1
Ejemplo resuelto de teorema del binomio
$\left(x+3\right)^5$
2
Podemos expandir la expresi贸n $\left(x+3\right)^5$ usando el binomio de Newton, el cual es una f贸rmula que nos permite obtener la forma expandida de un binomio elevado a un n煤mero entero $n$. La f贸rmula tal cual es: $\displaystyle(a\pm b)^n=\sum_{k=0}^{n}\left(\begin{matrix}n\\k\end{matrix}\right)a^{n-k}b^k=\left(\begin{matrix}n\\0\end{matrix}\right)a^n\pm\left(\begin{matrix}n\\1\end{matrix}\right)a^{n-1}b+\left(\begin{matrix}n\\2\end{matrix}\right)a^{n-2}b^2\pm\dots\pm\left(\begin{matrix}n\\n\end{matrix}\right)b^n$. El n煤mero de t茅rminos que resultan de la expansi贸n es siempre igual a $n+1$. Los coeficientes $\left(\begin{matrix}n\\k\end{matrix}\right)$ son n煤meros combinatorios los cuales corresponden a la fila en茅sima del tri谩ngulo de Tartaglia (o tri谩ngulo de Pascal). En la f贸rmula, podemos observar que el exponente de $a$ va disminuyendo, de $n$ a $0$, mientras que el exponente de $b$ va aumentando, de $0$ a $n$. Si uno de los t茅rminos del binomio es negativo, se alternan los signos positivos y negativos.
$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 3^{0}x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3^{1}x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 3^{2}x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 3^{3}x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 3^{4}x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 3^{5}x^{0}$
3
Calcular la potencia $3^{0}$
$\left(\begin{matrix}5\\0\end{matrix}\right)x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3^{1}x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 3^{2}x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 3^{3}x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 3^{4}x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 3^{5}x^{0}$
4
Calcular la potencia $3^{1}$
$\left(\begin{matrix}5\\0\end{matrix}\right)x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 3^{2}x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 3^{3}x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 3^{4}x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 3^{5}x^{0}$
5
Calcular la potencia $3^{2}$
$\left(\begin{matrix}5\\0\end{matrix}\right)x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 3^{3}x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 3^{4}x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 3^{5}x^{0}$
6
Calcular la potencia $3^{3}$
$\left(\begin{matrix}5\\0\end{matrix}\right)x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 3^{4}x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 3^{5}x^{0}$
7
Calcular la potencia $3^{4}$
$\left(\begin{matrix}5\\0\end{matrix}\right)x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 81x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 3^{5}x^{0}$
8
Calcular la potencia $3^{5}$
$\left(\begin{matrix}5\\0\end{matrix}\right)x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 81x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243x^{0}$
9
Cualquier expresi贸n elevada a la potencia uno es igual a esa misma expresi贸n
$\left(\begin{matrix}5\\0\end{matrix}\right)x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 81x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243x^{0}$
10
Cualquier expresi贸n matem谩tica elevada a la potencia $0$ es igual a $1$
$\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 1\cdot 243$
11
Cualquier expresi贸n algebraica multiplicada por uno es igual a esa misma expresi贸n
$\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243$
12
Calcular el coeficiente binomial $\left(\begin{matrix}5\\0\end{matrix}\right)$ aplicando la f贸rmula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}$
13
El factorial de $0$ es
$\frac{5!}{\left(5+0\right)!}x^{5}$
14
El factorial de $5$ es
$\frac{120}{\left(5+0\right)!}x^{5}$
15
Calcular el coeficiente binomial $\left(\begin{matrix}5\\0\end{matrix}\right)$ aplicando la f贸rmula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}$
16
El factorial de $0$ es
$\frac{5!}{\left(5+0\right)!}x^{5}$
17
El factorial de $5$ es
$\frac{120}{\left(5+0\right)!}x^{5}$
18
Calcular el coeficiente binomial $\left(\begin{matrix}5\\1\end{matrix}\right)$ aplicando la f贸rmula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(1!\right)\left(5-1\right)!}\cdot 3x^{4}$
19
El factorial de $1$ es
$\frac{5!}{\left(5-1\right)!}\cdot 3x^{4}$
20
El factorial de $5$ es
$\frac{120}{\left(5-1\right)!}\cdot 3x^{4}$
21
Calcular el coeficiente binomial $\left(\begin{matrix}5\\0\end{matrix}\right)$ aplicando la f贸rmula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}$
22
El factorial de $0$ es
$\frac{5!}{\left(5+0\right)!}x^{5}$
23
El factorial de $5$ es
$\frac{120}{\left(5+0\right)!}x^{5}$
24
Calcular el coeficiente binomial $\left(\begin{matrix}5\\1\end{matrix}\right)$ aplicando la f贸rmula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(1!\right)\left(5-1\right)!}\cdot 3x^{4}$
25
El factorial de $1$ es
$\frac{5!}{\left(5-1\right)!}\cdot 3x^{4}$
26
El factorial de $5$ es
$\frac{120}{\left(5-1\right)!}\cdot 3x^{4}$
27
Calcular el coeficiente binomial $\left(\begin{matrix}5\\2\end{matrix}\right)$ aplicando la f贸rmula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(2!\right)\left(5-2\right)!}\cdot 9x^{3}$
28
El factorial de $2$ es
$\frac{5!}{2\left(5-2\right)!}\cdot 9x^{3}$
29
El factorial de $5$ es
$\frac{120}{2\left(5-2\right)!}\cdot 9x^{3}$
30
Calcular el coeficiente binomial $\left(\begin{matrix}5\\0\end{matrix}\right)$ aplicando la f贸rmula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}$
31
El factorial de $0$ es
$\frac{5!}{\left(5+0\right)!}x^{5}$
32
El factorial de $5$ es
$\frac{120}{\left(5+0\right)!}x^{5}$
33
Calcular el coeficiente binomial $\left(\begin{matrix}5\\1\end{matrix}\right)$ aplicando la f贸rmula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(1!\right)\left(5-1\right)!}\cdot 3x^{4}$
34
El factorial de $1$ es
$\frac{5!}{\left(5-1\right)!}\cdot 3x^{4}$
35
El factorial de $5$ es
$\frac{120}{\left(5-1\right)!}\cdot 3x^{4}$
36
Calcular el coeficiente binomial $\left(\begin{matrix}5\\2\end{matrix}\right)$ aplicando la f贸rmula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(2!\right)\left(5-2\right)!}\cdot 9x^{3}$
37
El factorial de $2$ es
$\frac{5!}{2\left(5-2\right)!}\cdot 9x^{3}$
38
El factorial de $5$ es
$\frac{120}{2\left(5-2\right)!}\cdot 9x^{3}$
39
Calcular el coeficiente binomial $\left(\begin{matrix}5\\3\end{matrix}\right)$ aplicando la f贸rmula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(3!\right)\left(5-3\right)!}\cdot 27x^{2}$
40
El factorial de $3$ es
$\frac{5!}{6\left(5-3\right)!}\cdot 27x^{2}$
41
El factorial de $5$ es
$\frac{120}{6\left(5-3\right)!}\cdot 27x^{2}$
42
Calcular el coeficiente binomial $\left(\begin{matrix}5\\0\end{matrix}\right)$ aplicando la f贸rmula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}$
43
El factorial de $0$ es
$\frac{5!}{\left(5+0\right)!}x^{5}$
44
El factorial de $5$ es
$\frac{120}{\left(5+0\right)!}x^{5}$
45
Calcular el coeficiente binomial $\left(\begin{matrix}5\\1\end{matrix}\right)$ aplicando la f贸rmula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(1!\right)\left(5-1\right)!}\cdot 3x^{4}$
46
El factorial de $1$ es
$\frac{5!}{\left(5-1\right)!}\cdot 3x^{4}$
47
El factorial de $5$ es
$\frac{120}{\left(5-1\right)!}\cdot 3x^{4}$
48
Calcular el coeficiente binomial $\left(\begin{matrix}5\\2\end{matrix}\right)$ aplicando la f贸rmula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(2!\right)\left(5-2\right)!}\cdot 9x^{3}$
49
El factorial de $2$ es
$\frac{5!}{2\left(5-2\right)!}\cdot 9x^{3}$
50
El factorial de $5$ es
$\frac{120}{2\left(5-2\right)!}\cdot 9x^{3}$
51
Calcular el coeficiente binomial $\left(\begin{matrix}5\\3\end{matrix}\right)$ aplicando la f贸rmula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(3!\right)\left(5-3\right)!}\cdot 27x^{2}$
52
El factorial de $3$ es
$\frac{5!}{6\left(5-3\right)!}\cdot 27x^{2}$
53
El factorial de $5$ es
$\frac{120}{6\left(5-3\right)!}\cdot 27x^{2}$
54
Calcular el coeficiente binomial $\left(\begin{matrix}5\\4\end{matrix}\right)$ aplicando la f贸rmula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(4!\right)\left(5-4\right)!}\cdot 81x$
55
El factorial de $4$ es
$\frac{5!}{24\left(5-4\right)!}\cdot 81x$
56
El factorial de $5$ es
$\frac{120}{24\left(5-4\right)!}\cdot 81x$
57
Calcular el coeficiente binomial $\left(\begin{matrix}5\\0\end{matrix}\right)$ aplicando la f贸rmula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}$
58
El factorial de $0$ es
$\frac{5!}{\left(5+0\right)!}x^{5}$
59
El factorial de $5$ es
$\frac{120}{\left(5+0\right)!}x^{5}$
60
Calcular el coeficiente binomial $\left(\begin{matrix}5\\1\end{matrix}\right)$ aplicando la f贸rmula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(1!\right)\left(5-1\right)!}\cdot 3x^{4}$
61
El factorial de $1$ es
$\frac{5!}{\left(5-1\right)!}\cdot 3x^{4}$
62
El factorial de $5$ es
$\frac{120}{\left(5-1\right)!}\cdot 3x^{4}$
63
Calcular el coeficiente binomial $\left(\begin{matrix}5\\2\end{matrix}\right)$ aplicando la f贸rmula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(2!\right)\left(5-2\right)!}\cdot 9x^{3}$
64
El factorial de $2$ es
$\frac{5!}{2\left(5-2\right)!}\cdot 9x^{3}$
65
El factorial de $5$ es
$\frac{120}{2\left(5-2\right)!}\cdot 9x^{3}$
66
Calcular el coeficiente binomial $\left(\begin{matrix}5\\3\end{matrix}\right)$ aplicando la f贸rmula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(3!\right)\left(5-3\right)!}\cdot 27x^{2}$
67
El factorial de $3$ es
$\frac{5!}{6\left(5-3\right)!}\cdot 27x^{2}$
68
El factorial de $5$ es
$\frac{120}{6\left(5-3\right)!}\cdot 27x^{2}$
69
Calcular el coeficiente binomial $\left(\begin{matrix}5\\4\end{matrix}\right)$ aplicando la f贸rmula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(4!\right)\left(5-4\right)!}\cdot 81x$
70
El factorial de $4$ es
$\frac{5!}{24\left(5-4\right)!}\cdot 81x$
71
El factorial de $5$ es
$\frac{120}{24\left(5-4\right)!}\cdot 81x$
72
Calcular el coeficiente binomial $\left(\begin{matrix}5\\5\end{matrix}\right)$ aplicando la f贸rmula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\left(\frac{5!}{\left(5!\right)\left(5-5\right)!}\right)\cdot 243$
73
Simplificar la fracci贸n $\frac{5!}{\left(5!\right)\left(5-5\right)!}$ por $5!$
$\left(\frac{1}{\left(5-5\right)!}\right)\cdot 243$
74
Restar los valores $5$ y $-1$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(5-2\right)!}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(5-3\right)!}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(5-4\right)!}x+\frac{243\left(5!\right)}{\left(5!\right)\left(5-5\right)!}$
75
Restar los valores $5$ y $-2$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(5-3\right)!}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(5-4\right)!}x+\frac{243\left(5!\right)}{\left(5!\right)\left(5-5\right)!}$
76
Restar los valores $5$ y $-3$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(5-4\right)!}x+\frac{243\left(5!\right)}{\left(5!\right)\left(5-5\right)!}$
77
Restar los valores $5$ y $-4$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243\left(5!\right)}{\left(5!\right)\left(5-5\right)!}$
78
Restar los valores $5$ y $-5$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243\left(5!\right)}{\left(5!\right)\left(0!\right)}$
79
Sumar los valores $5$ y $0$
$\frac{5!}{\left(0!\right)\left(5!\right)}x^{5}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243\left(5!\right)}{\left(5!\right)\left(0!\right)}$
80
Simplificar la fracci贸n $\frac{5!}{\left(0!\right)\left(5!\right)}$ por $5!$
$\frac{1}{0!}x^{5}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243\left(5!\right)}{\left(5!\right)\left(0!\right)}$
81
Simplificar la fracci贸n $\frac{243\left(5!\right)}{\left(5!\right)\left(0!\right)}$ por $5!$
$\frac{1}{0!}x^{5}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
82
Multiplicar la fracci贸n por el t茅rmino
$\frac{x^{5}}{0!}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
83
Multiplicando la fracci贸n por el t茅rmino $x^{4}$
$\frac{x^{5}}{0!}+\frac{3\left(5!\right)\left(x^{4}\right)}{\left(1!\right)\left(4!\right)}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
84
Multiplicando la fracci贸n por el t茅rmino $x^{3}$
$\frac{x^{5}}{0!}+\frac{3\left(5!\right)\left(x^{4}\right)}{\left(1!\right)\left(4!\right)}+\frac{9\left(5!\right)\left(x^{3}\right)}{\left(2!\right)\left(3!\right)}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
85
Multiplicando la fracci贸n por el t茅rmino $x^{2}$
$\frac{x^{5}}{0!}+\frac{3\left(5!\right)\left(x^{4}\right)}{\left(1!\right)\left(4!\right)}+\frac{9\left(5!\right)\left(x^{3}\right)}{\left(2!\right)\left(3!\right)}+\frac{27\left(5!\right)\left(x^{2}\right)}{\left(3!\right)\left(2!\right)}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
86
Multiplicando la fracci贸n por el t茅rmino $x$
$\frac{x^{5}}{0!}+\frac{3\left(5!\right)\left(x^{4}\right)}{\left(1!\right)\left(4!\right)}+\frac{9\left(5!\right)\left(x^{3}\right)}{\left(2!\right)\left(3!\right)}+\frac{27\left(5!\right)\left(x^{2}\right)}{\left(3!\right)\left(2!\right)}+\frac{81\left(5!\right)x}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
87
El factorial de $0$ es
$\frac{x^{5}}{1}+\frac{3\left(5!\right)\left(x^{4}\right)}{\left(1!\right)\left(4!\right)}+\frac{9\left(5!\right)\left(x^{3}\right)}{\left(2!\right)\left(3!\right)}+\frac{27\left(5!\right)\left(x^{2}\right)}{\left(3!\right)\left(2!\right)}+\frac{81\left(5!\right)x}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
88
El factorial de $1$ es
$\frac{x^{5}}{1}+\frac{3\left(5!\right)\left(x^{4}\right)}{4!}+\frac{9\left(5!\right)\left(x^{3}\right)}{\left(2!\right)\left(3!\right)}+\frac{27\left(5!\right)\left(x^{2}\right)}{\left(3!\right)\left(2!\right)}+\frac{81\left(5!\right)x}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
89
El factorial de $4$ es
$\frac{x^{5}}{1}+\frac{3\left(5!\right)\left(x^{4}\right)}{24}+\frac{9\left(5!\right)\left(x^{3}\right)}{\left(2!\right)\left(3!\right)}+\frac{27\left(5!\right)\left(x^{2}\right)}{\left(3!\right)\left(2!\right)}+\frac{81\left(5!\right)x}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
90
El factorial de $5$ es
$\frac{x^{5}}{1}+\frac{3\cdot 120x^{4}}{24}+\frac{9\left(5!\right)\left(x^{3}\right)}{\left(2!\right)\left(3!\right)}+\frac{27\left(5!\right)\left(x^{2}\right)}{\left(3!\right)\left(2!\right)}+\frac{81\left(5!\right)x}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
91
El factorial de $2$ es
$\frac{x^{5}}{1}+\frac{3\cdot 120x^{4}}{24}+\frac{9\left(5!\right)\left(x^{3}\right)}{2\left(3!\right)}+\frac{27\left(5!\right)\left(x^{2}\right)}{\left(3!\right)\left(2!\right)}+\frac{81\left(5!\right)x}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
92
El factorial de $3$ es
$\frac{x^{5}}{1}+\frac{3\cdot 120x^{4}}{24}+\frac{9\left(5!\right)\left(x^{3}\right)}{2\cdot 6}+\frac{27\left(5!\right)\left(x^{2}\right)}{\left(3!\right)\left(2!\right)}+\frac{81\left(5!\right)x}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
93
El factorial de $5$ es
$\frac{x^{5}}{1}+\frac{3\cdot 120x^{4}}{24}+\frac{9\cdot 120x^{3}}{2\cdot 6}+\frac{27\left(5!\right)\left(x^{2}\right)}{\left(3!\right)\left(2!\right)}+\frac{81\left(5!\right)x}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
94
El factorial de $3$ es
$\frac{x^{5}}{1}+\frac{3\cdot 120x^{4}}{24}+\frac{9\cdot 120x^{3}}{2\cdot 6}+\frac{27\left(5!\right)\left(x^{2}\right)}{6\left(2!\right)}+\frac{81\left(5!\right)x}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
95
El factorial de $2$ es
$\frac{x^{5}}{1}+\frac{3\cdot 120x^{4}}{24}+\frac{9\cdot 120x^{3}}{2\cdot 6}+\frac{27\left(5!\right)\left(x^{2}\right)}{6\cdot 2}+\frac{81\left(5!\right)x}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
96
El factorial de $5$ es
$\frac{x^{5}}{1}+\frac{3\cdot 120x^{4}}{24}+\frac{9\cdot 120x^{3}}{2\cdot 6}+\frac{27\cdot 120x^{2}}{6\cdot 2}+\frac{81\left(5!\right)x}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
97
Multiplicar $3$ por $120$
$\frac{x^{5}}{1}+\frac{360x^{4}}{24}+\frac{9\cdot 120x^{3}}{2\cdot 6}+\frac{27\cdot 120x^{2}}{6\cdot 2}+\frac{81\left(5!\right)x}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
98
Multiplicar $2$ por $6$
$\frac{x^{5}}{1}+\frac{360x^{4}}{24}+\frac{9\cdot 120x^{3}}{12}+\frac{27\cdot 120x^{2}}{6\cdot 2}+\frac{81\left(5!\right)x}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
99
Multiplicar $9$ por $120$
$\frac{x^{5}}{1}+\frac{360x^{4}}{24}+\frac{1080x^{3}}{12}+\frac{27\cdot 120x^{2}}{6\cdot 2}+\frac{81\left(5!\right)x}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
100
Multiplicar $6$ por $2$
$\frac{x^{5}}{1}+\frac{360x^{4}}{24}+\frac{1080x^{3}}{12}+\frac{27\cdot 120x^{2}}{12}+\frac{81\left(5!\right)x}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
101
Multiplicar $27$ por $120$
$\frac{x^{5}}{1}+\frac{360x^{4}}{24}+\frac{1080x^{3}}{12}+\frac{3240x^{2}}{12}+\frac{81\left(5!\right)x}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
102
Cualquier expresi贸n matem谩tica dividida por uno ($1$) es igual a esa misma expresi贸n
$x^{5}+\frac{360x^{4}}{24}+\frac{1080x^{3}}{12}+\frac{3240x^{2}}{12}+\frac{81\left(5!\right)x}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
103
Sacar el $\frac{360}{24}$ de la fracci贸n
$x^{5}+15x^{4}+\frac{1080x^{3}}{12}+\frac{3240x^{2}}{12}+\frac{81\left(5!\right)x}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
104
Sacar el $\frac{1080}{12}$ de la fracci贸n
$x^{5}+15x^{4}+90x^{3}+\frac{3240x^{2}}{12}+\frac{81\left(5!\right)x}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
105
Sacar el $\frac{3240}{12}$ de la fracci贸n
$x^{5}+15x^{4}+90x^{3}+270x^{2}+\frac{81\left(5!\right)x}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
106
El factorial de $4$ es
$x^{5}+15x^{4}+90x^{3}+270x^{2}+\frac{81\left(5!\right)x}{24\left(1!\right)}+\frac{243}{0!}$
107
El factorial de $1$ es
$x^{5}+15x^{4}+90x^{3}+270x^{2}+\frac{81\left(5!\right)x}{24}+\frac{243}{0!}$
108
El factorial de $5$ es
$x^{5}+15x^{4}+90x^{3}+270x^{2}+\frac{81\cdot 120x}{24}+\frac{243}{0!}$
109
El factorial de $0$ es
$x^{5}+15x^{4}+90x^{3}+270x^{2}+\frac{81\cdot 120x}{24}+\frac{243}{1}$
110
Multiplicar $81$ por $120$
$x^{5}+15x^{4}+90x^{3}+270x^{2}+\frac{9720x}{24}+\frac{243}{1}$
111
Dividir $243$ entre $1$
$x^{5}+15x^{4}+90x^{3}+270x^{2}+\frac{9720x}{24}+243$
112
Sacar el $\frac{9720}{24}$ de la fracci贸n
$x^{5}+15x^{4}+90x^{3}+270x^{2}+405x+243$
Respuesta Final
$x^{5}+15x^{4}+90x^{3}+270x^{2}+405x+243$