1
Aqui apresentamos um exemplo resolvido passo a passo de teorema binomial. Esta solução foi gerada automaticamente pela nossa calculadora inteligente:
$\left(x+3\right)^5$
2
Aplicamos a regra: $\left(a+b\right)^n$$=newton\left(\left(a+b\right)^n\right)$, onde $a=x$, $b=3$, $a+b=x+3$ e $n=5$
$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 3^{0}x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3^{1}x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 3^{2}x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 3^{3}x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 3^{4}x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 3^{5}x^{0}$
3
Aplicamos a regra: $a^b$$=a^b$, onde $a=3$, $b=0$ e $a^b=3^{0}$
$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 1x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3^{1}x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 3^{2}x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 3^{3}x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 3^{4}x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 3^{5}x^{0}$
4
Aplicamos a regra: $a^b$$=a^b$, onde $a=3$, $b=1$ e $a^b=3^{1}$
$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 1x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 3^{2}x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 3^{3}x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 3^{4}x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 3^{5}x^{0}$
5
Aplicamos a regra: $a^b$$=a^b$, onde $a=3$, $b=2$ e $a^b=3^{2}$
$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 1x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 3^{3}x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 3^{4}x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 3^{5}x^{0}$
6
Aplicamos a regra: $a^b$$=a^b$, onde $a=3$, $b=3$ e $a^b=3^{3}$
$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 1x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 3^{4}x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 3^{5}x^{0}$
7
Aplicamos a regra: $a^b$$=a^b$, onde $a=3$, $b=4$ e $a^b=3^{4}$
$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 1x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 81x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 3^{5}x^{0}$
8
Aplicamos a regra: $a^b$$=a^b$, onde $a=3$, $b=5$ e $a^b=3^{5}$
$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 1x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 81x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243x^{0}$
9
Aplicamos a regra: $x^1$$=x$
$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 1x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 81x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243x^{0}$
10
Aplicamos a regra: $1x$$=x$, onde $x=\left(\begin{matrix}5\\0\end{matrix}\right)x^{5}$
$\left(\begin{matrix}5\\0\end{matrix}\right)x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 81x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243x^{0}$
11
Aplicamos a regra: $x^0$$=1$
$\left(\begin{matrix}5\\0\end{matrix}\right)x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 81x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243$
12
Aplicamos a regra: $\left(\begin{matrix}a\\b\end{matrix}\right)$$=\frac{a!}{\left(b!\right)\left(a-b\right)!}$, onde $a=5$, $b=0$, $a,b=5,0$ e $bicoefa,b=\left(\begin{matrix}5\\0\end{matrix}\right)$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}$
13
Aplicamos a regra: $x!$$=x!$, onde $factx=0!$ e $x=0$
$\frac{5!}{1\cdot 1}x^{5}$
14
Aplicamos a regra: $x!$$=x!$, onde $factx=5!$ e $x=5$
$\frac{120}{1\cdot 1}x^{5}$
15
Aplicamos a regra: $1x$$=x$, onde $x=1$
$\frac{120}{1}x^{5}$
16
Aplicamos a regra: $\frac{x}{1}$$=x$, onde $x=120$
$120x^{5}$
17
Aplicamos a regra: $\left(\begin{matrix}a\\b\end{matrix}\right)$$=\frac{a!}{\left(b!\right)\left(a-b\right)!}$, onde $a=5$, $b=0$, $a,b=5,0$ e $bicoefa,b=\left(\begin{matrix}5\\0\end{matrix}\right)$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}$
18
Aplicamos a regra: $x!$$=x!$, onde $factx=0!$ e $x=0$
$\frac{5!}{1\cdot 1}x^{5}$
19
Aplicamos a regra: $x!$$=x!$, onde $factx=5!$ e $x=5$
$\frac{120}{1\cdot 1}x^{5}$
20
Aplicamos a regra: $1x$$=x$, onde $x=1$
$\frac{120}{1}x^{5}$
21
Aplicamos a regra: $\frac{x}{1}$$=x$, onde $x=120$
$120x^{5}$
22
Aplicamos a regra: $\left(\begin{matrix}a\\b\end{matrix}\right)$$=\frac{a!}{\left(b!\right)\left(a-b\right)!}$, onde $a=5$, $b=1$, $a,b=5,1$ e $bicoefa,b=\left(\begin{matrix}5\\1\end{matrix}\right)$
$\frac{5!}{\left(1!\right)\left(5-1\right)!}\cdot 3x^{4}$
23
Aplicamos a regra: $x!$$=x!$, onde $factx=1!$ e $x=1$
$\frac{5!}{1\cdot 1}\cdot 3x^{4}$
24
Aplicamos a regra: $x!$$=x!$, onde $factx=5!$ e $x=5$
$\frac{120}{1\cdot 1}\cdot 3x^{4}$
25
Aplicamos a regra: $1x$$=x$, onde $x=1$
$\frac{120}{1}\cdot 3x^{4}$
26
Aplicamos a regra: $\frac{x}{1}$$=x$, onde $x=120$
$120\cdot 3x^{4}$
27
Aplicamos a regra: $\left(\begin{matrix}a\\b\end{matrix}\right)$$=\frac{a!}{\left(b!\right)\left(a-b\right)!}$, onde $a=5$, $b=0$, $a,b=5,0$ e $bicoefa,b=\left(\begin{matrix}5\\0\end{matrix}\right)$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}$
28
Aplicamos a regra: $x!$$=x!$, onde $factx=0!$ e $x=0$
$\frac{5!}{1\cdot 1}x^{5}$
29
Aplicamos a regra: $x!$$=x!$, onde $factx=5!$ e $x=5$
$\frac{120}{1\cdot 1}x^{5}$
30
Aplicamos a regra: $1x$$=x$, onde $x=1$
$\frac{120}{1}x^{5}$
31
Aplicamos a regra: $\frac{x}{1}$$=x$, onde $x=120$
$120x^{5}$
32
Aplicamos a regra: $\left(\begin{matrix}a\\b\end{matrix}\right)$$=\frac{a!}{\left(b!\right)\left(a-b\right)!}$, onde $a=5$, $b=1$, $a,b=5,1$ e $bicoefa,b=\left(\begin{matrix}5\\1\end{matrix}\right)$
$\frac{5!}{\left(1!\right)\left(5-1\right)!}\cdot 3x^{4}$
33
Aplicamos a regra: $x!$$=x!$, onde $factx=1!$ e $x=1$
$\frac{5!}{1\cdot 1}\cdot 3x^{4}$
34
Aplicamos a regra: $x!$$=x!$, onde $factx=5!$ e $x=5$
$\frac{120}{1\cdot 1}\cdot 3x^{4}$
35
Aplicamos a regra: $1x$$=x$, onde $x=1$
$\frac{120}{1}\cdot 3x^{4}$
36
Aplicamos a regra: $\frac{x}{1}$$=x$, onde $x=120$
$120\cdot 3x^{4}$
37
Aplicamos a regra: $\left(\begin{matrix}a\\b\end{matrix}\right)$$=\frac{a!}{\left(b!\right)\left(a-b\right)!}$, onde $a=5$, $b=2$, $a,b=5,2$ e $bicoefa,b=\left(\begin{matrix}5\\2\end{matrix}\right)$
$\frac{5!}{\left(2!\right)\left(5-2\right)!}\cdot 9x^{3}$
38
Aplicamos a regra: $x!$$=x!$, onde $factx=2!$ e $x=2$
$\frac{5!}{2\cdot 1}\cdot 9x^{3}$
39
Aplicamos a regra: $x!$$=x!$, onde $factx=5!$ e $x=5$
$\frac{120}{2\cdot 1}\cdot 9x^{3}$
40
Aplicamos a regra: $1x$$=x$, onde $x=2$
$\frac{120}{2}\cdot 9x^{3}$
41
Aplicamos a regra: $\left(\begin{matrix}a\\b\end{matrix}\right)$$=\frac{a!}{\left(b!\right)\left(a-b\right)!}$, onde $a=5$, $b=0$, $a,b=5,0$ e $bicoefa,b=\left(\begin{matrix}5\\0\end{matrix}\right)$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}$
42
Aplicamos a regra: $x!$$=x!$, onde $factx=0!$ e $x=0$
$\frac{5!}{1\cdot 1}x^{5}$
43
Aplicamos a regra: $x!$$=x!$, onde $factx=5!$ e $x=5$
$\frac{120}{1\cdot 1}x^{5}$
44
Aplicamos a regra: $1x$$=x$, onde $x=1$
$\frac{120}{1}x^{5}$
45
Aplicamos a regra: $\frac{x}{1}$$=x$, onde $x=120$
$120x^{5}$
46
Aplicamos a regra: $\left(\begin{matrix}a\\b\end{matrix}\right)$$=\frac{a!}{\left(b!\right)\left(a-b\right)!}$, onde $a=5$, $b=1$, $a,b=5,1$ e $bicoefa,b=\left(\begin{matrix}5\\1\end{matrix}\right)$
$\frac{5!}{\left(1!\right)\left(5-1\right)!}\cdot 3x^{4}$
47
Aplicamos a regra: $x!$$=x!$, onde $factx=1!$ e $x=1$
$\frac{5!}{1\cdot 1}\cdot 3x^{4}$
48
Aplicamos a regra: $x!$$=x!$, onde $factx=5!$ e $x=5$
$\frac{120}{1\cdot 1}\cdot 3x^{4}$
49
Aplicamos a regra: $1x$$=x$, onde $x=1$
$\frac{120}{1}\cdot 3x^{4}$
50
Aplicamos a regra: $\frac{x}{1}$$=x$, onde $x=120$
$120\cdot 3x^{4}$
51
Aplicamos a regra: $\left(\begin{matrix}a\\b\end{matrix}\right)$$=\frac{a!}{\left(b!\right)\left(a-b\right)!}$, onde $a=5$, $b=2$, $a,b=5,2$ e $bicoefa,b=\left(\begin{matrix}5\\2\end{matrix}\right)$
$\frac{5!}{\left(2!\right)\left(5-2\right)!}\cdot 9x^{3}$
52
Aplicamos a regra: $x!$$=x!$, onde $factx=2!$ e $x=2$
$\frac{5!}{2\cdot 1}\cdot 9x^{3}$
53
Aplicamos a regra: $x!$$=x!$, onde $factx=5!$ e $x=5$
$\frac{120}{2\cdot 1}\cdot 9x^{3}$
54
Aplicamos a regra: $1x$$=x$, onde $x=2$
$\frac{120}{2}\cdot 9x^{3}$
55
Aplicamos a regra: $\left(\begin{matrix}a\\b\end{matrix}\right)$$=\frac{a!}{\left(b!\right)\left(a-b\right)!}$, onde $a=5$, $b=3$, $a,b=5,3$ e $bicoefa,b=\left(\begin{matrix}5\\3\end{matrix}\right)$
$\frac{5!}{\left(3!\right)\left(5-3\right)!}\cdot 27x^{2}$
56
Aplicamos a regra: $x!$$=x!$, onde $factx=3!$ e $x=3$
$\frac{5!}{6\cdot 1}\cdot 27x^{2}$
57
Aplicamos a regra: $x!$$=x!$, onde $factx=5!$ e $x=5$
$\frac{120}{6\cdot 1}\cdot 27x^{2}$
58
Aplicamos a regra: $1x$$=x$, onde $x=6$
$\frac{120}{6}\cdot 27x^{2}$
59
Aplicamos a regra: $\left(\begin{matrix}a\\b\end{matrix}\right)$$=\frac{a!}{\left(b!\right)\left(a-b\right)!}$, onde $a=5$, $b=0$, $a,b=5,0$ e $bicoefa,b=\left(\begin{matrix}5\\0\end{matrix}\right)$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}$
60
Aplicamos a regra: $x!$$=x!$, onde $factx=0!$ e $x=0$
$\frac{5!}{1\cdot 1}x^{5}$
61
Aplicamos a regra: $x!$$=x!$, onde $factx=5!$ e $x=5$
$\frac{120}{1\cdot 1}x^{5}$
62
Aplicamos a regra: $1x$$=x$, onde $x=1$
$\frac{120}{1}x^{5}$
63
Aplicamos a regra: $\frac{x}{1}$$=x$, onde $x=120$
$120x^{5}$
64
Aplicamos a regra: $\left(\begin{matrix}a\\b\end{matrix}\right)$$=\frac{a!}{\left(b!\right)\left(a-b\right)!}$, onde $a=5$, $b=1$, $a,b=5,1$ e $bicoefa,b=\left(\begin{matrix}5\\1\end{matrix}\right)$
$\frac{5!}{\left(1!\right)\left(5-1\right)!}\cdot 3x^{4}$
65
Aplicamos a regra: $x!$$=x!$, onde $factx=1!$ e $x=1$
$\frac{5!}{1\cdot 1}\cdot 3x^{4}$
66
Aplicamos a regra: $x!$$=x!$, onde $factx=5!$ e $x=5$
$\frac{120}{1\cdot 1}\cdot 3x^{4}$
67
Aplicamos a regra: $1x$$=x$, onde $x=1$
$\frac{120}{1}\cdot 3x^{4}$
68
Aplicamos a regra: $\frac{x}{1}$$=x$, onde $x=120$
$120\cdot 3x^{4}$
69
Aplicamos a regra: $\left(\begin{matrix}a\\b\end{matrix}\right)$$=\frac{a!}{\left(b!\right)\left(a-b\right)!}$, onde $a=5$, $b=2$, $a,b=5,2$ e $bicoefa,b=\left(\begin{matrix}5\\2\end{matrix}\right)$
$\frac{5!}{\left(2!\right)\left(5-2\right)!}\cdot 9x^{3}$
70
Aplicamos a regra: $x!$$=x!$, onde $factx=2!$ e $x=2$
$\frac{5!}{2\cdot 1}\cdot 9x^{3}$
71
Aplicamos a regra: $x!$$=x!$, onde $factx=5!$ e $x=5$
$\frac{120}{2\cdot 1}\cdot 9x^{3}$
72
Aplicamos a regra: $1x$$=x$, onde $x=2$
$\frac{120}{2}\cdot 9x^{3}$
73
Aplicamos a regra: $\left(\begin{matrix}a\\b\end{matrix}\right)$$=\frac{a!}{\left(b!\right)\left(a-b\right)!}$, onde $a=5$, $b=3$, $a,b=5,3$ e $bicoefa,b=\left(\begin{matrix}5\\3\end{matrix}\right)$
$\frac{5!}{\left(3!\right)\left(5-3\right)!}\cdot 27x^{2}$
74
Aplicamos a regra: $x!$$=x!$, onde $factx=3!$ e $x=3$
$\frac{5!}{6\cdot 1}\cdot 27x^{2}$
75
Aplicamos a regra: $x!$$=x!$, onde $factx=5!$ e $x=5$
$\frac{120}{6\cdot 1}\cdot 27x^{2}$
76
Aplicamos a regra: $1x$$=x$, onde $x=6$
$\frac{120}{6}\cdot 27x^{2}$
77
Aplicamos a regra: $\left(\begin{matrix}a\\b\end{matrix}\right)$$=\frac{a!}{\left(b!\right)\left(a-b\right)!}$, onde $a=5$, $b=4$, $a,b=5,4$ e $bicoefa,b=\left(\begin{matrix}5\\4\end{matrix}\right)$
$\frac{5!}{\left(4!\right)\left(5-4\right)!}\cdot 81x$
78
Aplicamos a regra: $x!$$=x!$, onde $factx=4!$ e $x=4$
$\frac{5!}{24\cdot 1}\cdot 81x$
79
Aplicamos a regra: $x!$$=x!$, onde $factx=5!$ e $x=5$
$\frac{120}{24\cdot 1}\cdot 81x$
80
Aplicamos a regra: $1x$$=x$, onde $x=24$
$\frac{120}{24}\cdot 81x$
81
Aplicamos a regra: $a+b$$=a+b$, onde $a=5$, $b=-1$ e $a+b=5-1$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(5-2\right)!}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(5-3\right)!}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(5-4\right)!}x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243$
82
Aplicamos a regra: $a+b$$=a+b$, onde $a=5$, $b=-2$ e $a+b=5-2$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(5-3\right)!}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(5-4\right)!}x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243$
83
Aplicamos a regra: $a+b$$=a+b$, onde $a=5$, $b=-3$ e $a+b=5-3$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(5-4\right)!}x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243$
84
Aplicamos a regra: $a+b$$=a+b$, onde $a=5$, $b=-4$ e $a+b=5-4$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243$
85
Aplicamos a regra: $a+b$$=a+b$, onde $a=5$, $b=0$ e $a+b=5+0$
$\frac{5!}{\left(0!\right)\left(5!\right)}x^{5}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243$
86
Aplicamos a regra: $\frac{a}{a}$$=1$, onde $a/a=\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}$
$\frac{1}{0!}x^{5}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243$
87
Aplicamos a regra: $a\frac{b}{x}$$=\frac{ab}{x}$
$\frac{1x^{5}}{0!}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243$
88
Aplicamos a regra: $1x$$=x$, onde $x=x^{5}$
$\frac{x^{5}}{0!}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243$
89
Aplicamos a regra: $\left(\begin{matrix}a\\b\end{matrix}\right)$$=\frac{a!}{\left(b!\right)\left(a-b\right)!}$, onde $a=5$, $b=0$, $a,b=5,0$ e $bicoefa,b=\left(\begin{matrix}5\\0\end{matrix}\right)$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}$
90
Aplicamos a regra: $x!$$=x!$, onde $factx=0!$ e $x=0$
$\frac{5!}{1\cdot 1}x^{5}$
91
Aplicamos a regra: $x!$$=x!$, onde $factx=5!$ e $x=5$
$\frac{120}{1\cdot 1}x^{5}$
92
Aplicamos a regra: $1x$$=x$, onde $x=1$
$\frac{120}{1}x^{5}$
93
Aplicamos a regra: $\frac{x}{1}$$=x$, onde $x=120$
$120x^{5}$
94
Aplicamos a regra: $\left(\begin{matrix}a\\b\end{matrix}\right)$$=\frac{a!}{\left(b!\right)\left(a-b\right)!}$, onde $a=5$, $b=1$, $a,b=5,1$ e $bicoefa,b=\left(\begin{matrix}5\\1\end{matrix}\right)$
$\frac{5!}{\left(1!\right)\left(5-1\right)!}\cdot 3x^{4}$
95
Aplicamos a regra: $x!$$=x!$, onde $factx=1!$ e $x=1$
$\frac{5!}{1\cdot 1}\cdot 3x^{4}$
96
Aplicamos a regra: $x!$$=x!$, onde $factx=5!$ e $x=5$
$\frac{120}{1\cdot 1}\cdot 3x^{4}$
97
Aplicamos a regra: $1x$$=x$, onde $x=1$
$\frac{120}{1}\cdot 3x^{4}$
98
Aplicamos a regra: $\frac{x}{1}$$=x$, onde $x=120$
$120\cdot 3x^{4}$
99
Aplicamos a regra: $\left(\begin{matrix}a\\b\end{matrix}\right)$$=\frac{a!}{\left(b!\right)\left(a-b\right)!}$, onde $a=5$, $b=2$, $a,b=5,2$ e $bicoefa,b=\left(\begin{matrix}5\\2\end{matrix}\right)$
$\frac{5!}{\left(2!\right)\left(5-2\right)!}\cdot 9x^{3}$
100
Aplicamos a regra: $x!$$=x!$, onde $factx=2!$ e $x=2$
$\frac{5!}{2\cdot 1}\cdot 9x^{3}$
101
Aplicamos a regra: $x!$$=x!$, onde $factx=5!$ e $x=5$
$\frac{120}{2\cdot 1}\cdot 9x^{3}$
102
Aplicamos a regra: $1x$$=x$, onde $x=2$
$\frac{120}{2}\cdot 9x^{3}$
103
Aplicamos a regra: $\left(\begin{matrix}a\\b\end{matrix}\right)$$=\frac{a!}{\left(b!\right)\left(a-b\right)!}$, onde $a=5$, $b=3$, $a,b=5,3$ e $bicoefa,b=\left(\begin{matrix}5\\3\end{matrix}\right)$
$\frac{5!}{\left(3!\right)\left(5-3\right)!}\cdot 27x^{2}$
104
Aplicamos a regra: $x!$$=x!$, onde $factx=3!$ e $x=3$
$\frac{5!}{6\cdot 1}\cdot 27x^{2}$
105
Aplicamos a regra: $x!$$=x!$, onde $factx=5!$ e $x=5$
$\frac{120}{6\cdot 1}\cdot 27x^{2}$
106
Aplicamos a regra: $1x$$=x$, onde $x=6$
$\frac{120}{6}\cdot 27x^{2}$
107
Aplicamos a regra: $\left(\begin{matrix}a\\b\end{matrix}\right)$$=\frac{a!}{\left(b!\right)\left(a-b\right)!}$, onde $a=5$, $b=4$, $a,b=5,4$ e $bicoefa,b=\left(\begin{matrix}5\\4\end{matrix}\right)$
$\frac{5!}{\left(4!\right)\left(5-4\right)!}\cdot 81x$
108
Aplicamos a regra: $x!$$=x!$, onde $factx=4!$ e $x=4$
$\frac{5!}{24\cdot 1}\cdot 81x$
109
Aplicamos a regra: $x!$$=x!$, onde $factx=5!$ e $x=5$
$\frac{120}{24\cdot 1}\cdot 81x$
110
Aplicamos a regra: $1x$$=x$, onde $x=24$
$\frac{120}{24}\cdot 81x$
111
Aplicamos a regra: $\left(\begin{matrix}a\\b\end{matrix}\right)$$=\frac{a!}{\left(b!\right)\left(a-b\right)!}$, onde $a=5$, $b=5$, $a,b=5,5$ e $bicoefa,b=\left(\begin{matrix}5\\5\end{matrix}\right)$
$\left(\frac{5!}{\left(5!\right)\left(5-5\right)!}\right)\cdot 243$
112
Aplicamos a regra: $\frac{a}{a}$$=1$, onde $a/a=\frac{1}{\left(5-5\right)!}$
$\left(\frac{1}{\left(5-5\right)!}\right)\cdot 243$
113
Aplicamos a regra: $a+b$$=a+b$, onde $a=5$, $b=-5$ e $a+b=5-5$
$\frac{x^{5}}{0!}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243\left(5!\right)}{\left(5!\right)\left(0!\right)}$
114
Aplicamos a regra: $\frac{a}{a}$$=1$, onde $a/a=\frac{243}{0!}$
$\frac{x^{5}}{0!}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
115
Aplicamos a regra: $x!$$=x!$, onde $factx=0!$ e $x=0$
$\frac{x^{5}}{1}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
116
Aplicamos a regra: $x!$$=x!$, onde $factx=1!$ e $x=1$
$\frac{x^{5}}{1}+\frac{3\left(5!\right)}{1\cdot 24}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
117
Aplicamos a regra: $x!$$=x!$, onde $factx=5!$ e $x=5$
$\frac{x^{5}}{1}+\frac{3\cdot 120}{1\cdot 24}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
118
Aplicamos a regra: $x!$$=x!$, onde $factx=2!$ e $x=2$
$\frac{x^{5}}{1}+\frac{3\cdot 120}{1\cdot 24}x^{4}+\frac{9\left(5!\right)}{2\cdot 6}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
119
Aplicamos a regra: $x!$$=x!$, onde $factx=5!$ e $x=5$
$\frac{x^{5}}{1}+\frac{3\cdot 120}{1\cdot 24}x^{4}+\frac{9\cdot 120}{2\cdot 6}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
120
Aplicamos a regra: $ab$$=ab$, onde $ab=1\cdot 24$, $a=1$ e $b=24$
$\frac{x^{5}}{1}+\frac{3\cdot 120}{24}x^{4}+\frac{9\cdot 120}{2\cdot 6}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
121
Aplicamos a regra: $ab$$=ab$, onde $ab=3\cdot 120$, $a=3$ e $b=120$
$\frac{x^{5}}{1}+\frac{360}{24}x^{4}+\frac{9\cdot 120}{2\cdot 6}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
122
Aplicamos a regra: $ab$$=ab$, onde $ab=2\cdot 6$, $a=2$ e $b=6$
$\frac{x^{5}}{1}+\frac{360}{24}x^{4}+\frac{9\cdot 120}{12}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
123
Aplicamos a regra: $ab$$=ab$, onde $ab=9\cdot 120$, $a=9$ e $b=120$
$\frac{x^{5}}{1}+\frac{360}{24}x^{4}+\frac{1080}{12}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
124
Aplicamos a regra: $\frac{a}{b}$$=\frac{a}{b}$, onde $a=360$, $b=24$ e $a/b=\frac{360}{24}$
$\frac{x^{5}}{1}+15x^{4}+\frac{1080}{12}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
125
Aplicamos a regra: $\frac{a}{b}$$=\frac{a}{b}$, onde $a=1080$, $b=12$ e $a/b=\frac{1080}{12}$
$\frac{x^{5}}{1}+15x^{4}+90x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
126
Aplicamos a regra: $\frac{x}{1}$$=x$, onde $x=x^{5}$
$x^{5}+15x^{4}+90x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
127
Aplicamos a regra: $x!$$=x!$, onde $factx=3!$ e $x=3$
$x^{5}+15x^{4}+90x^{3}+\frac{27\left(5!\right)}{6\cdot 2}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
128
Aplicamos a regra: $x!$$=x!$, onde $factx=5!$ e $x=5$
$x^{5}+15x^{4}+90x^{3}+\frac{27\cdot 120}{6\cdot 2}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
129
Aplicamos a regra: $x!$$=x!$, onde $factx=4!$ e $x=4$
$x^{5}+15x^{4}+90x^{3}+\frac{27\cdot 120}{6\cdot 2}x^{2}+\frac{81\left(5!\right)}{24\cdot 1}x+\frac{243}{0!}$
130
Aplicamos a regra: $x!$$=x!$, onde $factx=5!$ e $x=5$
$x^{5}+15x^{4}+90x^{3}+\frac{27\cdot 120}{6\cdot 2}x^{2}+\frac{81\cdot 120}{24\cdot 1}x+\frac{243}{0!}$
131
Aplicamos a regra: $x!$$=x!$, onde $factx=0!$ e $x=0$
$x^{5}+15x^{4}+90x^{3}+\frac{27\cdot 120}{6\cdot 2}x^{2}+\frac{81\cdot 120}{24\cdot 1}x+\frac{243}{1}$
132
Aplicamos a regra: $ab$$=ab$, onde $ab=6\cdot 2$, $a=6$ e $b=2$
$x^{5}+15x^{4}+90x^{3}+\frac{27\cdot 120}{12}x^{2}+\frac{81\cdot 120}{24\cdot 1}x+\frac{243}{1}$
133
Aplicamos a regra: $ab$$=ab$, onde $ab=27\cdot 120$, $a=27$ e $b=120$
$x^{5}+15x^{4}+90x^{3}+\frac{3240}{12}x^{2}+\frac{81\cdot 120}{24\cdot 1}x+\frac{243}{1}$
134
Aplicamos a regra: $ab$$=ab$, onde $ab=24\cdot 1$, $a=24$ e $b=1$
$x^{5}+15x^{4}+90x^{3}+\frac{3240}{12}x^{2}+\frac{81\cdot 120}{24}x+\frac{243}{1}$
135
Aplicamos a regra: $ab$$=ab$, onde $ab=81\cdot 120$, $a=81$ e $b=120$
$x^{5}+15x^{4}+90x^{3}+\frac{3240}{12}x^{2}+\frac{9720}{24}x+\frac{243}{1}$
136
Aplicamos a regra: $\frac{a}{b}$$=\frac{a}{b}$, onde $a=3240$, $b=12$ e $a/b=\frac{3240}{12}$
$x^{5}+15x^{4}+90x^{3}+270x^{2}+\frac{9720}{24}x+\frac{243}{1}$
137
Aplicamos a regra: $\frac{a}{b}$$=\frac{a}{b}$, onde $a=9720$, $b=24$ e $a/b=\frac{9720}{24}$
$x^{5}+15x^{4}+90x^{3}+270x^{2}+405x+\frac{243}{1}$
138
Aplicamos a regra: $\frac{a}{b}$$=\frac{a}{b}$, onde $a=243$, $b=1$ e $a/b=\frac{243}{1}$
$x^{5}+15x^{4}+90x^{3}+270x^{2}+405x+243$
Resposta final para o problema
$x^{5}+15x^{4}+90x^{3}+270x^{2}+405x+243$