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CAPITULO XXVIII

Problemas sobre ecuaciones simultaneas

Ejercicio 211

Desarrollar, hallando los coeficientes por el triángulo de Pascal:

Para resolver este grupo de ejercicios, hay que tener en cuenta el tringulo de Pascal que se detalla a continuación:

\begin{matrix} 1 \\ 1 & 1 \\ 1 & 2 & 1 \\ 1 & 3 & 3 & 1 \\ 1 & 4 & 6 & 4 & 1 \\ 1 & 5 & 10 & 10 & 5 & 1 \\ 1 & 6 & 15 & 20 & 15 & 6 & 1 \\ 1 & 7 & 21 & 35 & 35 & 21 & 7 & 1 \\ 1 & 8 & 28 & 56 & 70 & 56 & 28 & 8 & 1 \\ 1 & 9 & 36 & 84 & 126 & 126 & 84 & 36 & 9 & 1 \\ 1 & 10 & 45 & 120 & 210 & 252 & 210 & 120 & 45 & 10 & 1 \end{matrix}

$\begin{matrix} {(a + 2 b)}^{6} & = a^{6} + 6 a^{5} (2 b) + 15 a^{4} {(2 b)}^{2} + 20 a^{3} {(2 b)}^{3} + 15 a^{2} {(2 b)}^{4} + 6 a {(2 b)}^{5} + {(2 b)}^{6} \\ = a^{6} + 12 a^{5} b + 60 a^{4} b^{2} + 160 a^{3} b^{3} + 240 a^{2} b^{4} + 192 a b^{5} + 64 b^{6} \end{matrix}$
$\begin{matrix} {(2 m^{2} - 3 n^{3})}^{5} & = {(2 m^{2})}^{5} - 5 {(2 m^{2})}^{4} (3 n^{3}) + 10 {(2 m^{2})}^{3} {(3 n^{3})}^{2} - 10 {(2 m^{2})}^{2} {(3 n^{3})}^{3} + 5 (2 m^{2}) {(3 n^{3})}^{4} - {(3 n^{3})}^{5} \\ = 32 m^{10} - 240 m^{8} n^{3} + 720 m^{6} n^{6} - 1080 m^{4} n^{9} + 810 m^{2} n^{12} - 243 n^{5} \end{matrix}$
$\begin{matrix} {(x^{2} + y^{3})}^{6} & = {(x^{2})}^{6} + 6 {(x^{2})}^{5} (y^{3}) + 15 {(x^{2})}^{4} {(y^{3})}^{2} + 20 {(x^{2})}^{3} {(y^{3})}^{3} + 15 {(x^{2})}^{2} {(y^{3})}^{4} + 6 (x^{2}) {(y^{3})}^{5} + {(y^{3})}^{6} \\ = x^{12} + 6 x^{10} y^{3} + 15 x^{8} y^{6} + 20 x^{6} y^{9} + 15 x^{4} y^{12} + 6 x^{2} y^{15} + y^{18} \end{matrix}$
$\begin{matrix} {(3 - y^{7})}^{7} & = {(3)}^{7} - 7 {(3)}^{6} (y^{7}) + 21 {(3)}^{5} {(y^{7})}^{2} - 35 {(3)}^{4} {(y^{7})}^{3} + 35 {(3)}^{3} {(y^{7})}^{4} - 21 {(3)}^{2} {(y^{7})}^{5} + 7 (3) {(y^{7})}^{6} - {(y^{7})}^{7} \\ = 2187 - 5103 y^{7} + 5103 y^{14} - 2835 y^{21} + 945 y^{28} - 189 y^{35} + 21 y^{42} - y^{49} \end{matrix}$
$\begin{matrix} {(2 x^{3} - 3 y^{4})}^{6} & = {(2 x^{3})}^{6} - 6 {(2 x^{3})}^{5} (3 y^{4}) + 15 {(2 x^{3})}^{4} {(3 y^{4})}^{2} - 20 {(2 x^{3})}^{3} {(3 y^{4})}^{3} + 15 {(2 x^{3})}^{2} {(3 y^{4})}^{4} - 6 (2 x^{3}) {(3 y^{4})}^{5} + {(3 y^{4})}^{6} \\ = 64 x^{18} - 576 x^{15} y^{4} + 2160 x^{12} y^{8} - 4320 x^{9} y^{12} + 4860 x^{6} y^{16} - 2916 x^{3} y^{20} + 729 y^{24} \end{matrix}$
$\begin{matrix} {(\frac{x^{2}}{2} + y^{3})}^{5} & = {(\frac{x^{2}}{2})}^{5} + 5 {(\frac{x^{2}}{2})}^{4} (y^{3}) + 10 {(\frac{x^{2}}{2})}^{3} {(y^{3})}^{2} + 10 {(\frac{x^{2}}{2})}^{2} {(y^{3})}^{3} + 5 (\frac{x^{2}}{2}) {(y^{3})}^{4} + {(y^{3})}^{5} \\ = \frac{x^{10}}{32} + \frac{5 x^{8} y^{3}}{16} + \frac{5 x^{6} y^{6}}{4} + \frac{5 x^{4} y^{9}}{2} + \frac{5 x^{2} y^{12}}{2} + y^{15} \end{matrix}$
$\begin{matrix} {(\frac{a}{3} - \frac{3}{b})}^{6} & = {(\frac{a}{3})}^{6} - 6 {(\frac{a}{3})}^{5} (\frac{3}{b}) + 15 {(\frac{a}{3})}^{4} {(\frac{3}{b})}^{2} - 20 {(\frac{a}{3})}^{3} {(\frac{3}{b})}^{3} + 15 {(\frac{a}{3})}^{2} {(\frac{3}{b})}^{4} - 6 (\frac{a}{3}) {(\frac{3}{b})}^{5} + {(\frac{3}{b})}^{6} \\ = \frac{a^{6}}{729} - \frac{2 a^{5}}{27 b} + \frac{5 a^{4}}{3 b^{2}} - \frac{20 a^{3}}{b^{3}} + \frac{135 a^{2}}{b^{4}} - \frac{486 a}{b^{5}} + \frac{729}{b^{6}} \end{matrix}$
$\begin{matrix} {(1 - x^{4})}^{8} & = {(1)}^{8} - 8 {(1)}^{7} (x^{4}) + 28 {(1)}^{6} {(x^{4})}^{2} - 56 {(1)}^{5} {(x^{4})}^{3} + 70 {(1)}^{4} {(x^{4})}^{4} - 56 {(1)}^{3} {(x^{4})}^{5} + 28 {(1)}^{2} {(x^{4})}^{6} - 8 (1) {(x^{4})}^{7} + {(x^{4})}^{8} \\ = 1 - 8 x^{4} + 28 x^{8} - 56 x^{12} + 70 x^{16} - 56 x^{20} + 28 x^{24} - 8 x^{28} + x^{32} \end{matrix}$
$\begin{matrix} {(\frac{2}{3 x} - \frac{3}{2 y})}^{7} & = {(\frac{2}{3 x})}^{7} - 7 {(\frac{2}{3 x})}^{6} (\frac{3}{2 y}) + 21 {(\frac{2}{3 x})}^{5} {(\frac{3}{2 y})}^{2} - 35 {(\frac{2}{3 x})}^{4} {(\frac{3}{2 y})}^{3} + 35 {(\frac{2}{3 x})}^{3} {(\frac{3}{2 y})}^{4} \\ - 21 {(\frac{2}{3 x})}^{2} {(\frac{3}{2 y})}^{5} + 7 (\frac{2}{3 x}) {(\frac{3}{2 y})}^{6} - {(\frac{3}{2 y})}^{7} \\ = \frac{128}{2187 x^{7}} - \frac{224}{243 x^{6} y} + \frac{56}{9 x^{5} y^{2}} - \frac{70}{3 x^{4} y^{3}} + \frac{105}{2 x^{3} y^{4}} - \frac{567}{8 x^{2} y^{5}} + \frac{1701}{32 x y^{6}} - \frac{2187}{128 y^{7}} \end{matrix}$
$\begin{matrix} {(\frac{2}{m} - \frac{m^{2}}{2})}^{7} & = {(\frac{2}{m})}^{7} - 7 {(\frac{2}{m})}^{6} (\frac{m^{2}}{2}) + 21 {(\frac{2}{m})}^{5} {(\frac{m^{2}}{2})}^{2} - 35 {(\frac{2}{m})}^{4} {(\frac{m^{2}}{2})}^{3} + 35 {(\frac{2}{m})}^{3} {(\frac{m^{2}}{2})}^{4} - 21 {(\frac{2}{m})}^{2} {(\frac{m^{2}}{2})}^{5} \\ + 7 (\frac{2}{m}) {(\frac{m^{2}}{2})}^{6} - {(\frac{m^{2}}{2})}^{7} \\ = \frac{128}{m^{7}} - \frac{224}{m^{4}} + \frac{168}{m} - 70 m^{2} + \frac{35 m^{2}}{2} - \frac{21 m^{8}}{8} + \frac{7 m^{11}}{32} - \frac{m^{14}}{128} \end{matrix}$
$\begin{matrix} {(x^{3} + m n)}^{8} & = {(x^{3})}^{8} - 8 {(x^{3})}^{7} (m n) + 28 {(x^{3})}^{6} {(m n)}^{2} - 56 {(x^{3})}^{5} {(m n)}^{3} + 70 {(x^{3})}^{4} {(m n)}^{4} - 56 {(x^{3})}^{3} {(m n)}^{5} \\ + 28 {(x^{3})}^{2} {(m n)}^{6} - 8 (x^{3}) {(m n)}^{7} + {(m n)}^{8} \\ = x^{18} - 8 x^{21} m n + 28 x^{18} m^{2} n^{2} - 56 x^{15} m^{3} n^{3} + 70 x^{12} m^{4} n^{4} - 56 x^{9} m^{5} n^{5} + 28 x^{6} m^{6} n^{6} - 8 x^{3} m^{7} n^{7} + m^{8} n^{8} \end{matrix}$
$\begin{matrix} {(3 - \frac{b^{2}}{3})}^{9} & = {(3)}^{9} - 9 {(3)}^{8} (\frac{b^{2}}{3}) + 36 {(3)}^{7} {(\frac{b^{2}}{3})}^{2} - 84 {(3)}^{6} {(\frac{b^{2}}{3})}^{3} + 126 {(3)}^{5} {(\frac{b^{2}}{3})}^{4} - 126 {(3)}^{4} {(\frac{b^{2}}{3})}^{5} \\ + 84 {(3)}^{3} {(\frac{b^{2}}{3})}^{6} - 36 {(3)}^{2} {(\frac{b^{2}}{3})}^{7} + 9 (3) {(\frac{b^{2}}{3})}^{8} - {(\frac{b^{2}}{3})}^{9} \\ = 19683 - 19683 b^{2} + 8748 b^{4} - 2268 b^{6} + 378 b^{8} - 42 b^{10} + \frac{28 b^{12}}{9} - \frac{4 b^{14}}{27} + \frac{b^{16}}{243} - \frac{b^{18}}{19683} \end{matrix}$
$\begin{matrix} {(1 - \frac{1}{x})}^{10} & = {(1)}^{10} - 10 {(1)}^{9} (\frac{1}{x}) + 45 {(1)}^{8} {(\frac{1}{x})}^{2} - 120 {(1)}^{7} {(\frac{1}{x})}^{3} + 210 {(1)}^{6} {(\frac{1}{x})}^{4} - 252 {(1)}^{5} {(\frac{1}{x})}^{5} + 210 {(1)}^{4} {(\frac{1}{x})}^{6} \\ - 120 {(1)}^{3} {(\frac{1}{x})}^{7} + 45 {(1)}^{2} {(\frac{1}{x})}^{8} - 10 (1) {(\frac{1}{x})}^{9} + {(\frac{1}{x})}^{10} \\ = 1 - \frac{10}{x} + \frac{45}{x^{2}} - \frac{120}{x^{3}} + \frac{210}{x^{4}} - \frac{252}{x^{5}} + \frac{210}{x^{6}} - \frac{120}{x^{7}} + \frac{45}{x^{8}} - \frac{10}{x^{9}} + \frac{1}{x^{10}} \end{matrix}$
$\begin{matrix} {(2 m^{2} - 5 n^{5})}^{6} & = {(2 m^{2})}^{6} - 6 {(2 m^{2})}^{5} (5 n^{5}) + 15 {(2 m^{2})}^{4} {(5 n^{5})}^{2} - 20 {(2 m^{2})}^{3} {(5 n^{5})}^{3} + 15 {(2 m^{2})}^{2} {(5 n^{5})}^{4} - 6 (2 m^{2}) {(5 n^{5})}^{5} + {(5 n^{5})}^{6} \\ = 64 m^{12} - 960 m^{10} n^{5} + 6000 m^{8} n^{10} - 20000 m^{6} n^{15} + 37500 m^{4} n^{20} - 37500 m^{2} n^{25} + 15625 n^{30} \end{matrix}$
$\begin{matrix} {(4 - \frac{x^{5}}{4})}^{7} & = {(4)}^{7} - 7 {(4)}^{6} (\frac{x^{5}}{4}) + 21 {(4)}^{5} {(\frac{x^{5}}{4})}^{2} - 35 {(4)}^{4} {(\frac{x^{5}}{4})}^{3} + 35 {(4)}^{3} {(\frac{x^{5}}{4})}^{4} - 21 {(4)}^{2} {(\frac{x^{5}}{4})}^{5} + 7 (4) {(\frac{x^{5}}{4})}^{6} - {(\frac{x^{5}}{4})}^{7} \\ = 16384 - 7168 x^{5} + 1344 x^{10} - 140 x^{15} + \frac{35 x^{20}}{4} - \frac{21 x^{25}}{64} + \frac{7 x^{30}}{1024} - \frac{x^{35}}{16384} \end{matrix}$