{matrix}$
$3 x^{2} + 3 x - 60, 6 x^{2} - 18 x - 24$
$\begin{matrix} 3 x^{2} + 3 x - 60 = 3 (x^{2} + x - 20) = 3 (x + 5) (x - 4) \\ 6 x^{2} - 18 x - 24 = 6 (x^{2} - 3 x - 4) = 6 (x - 4) (x + 1) \\ m . c . d = 3 (x - 4) \end{matrix}$
$8 x^{3} + y^{3}, 4 a x^{2} - a y^{2}$
$\begin{matrix} 8 x^{3} + y^{3} = (2 x + y) (4 x^{2} - 2 x y + y^{2}) \\ 4 a x^{2} - a y^{2} = a (4 x^{2} - y^{2}) = a (2 x - y) (2 x + y) \\ m . c . d = 2 x + y \end{matrix}$
$2 a^{3} - 12 a^{2} b + 18 a b^{2}, a^{3} x - 9 a b^{2} x$
$\begin{matrix} 2 a^{3} - 12 a^{2} b + 18 a b^{2} = & 2 a (a^{2} - 6 a b + 9 b^{2}) \\ = & 2 a {(a - 3 b)}^{2} \\ a^{3} x - 9 a b^{2} x & = a x (a^{2} - 9 b^{2}) \\ = a x (a - 3 b) (a + 3 b) \\ m . c . d = a (a - 3 b) \end{matrix}$
$a c + a d - 2 b c - 2 b d, 2 c^{2} + 4 c d + 2 d^{2}$
$\begin{matrix} a c + a d - 2 b c - 2 b d & = a (c + d) - 2 b (c + d) \\ = (c + d) (a - 2 b) \\ 2 c^{2} + 4 c d + 2 d^{2} & = 2 (c^{2} + 2 c d + d^{2}) \\ = 2 {(c + d)}^{2} \\ m . c . d = c + d \end{matrix}$
$3 a^{2} m^{2} + 6 a^{2} m - 45 a^{2}, 6 a m^{2} x + 24 a m x - 30 a x$
$\begin{matrix} 3 a^{2} m^{2} + 6 a^{2} m - 45 a^{2} & = 3 a^{2} (m^{2} + 2 m - 15) \\ = 3 a^{2} (m + 5) (m - 3) \\ 6 a m^{2} x + 24 a m x - 30 a x & = 6 a x (m^{2} + 4 m - 5) \\ = 6 a x (m + 5) (m - 1) \\ m . c . d = 3 a (m + 5) \end{matrix}$
$4 x^{4} - y^{2}, {(2 x^{2} - y)}^{2}$
$\begin{matrix} 4 x^{4} - y^{2} & = (2 x^{2} - y) (2 x^{2} + y) \\ {(2 x^{2} - y)}^{2} & = {(2 x^{2} - y)}^{2} \\ m . c . d = 2 x^{2} - y \end{matrix}$
$3 x^{5} - 3 x, 9 x^{3} - 9 x$
$\begin{matrix} 3 x^{5} - 3 x & = 3 x (x^{4} - 1) \\ = 3 x (x^{2} - 1) (x^{2} + 1) \\ = 3 x (x - 1) (x + 1) (x^{2} + 1) \\ 9 x^{3} - 9 x & = 9 x (x^{2} - 1) \\ = 9 x (x - 1) (x + 1) \\ m . c . d = 3 x (x - 1) \end{matrix}$
$a^{2} + a b, a b + b^{2}, a^{3} + a^{2} b$
$\begin{matrix} a^{2} + a b & = a (a + b) \\ a b + b^{2} & = b (a + b) \\ a^{3} + a^{2} b & = a^{2} (a + b) \\ m . c . d = a (a + b) \end{matrix}$
$2 x^{3} - 2 x^{2}, 3 x^{2} - 3 x, 4 x^{3} - 4 x^{2}$
$\begin{matrix} 2 x^{3} - 2 x^{2} & = 2 x^{2} (x - 1) \\ 3 x^{2} - 3 x & = 3 x (x - 1) \\ 4 x^{3} - 4 x^{2} & = 4 x^{2} (x - 1) \\ m . c . d = x (x - 1) \end{matrix}$
$x^{4} - 9 x^{2}, x^{4} - 5 x^{3} + 6 x^{2}, x^{4} - 6 x^{3} + 9 x^{2}$
$\begin{matrix} x^{4} - 9 x^{2} & = x^{2} (x^{2} - 9) \\ = x^{2} (x - 3) (x + 3) \\ x^{4} - 5 x^{3} + 6 x^{2} & = x^{2} (x^{2} - 5 x + 6) \\ = x^{2} (x - 3) (x - 2) \\ x^{4} - 6 x^{3} + 9 x^{2} & = x^{2} (x^{2} - 6 x + 9) \\ = x^{2} {(x - 3)}^{2} \\ m . c . d = x^{2} (x - 3) \end{matrix}$
$a^{3} b + 2 a^{2} b^{2} + a b^{3}, a^{4} b - a^{2} b^{3}$
$\begin{matrix} a^{3} b + 2 a^{2} b^{2} + a b^{3} & = a b (a^{2} + 2 a b + b^{2}) \\ = a b {(a + b)}^{2} \\ a^{4} b - a^{2} b^{3} & = a^{2} b (a^{2} - b^{2}) \\ = a^{2} b (a + b) (a - b) \\ m . c . d = a b (a + b) \end{matrix}$
$2 x^{2} + 2 x - 4, 2 x^{2} - 8 x + 6, 2 x^{3} - 2$
$\begin{matrix} 2 x^{2} + 2 x - 4 & = 2 (x^{2} + x - 2) \\ = 2 (x + 2) (x - 1) \\ 2 x^{2} - 8 x + 6 & = 2 (x^{2} - 4 x + 3) \\ = 2 (x - 3) (x - 1) \\ 2 x^{3} - 2 & = 2 (x^{3} - 1) \\ = 2 (x - 1) (x^{2} + x + 1) \\ m . c . d = 2 (x - 1) \end{matrix}$
$a x^{3} - 2 a x^{2} - 8 a x, a x^{2} - a x - 6 a, a^{2} x^{3} - 3 a^{2} x^{2} - 10 a^{2} x$
$\begin{matrix} a x^{3} - 2 a x^{2} - 8 a x & = a x (x^{2} - 2 x - 8) \\ = a x (x - 4) (x + 2) \\ a x^{2} - a x - 6 a & = a (x^{2} - x - 6) \\ = a (x - 3) (x + 2) \\ a^{2} x^{3} - 3 a^{2} x^{2} - 10 a^{2} x & = a^{2} x (x^{2} - 3 x - 10) \\ = a^{2} x (x - 5) (x + 2) \\ m . c . d = a (x + 2) \end{matrix}$
$2 a n^{4} - 16 a n^{2} + 32 a, 2 a n^{3} - 8 a n, 2 a^{2} n^{3} + 16 a^{2}$
$\begin{matrix} 2 a n^{4} - 16 a n^{2} + 32 a & = 2 a (n^{4} - 8 n^{2} + 16) \\ = 2 a {(n^{2} - 4)}^{2} \\ = 2 a {[(n - 2) (n + 2)]}^{2} \\ = 2 a {(n - 2)}^{2} {(n + 2)}^{2} \\ 2 a n^{3} - 8 a n & = 2 a n (n^{2} - 4) \\ = 2 a n (n - 2) (n + 2) \\ 2 a^{2} n^{3} + 16 a^{2} & = 2 a^{2} (n^{3} + 8) \\ = 2 a^{2} (n + 2) (n^{2} - 2 n + 4) \\ m . c . d = 2 a (n + 2) \end{matrix}$
$4 a^{2} + 8 a - 12, 2 a^{2} - 6 a + 4, 6 a^{2} + 18 a - 24$
$\begin{matrix} 4 a^{2} + 8 a - 12 & = 4 (a^{2} + 2 a - 3) \\ = 4 (a + 3) (a - 1) \\ 2 a^{2} - 6 a + 4 & = 2 (a^{2} - 3 a + 2) \\ = 2 (a - 2) (a - 1) \\ 6 a^{2} + 18 a - 24 & = 6 (a^{2} + 3 a - 4) \\ = 6 (a + 4) (a - 1) \\ m . c . d = 2 (a - 1) \end{matrix}$
$4 a^{2} - b^{2}, 8 a^{3} + b^{3}, 4 a^{2} + 4 a b + b^{2}$
$\begin{matrix} 4 a^{2} - b^{2} & = (2 a - b) (2 a + b) \\ 8 a^{3} + b^{3} & = (2 a + b) (4 a^{2} - 2 a b + b^{2}) \\ 4 a^{2} + 4 a b + b^{2} & = {(2 a + b)}^{2} \\ m . c . d = 2 a + b \end{matrix}$
$x^{2} - 2 x - 8, x^{2} - x - 12, x^{3} - 9 x^{2} + 20 x$
$\begin{matrix} x^{2} - 2 x - 8 & = (x - 4) (x + 2) \\ x^{2} - x - 12 & = (x - 4) (x + 3) \\ x^{3} - 9 x^{2} + 20 x & = x (x^{2} - 9 x + 20) \\ = x (x - 5) (x - 4) \\ m . c . d = x - 4 \end{matrix}$
$a^{2} + a, a^{3} - 6 a^{2} - 7 a, a^{6} + a$
$\begin{matrix} a^{2} + a & = a (a + 1) \\ a^{3} - 6 a^{2} - 7 a & = a (a^{2} - 6 a - 7) \\ = a (a - 7) (a + 1) \\ a^{6} + a & = a (a^{5} + 1) \\ = a (a + 1) (a^{4} - a^{3} + a^{2} - a + 1) \\ m . c . d = a (a + 1) \end{matrix}$
$x^{3} + 27, 2 x^{2} - 6 x + 18, x^{4} - 3 x^{3} + 9 x^{2}$
$\begin{matrix} x^{3} + 27 & = (x + 3) (x^{2} - 3 x + 9) \\ 2 x^{2} - 6 x + 18 & = 2 (x^{2} - 3 x + 9) \\ x^{4} - 3 x^{3} + 9 x^{2} & = x^{2} (x^{2} - 3 x + 9) \\ m . c . d = x^{2} - 3 x + 9 \end{matrix}$
$x^{2} + a x - 6 a^{2}, x^{2} + 2 a x - 3 a^{2}, x^{2} + 6 a x + 9 a^{2}$
$\begin{matrix} x^{2} + a x - 6 a^{2} & = (x + 3 a) (x - 2 a) \\ x^{2} + 2 a x - 3 a^{2} & = (x + 3 a) (x - a) \\ x^{2} + 6 a x + 9 a^{2} & = {(x + 3 a)}^{2} \\ m . c . d = x + 3 a \end{matrix}$
$54 x^{3} + 250, 18 a x^{2} - 50 a, 50 + 60 x + 18 x^{2}$
$\begin{matrix} 54 x^{3} + 250 & = 2 (27 x^{3} + 125) \\ = 2 (3 x + 5) (9 x^{2} - 15 x + 25) \\ 18 a x^{2} - 50 a & = 2 a (9 x^{2} - 25) \\ = 2 a (3 x + 5) (3 x - 5) \\ 50 + 60 x + 18 x^{2} & = 2 (9 x^{2} + 30 x + 25) \\ = 2 {(3 x + 5)}^{2} \\ m . c . d = 2 (3 x + 5) \end{matrix}$
${(x^{2} - 1)}^{2}, x^{2} - 4 x - 5, x^{4} - 1$
$\begin{matrix} {(x^{2} - 1)}^{2} & = {[(x + 1) (x - 1)]}^{2} \\ = {(x + 1)}^{2} {(x - 1)}^{2} \\ x^{2} - 4 x - 5 & = (x - 4) (x + 1) \\ x^{4} - 1 & = (x^{2} + 1) (x^{2} - 1) \\ = (x^{2} + 1) (x + 1) (x - 1) \\ m . c . d = x + 1 \end{matrix}$
$4 a x^{2} - 28 a x, a^{2} x^{3} - 8 a^{2} x^{2} + 7 a^{2} x, a x^{4} - 15 a x^{3} + 56 a x^{2}$
$\begin{matrix} 4 a x^{2} - 28 a x & = 4 a x (x - 7) \\ a^{2} x^{3} - 8 a^{2} x^{2} + 7 a^{2} x & = a^{2} x (x^{2} - 8 x + 7) \\ = a^{2} x (x - 7) (x - 1) \\ a x^{4} - 15 a x^{3} + 56 a x^{2} & = a x^{2} (x^{2} - 15 x + 56) \\ = a x^{2} (x - 8) (x - 7) \\ m . c . d = a x (x - 7) \end{matrix}$
$3 a^{2} - 6 a, a^{3} - 4 a, a^{2} b - 2 a b, a^{2} - a - 2$
$\begin{matrix} 3 a^{2} - 6 a & = 3 a (a - 2) \\ a^{3} - 4 a & = a (a^{2} - 4) \\ = a (a - 2) (a + 2) \\ a^{2} b - 2 a b & = a b (a - 2) \\ a^{2} - a - 2 & = (a - 2) (a + 1) \\ m . c . d = a - 2 \end{matrix}$
$3 x^{2} - x, 27 x^{3} - 1, 9 x^{2} - 6 x + 1, 3 a x - a + 6 x - 2$
$\begin{matrix} 3 x^{2} - x & = x (3 x - 1) \\ 27 x^{3} - 1 & = (3 x - 1) (9 x^{2} + 3 x + 1) \\ 9 x^{2} - 6 x + 1 & = {(3 x - 1)}^{2} \\ 3 a x - a + 6 x - 2 & = a (3 x - 1) + 2 (3 x - 1) \\ = (3 x - 1) (a + 2) \\ m . c . d = 3 x - 1 \end{matrix}$
$a^{4} - 1, a^{3} + a^{2} + a + 1, a^{3} x + a^{2} x + a x + x, a^{5} + a^{3} + a^{2} + 1$
$\begin{matrix} a^{4} - 1 & = (a^{2} + 1) (a^{2} - 1) \\ = (a^{2} + 1) (a + 1) (a - 1) \\ a^{3} + a^{2} + a + 1 & = a^{2} (a + 1) + (a + 1) \\ = (a + 1) (a^{2} + 1) \\ a^{3} x + a^{2} x + a x + x & = a^{2} x (a + 1) + x (a + 1) \\ = (a + 1) (a^{2} x + x) \\ = x (a + 1) (a^{2} + 1) \\ a^{5} + a^{3} + a^{2} + 1 & = a^{3} (a^{2} + 1) + (a^{2} + 1) \\ = (a^{2} + 1) (a^{3} + 1) \\ = (a^{2} + 1) (a + 1) (a^{2} - a + 1) \\ m . c . d = (a + 1) (a^{2} + 1) \end{matrix}$
$2 m^{2} + 4 m n + 2 n^{2}, m^{3} + m^{2} n + m n^{2} + n^{3}, m^{3} + n^{3}, m^{3} - m n^{2}$
$\begin{matrix} 2 m^{2} + 4 m n + 2 n^{2} & = 2 (m^{2} + 2 m n + n^{2}) \\ = 2 {(m + n)}^{2} \\ m^{3} + m^{2} n + m n^{2} + n^{3} & = m^{2} (m + n) + n^{2} (m + n) \\ = (m + n) (m^{2} + n^{2}) \\ m^{3} + n^{3} & = (m + n) (m^{2} - m n + n^{2}) \\ m^{3} - m n^{2} & = m (m^{2} - n^{2}) \\ = m (m - n) (m + n) \\ m . c . d = m + n \end{matrix}$
$a^{3} - 3 a^{2} + 3 a - 1, a^{2} - 2 a + 1, a^{3} - a, a^{2} - 4 a + 3$
$\begin{matrix} a^{3} - 3 a^{2} + 3 a - 1 & = (a^{3} - 1) - 3 a (a - 1) \\ = (a - 1) (a^{2} + a + 1) - 3 a (a - 1) \\ = (a - 1) (a^{2} + a + 1 - 3 a) \\ = (a - 1) (a^{2} - 2 a + 1) \\ a^{2} - 2 a + 1 & = {(a - 1)}^{2} \\ a^{3} - a & = a (a^{2} - 1) \\ = a (a - 1) (a + 1) \\ a^{2} - 4 a + 3 & = (a - 3) (a - 1) \\ m . c . d = a - 1 \end{matrix}$
$16 a^{3} x + 54 x, 12 a^{2} x^{2} - 42 a x^{2} - 90 x^{2}, 32 a^{3} x + 24 a^{2} x - 36 a x, 32 a^{4} x - 144 a^{2} x + 162 x$
$\begin{matrix} 16 a^{3} x + 54 x & = 2 x (8 a^{3} + 27) \\ = 2 x (2 a + 3) (4 a^{2} - 6 a + 9) \\ 12 a^{2} x^{2} - 42 a x^{2} - 90 x^{2} & = 6 x^{2} (2 a^{2} - 7 a - 15) \\ = 6 x^{2} (2 a^{2} - 10 a + 3 a - 15) \\ = 6 x^{2} [2 a (a - 5) + 3 (a - 5)] \\ = 6 x^{2} (a - 5) (2 a + 3) \\ 32 a^{3} x + 24 a^{2} x - 36 a x & = 4 a x (8 a^{2} + 6 a - 9) \\ = 4 a x (8 a^{2} + 12 a - 6 a - 9) \\ = 4 a x [4 a (2 a + 3) - 3 (2 a + 3)] \\ = 4 a x (2 a + 3) (4 a - 3) \\ 32 a^{4} x - 144 a^{2} x + 162 x & = 2 x (16 a^{4} - 72 a^{2} + 81) \\ = 2 x {(4 a^{2} - 9)}^{2} \\ = 2 x {[(2 a - 3) (2 a + 3)]}^{2} \\ = 2 x {(2 a - 3)}^{2} {(2 a + 3)}^{2} \\ m . c . d = 2 x (2 a + 3) \end{matrix}$
${(x y + y^{2})}^{2}, x^{2} y - 2 x y^{2} - 3 y^{3}, a x^{3} y + a y^{4}, x^{2} y - y^{3}$
$\begin{matrix} {(x y + y^{2})}^{2} & = {[y (x + y)]}^{2} \\ = y^{2} {(x + y)}^{2} \\ x^{2} y - 2 x y^{2} - 3 y^{3} & = y (x^{2} - 2 x y - 3 y^{2}) \\ = y (x - 3) (x + 1) \\ a x^{3} y + a y^{4} & = a y (x^{3} + y^{4}) \\ x^{2} y - y^{3} & = y (x^{2} - y^{2}) \\ = y (x - y) (x + y) \\ m . c . d = y \end{matrix}$
$2 a^{2} - a m + 4 a - 2 m, 2 a m^{2} - m^{3}, 6 a^{2} + 5 a m - 4 m^{2}, 16 a^{2} + 72 a m - 40 m^{2}$
$\begin{matrix} 2 a^{2} - a m + 4 a - 2 m & = a (2 a - m) + 2 (2 a - m) \\ = (2 a - m) (a + 2) \\ 2 a m^{2} - m^{3} & = m^{2} (2 a - m) \\ 6 a^{2} + 5 a m - 4 m^{2} & = 6 a^{2} - 3 a m + 8 a m - 4 m^{2} \\ = 3 a (2 a - m) + 4 m (2 a - m) \\ = (2 a - m) (3 a + 4 m) \\ 16 a^{2} + 72 a m - 40 m^{2} & = 8 (2 a^{2} + 9 a m - 5 m^{2}) \\ = 8 (2 a^{2} - a m + 10 a m - 5 m^{2}) \\ = 8 [a (2 a - m) + 5 m (2 a - m)] \\ = 8 (2 a - m) (a + 5 m) \\ m . c . d = 2 a - m \end{matrix}$
$12 a x - 6 a y + 24 b x - 12 b y, 3 a^{3} + 24 b^{3}, 9 a^{2} + 9 a b - 18 b^{2}, 12 a^{2} + 24 a b$
$\begin{matrix} 12 a x - 6 a y + 24 b x - 12 b y & = 6 a (2 x - y) + 12 b (2 x - y) \\ = (2 x - y) (6 a + 12 b) \\ = 6 (2 x - y) (a + 2 b) \\ 3 a^{3} + 24 b^{3} & = 3 (a^{3} + 8 b^{3}) \\ = 3 (a + 2 b) (a^{2} - 2 a b + 4 b^{2}) \\ 9 a^{2} + 9 a b - 18 b^{2} & = 9 (a^{2} + a b - 2 b^{2}) \\ = 9 (a + 2 b) (a - b) \\ 12 a^{2} + 24 a b & = 12 a (a + 2 b) \\ m . c . d = a + 2 b \end{matrix}$
$5 a^{2} + 5 a x + 5 a y + 5 x y, 15 a^{3} - 15 a x^{2} + 15 a^{2} y - 15 x^{2} y, 20 a^{3} - 20 a y^{2} + 20 a^{2} x - 20 x y^{2}, 5 a^{5} + 5 a^{4} x + 5 a^{2} y^{3} + 5 a x y^{3}$
$\begin{matrix} 5 a^{2} + 5 a x + 5 a y + 5 x y & = 5 a (a + x) + 5 y (a + x) \\ = (a + x) (5 a + 5 y) \\ = 5 (a + x) (a + y) \\ 15 a^{3} - 15 a x^{2} + 15 a^{2} y - 15 x^{2} y & = 15 a^{3} + 15 a^{2} y - 15 a x^{2} - 15 x^{2} y \\ = 15 a^{2} (a + y) - 15 x^{2} (a + y) \\ = (a + y) (15 a^{2} - 15 x^{2}) \\ = 15 (a + y) (a^{2} - x^{2}) \\ = 15 (a + y) (a + x) (a - x) \\ 20 a^{3} - 20 a y^{2} + 20 a^{2} x - 20 x y^{2} & = 20 a (a^{2} - y^{2}) + 20 x (a^{2} - y^{2}) \\ = (a^{2} - y^{2}) (20 a + 20 x) \\ = 20 (a + y) (a - y) (a + x) \\ 5 a^{5} + 5 a^{4} x + 5 a^{2} y^{3} + 5 a x y^{3} & = 5 a^{4} (a + x) + 5 a y^{3} (a + x) \\ = (a + x) (5 a^{4} + 5 a y^{3}) \\ = (a + x) 5 a (a^{3} + y^{3}) \\ = 5 a (a + x) (a + y) (a^{2} - a y + y^{2}) \\ m . c . d = 5 (a + x) (a + y) \end{matrix}$