1
Here, we show you a step-by-step solved example of binomial theorem. This solution was automatically generated by our smart calculator:
$\left(x+3\right)^5$
2
We can expand the expression $\left(x+3\right)^5$ using Newton's binomial theorem, which is a formula that allow us to find the expanded form of a binomial raised to a positive integer $n$. The formula is as follows: $\displaystyle(a\pm b)^n=\sum_{k=0}^{n}\left(\begin{matrix}n\\k\end{matrix}\right)a^{n-k}b^k=\left(\begin{matrix}n\\0\end{matrix}\right)a^n\pm\left(\begin{matrix}n\\1\end{matrix}\right)a^{n-1}b+\left(\begin{matrix}n\\2\end{matrix}\right)a^{n-2}b^2\pm\dots\pm\left(\begin{matrix}n\\n\end{matrix}\right)b^n$. The number of terms resulting from the expansion always equals $n + 1$. The coefficients $\left(\begin{matrix}n\\k\end{matrix}\right)$ are combinatorial numbers which correspond to the nth row of the Tartaglia triangle (or Pascal's triangle). In the formula, we can observe that the exponent of $a$ decreases, from $n$ to $0$, while the exponent of $b$ increases, from $0$ to $n$. If one of the binomial terms is negative, the positive and negative signs alternate.
$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 3^{0}x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3^{1}x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 3^{2}x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 3^{3}x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 3^{4}x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 3^{5}x^{0}$
3
Calculate the power $3^{0}$
$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 1x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3^{1}x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 3^{2}x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 3^{3}x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 3^{4}x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 3^{5}x^{0}$
4
Calculate the power $3^{1}$
$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 1x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 3^{2}x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 3^{3}x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 3^{4}x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 3^{5}x^{0}$
5
Calculate the power $3^{2}$
$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 1x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 3^{3}x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 3^{4}x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 3^{5}x^{0}$
6
Calculate the power $3^{3}$
$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 1x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 3^{4}x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 3^{5}x^{0}$
7
Calculate the power $3^{4}$
$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 1x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 81x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 3^{5}x^{0}$
8
Calculate the power $3^{5}$
$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 1x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 81x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243x^{0}$
9
Any expression to the power of $1$ is equal to that same expression
$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 1x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 81x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243x^{0}$
10
Any expression multiplied by $1$ is equal to itself
$\left(\begin{matrix}5\\0\end{matrix}\right)x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 81x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243x^{0}$
11
Any expression (except $0$ and $\infty$) to the power of $0$ is equal to $1$
$\left(\begin{matrix}5\\0\end{matrix}\right)x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 81x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243$
12
Calculate the binomial coefficient $\left(\begin{matrix}5\\0\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}$
13
The factorial of $0$ is
$\frac{5!}{1\cdot 1}x^{5}$
14
The factorial of $5$ is
$\frac{120}{1\cdot 1}x^{5}$
15
Any expression multiplied by $1$ is equal to itself
$\frac{120}{1}x^{5}$
16
Any expression divided by one ($1$) is equal to that same expression
$120x^{5}$
17
Calculate the binomial coefficient $\left(\begin{matrix}5\\0\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}$
18
The factorial of $0$ is
$\frac{5!}{1\cdot 1}x^{5}$
19
The factorial of $5$ is
$\frac{120}{1\cdot 1}x^{5}$
20
Any expression multiplied by $1$ is equal to itself
$\frac{120}{1}x^{5}$
21
Any expression divided by one ($1$) is equal to that same expression
$120x^{5}$
22
Calculate the binomial coefficient $\left(\begin{matrix}5\\1\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(1!\right)\left(5-1\right)!}\cdot 3x^{4}$
23
The factorial of $1$ is
$\frac{5!}{1\cdot 1}\cdot 3x^{4}$
24
The factorial of $5$ is
$\frac{120}{1\cdot 1}\cdot 3x^{4}$
25
Any expression multiplied by $1$ is equal to itself
$\frac{120}{1}\cdot 3x^{4}$
26
Any expression divided by one ($1$) is equal to that same expression
$120\cdot 3x^{4}$
27
Calculate the binomial coefficient $\left(\begin{matrix}5\\0\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}$
28
The factorial of $0$ is
$\frac{5!}{1\cdot 1}x^{5}$
29
The factorial of $5$ is
$\frac{120}{1\cdot 1}x^{5}$
30
Any expression multiplied by $1$ is equal to itself
$\frac{120}{1}x^{5}$
31
Any expression divided by one ($1$) is equal to that same expression
$120x^{5}$
32
Calculate the binomial coefficient $\left(\begin{matrix}5\\1\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(1!\right)\left(5-1\right)!}\cdot 3x^{4}$
33
The factorial of $1$ is
$\frac{5!}{1\cdot 1}\cdot 3x^{4}$
34
The factorial of $5$ is
$\frac{120}{1\cdot 1}\cdot 3x^{4}$
35
Any expression multiplied by $1$ is equal to itself
$\frac{120}{1}\cdot 3x^{4}$
36
Any expression divided by one ($1$) is equal to that same expression
$120\cdot 3x^{4}$
37
Calculate the binomial coefficient $\left(\begin{matrix}5\\2\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(2!\right)\left(5-2\right)!}\cdot 9x^{3}$
38
The factorial of $2$ is
$\frac{5!}{2\cdot 1}\cdot 9x^{3}$
39
The factorial of $5$ is
$\frac{120}{2\cdot 1}\cdot 9x^{3}$
40
Any expression multiplied by $1$ is equal to itself
$\frac{120}{2}\cdot 9x^{3}$
41
Calculate the binomial coefficient $\left(\begin{matrix}5\\0\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}$
42
The factorial of $0$ is
$\frac{5!}{1\cdot 1}x^{5}$
43
The factorial of $5$ is
$\frac{120}{1\cdot 1}x^{5}$
44
Any expression multiplied by $1$ is equal to itself
$\frac{120}{1}x^{5}$
45
Any expression divided by one ($1$) is equal to that same expression
$120x^{5}$
46
Calculate the binomial coefficient $\left(\begin{matrix}5\\1\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(1!\right)\left(5-1\right)!}\cdot 3x^{4}$
47
The factorial of $1$ is
$\frac{5!}{1\cdot 1}\cdot 3x^{4}$
48
The factorial of $5$ is
$\frac{120}{1\cdot 1}\cdot 3x^{4}$
49
Any expression multiplied by $1$ is equal to itself
$\frac{120}{1}\cdot 3x^{4}$
50
Any expression divided by one ($1$) is equal to that same expression
$120\cdot 3x^{4}$
51
Calculate the binomial coefficient $\left(\begin{matrix}5\\2\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(2!\right)\left(5-2\right)!}\cdot 9x^{3}$
52
The factorial of $2$ is
$\frac{5!}{2\cdot 1}\cdot 9x^{3}$
53
The factorial of $5$ is
$\frac{120}{2\cdot 1}\cdot 9x^{3}$
54
Any expression multiplied by $1$ is equal to itself
$\frac{120}{2}\cdot 9x^{3}$
55
Calculate the binomial coefficient $\left(\begin{matrix}5\\3\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(3!\right)\left(5-3\right)!}\cdot 27x^{2}$
56
The factorial of $3$ is
$\frac{5!}{6\cdot 1}\cdot 27x^{2}$
57
The factorial of $5$ is
$\frac{120}{6\cdot 1}\cdot 27x^{2}$
58
Any expression multiplied by $1$ is equal to itself
$\frac{120}{6}\cdot 27x^{2}$
59
Calculate the binomial coefficient $\left(\begin{matrix}5\\0\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}$
60
The factorial of $0$ is
$\frac{5!}{1\cdot 1}x^{5}$
61
The factorial of $5$ is
$\frac{120}{1\cdot 1}x^{5}$
62
Any expression multiplied by $1$ is equal to itself
$\frac{120}{1}x^{5}$
63
Any expression divided by one ($1$) is equal to that same expression
$120x^{5}$
64
Calculate the binomial coefficient $\left(\begin{matrix}5\\1\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(1!\right)\left(5-1\right)!}\cdot 3x^{4}$
65
The factorial of $1$ is
$\frac{5!}{1\cdot 1}\cdot 3x^{4}$
66
The factorial of $5$ is
$\frac{120}{1\cdot 1}\cdot 3x^{4}$
67
Any expression multiplied by $1$ is equal to itself
$\frac{120}{1}\cdot 3x^{4}$
68
Any expression divided by one ($1$) is equal to that same expression
$120\cdot 3x^{4}$
69
Calculate the binomial coefficient $\left(\begin{matrix}5\\2\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(2!\right)\left(5-2\right)!}\cdot 9x^{3}$
70
The factorial of $2$ is
$\frac{5!}{2\cdot 1}\cdot 9x^{3}$
71
The factorial of $5$ is
$\frac{120}{2\cdot 1}\cdot 9x^{3}$
72
Any expression multiplied by $1$ is equal to itself
$\frac{120}{2}\cdot 9x^{3}$
73
Calculate the binomial coefficient $\left(\begin{matrix}5\\3\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(3!\right)\left(5-3\right)!}\cdot 27x^{2}$
74
The factorial of $3$ is
$\frac{5!}{6\cdot 1}\cdot 27x^{2}$
75
The factorial of $5$ is
$\frac{120}{6\cdot 1}\cdot 27x^{2}$
76
Any expression multiplied by $1$ is equal to itself
$\frac{120}{6}\cdot 27x^{2}$
77
Calculate the binomial coefficient $\left(\begin{matrix}5\\4\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(4!\right)\left(5-4\right)!}\cdot 81x$
78
The factorial of $4$ is
$\frac{5!}{24\cdot 1}\cdot 81x$
79
The factorial of $5$ is
$\frac{120}{24\cdot 1}\cdot 81x$
80
Any expression multiplied by $1$ is equal to itself
$\frac{120}{24}\cdot 81x$
81
Subtract the values $5$ and $-1$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(5-2\right)!}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(5-3\right)!}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(5-4\right)!}x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243$
82
Subtract the values $5$ and $-2$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(5-3\right)!}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(5-4\right)!}x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243$
83
Subtract the values $5$ and $-3$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(5-4\right)!}x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243$
84
Subtract the values $5$ and $-4$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243$
85
Add the values $5$ and $0$
$\frac{5!}{\left(0!\right)\left(5!\right)}x^{5}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243$
$\frac{1}{0!}x^{5}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243$
87
Multiply the fraction by the term
$\frac{1x^{5}}{0!}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243$
88
Any expression multiplied by $1$ is equal to itself
$\frac{x^{5}}{0!}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243$
89
Calculate the binomial coefficient $\left(\begin{matrix}5\\0\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(0!\right)\left(5+0\right)!}x^{5}$
90
The factorial of $0$ is
$\frac{5!}{1\cdot 1}x^{5}$
91
The factorial of $5$ is
$\frac{120}{1\cdot 1}x^{5}$
92
Any expression multiplied by $1$ is equal to itself
$\frac{120}{1}x^{5}$
93
Any expression divided by one ($1$) is equal to that same expression
$120x^{5}$
94
Calculate the binomial coefficient $\left(\begin{matrix}5\\1\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(1!\right)\left(5-1\right)!}\cdot 3x^{4}$
95
The factorial of $1$ is
$\frac{5!}{1\cdot 1}\cdot 3x^{4}$
96
The factorial of $5$ is
$\frac{120}{1\cdot 1}\cdot 3x^{4}$
97
Any expression multiplied by $1$ is equal to itself
$\frac{120}{1}\cdot 3x^{4}$
98
Any expression divided by one ($1$) is equal to that same expression
$120\cdot 3x^{4}$
99
Calculate the binomial coefficient $\left(\begin{matrix}5\\2\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(2!\right)\left(5-2\right)!}\cdot 9x^{3}$
100
The factorial of $2$ is
$\frac{5!}{2\cdot 1}\cdot 9x^{3}$
101
The factorial of $5$ is
$\frac{120}{2\cdot 1}\cdot 9x^{3}$
102
Any expression multiplied by $1$ is equal to itself
$\frac{120}{2}\cdot 9x^{3}$
103
Calculate the binomial coefficient $\left(\begin{matrix}5\\3\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(3!\right)\left(5-3\right)!}\cdot 27x^{2}$
104
The factorial of $3$ is
$\frac{5!}{6\cdot 1}\cdot 27x^{2}$
105
The factorial of $5$ is
$\frac{120}{6\cdot 1}\cdot 27x^{2}$
106
Any expression multiplied by $1$ is equal to itself
$\frac{120}{6}\cdot 27x^{2}$
107
Calculate the binomial coefficient $\left(\begin{matrix}5\\4\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\frac{5!}{\left(4!\right)\left(5-4\right)!}\cdot 81x$
108
The factorial of $4$ is
$\frac{5!}{24\cdot 1}\cdot 81x$
109
The factorial of $5$ is
$\frac{120}{24\cdot 1}\cdot 81x$
110
Any expression multiplied by $1$ is equal to itself
$\frac{120}{24}\cdot 81x$
111
Calculate the binomial coefficient $\left(\begin{matrix}5\\5\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$\left(\frac{5!}{\left(5!\right)\left(5-5\right)!}\right)\cdot 243$
112
Simplify the fraction
$\left(\frac{1}{\left(5-5\right)!}\right)\cdot 243$
113
Subtract the values $5$ and $-5$
$\frac{x^{5}}{0!}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243\left(5!\right)}{\left(5!\right)\left(0!\right)}$
114
Simplify the fraction
$\frac{x^{5}}{0!}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
115
The factorial of $0$ is
$\frac{x^{5}}{1}+\frac{3\left(5!\right)}{\left(1!\right)\left(4!\right)}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
116
The factorial of $1$ is
$\frac{x^{5}}{1}+\frac{3\left(5!\right)}{1\cdot 24}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
117
The factorial of $5$ is
$\frac{x^{5}}{1}+\frac{3\cdot 120}{1\cdot 24}x^{4}+\frac{9\left(5!\right)}{\left(2!\right)\left(3!\right)}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
118
The factorial of $2$ is
$\frac{x^{5}}{1}+\frac{3\cdot 120}{1\cdot 24}x^{4}+\frac{9\left(5!\right)}{2\cdot 6}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
119
The factorial of $5$ is
$\frac{x^{5}}{1}+\frac{3\cdot 120}{1\cdot 24}x^{4}+\frac{9\cdot 120}{2\cdot 6}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
120
Multiply $1$ times $24$
$\frac{x^{5}}{1}+\frac{3\cdot 120}{24}x^{4}+\frac{9\cdot 120}{2\cdot 6}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
121
Multiply $3$ times $120$
$\frac{x^{5}}{1}+\frac{360}{24}x^{4}+\frac{9\cdot 120}{2\cdot 6}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
122
Multiply $2$ times $6$
$\frac{x^{5}}{1}+\frac{360}{24}x^{4}+\frac{9\cdot 120}{12}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
123
Multiply $9$ times $120$
$\frac{x^{5}}{1}+\frac{360}{24}x^{4}+\frac{1080}{12}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
$\frac{x^{5}}{1}+15x^{4}+\frac{1080}{12}x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
125
Divide $1080$ by $12$
$\frac{x^{5}}{1}+15x^{4}+90x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
126
Any expression divided by one ($1$) is equal to that same expression
$x^{5}+15x^{4}+90x^{3}+\frac{27\left(5!\right)}{\left(3!\right)\left(2!\right)}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
127
The factorial of $3$ is
$x^{5}+15x^{4}+90x^{3}+\frac{27\left(5!\right)}{6\cdot 2}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
128
The factorial of $5$ is
$x^{5}+15x^{4}+90x^{3}+\frac{27\cdot 120}{6\cdot 2}x^{2}+\frac{81\left(5!\right)}{\left(4!\right)\left(1!\right)}x+\frac{243}{0!}$
129
The factorial of $4$ is
$x^{5}+15x^{4}+90x^{3}+\frac{27\cdot 120}{6\cdot 2}x^{2}+\frac{81\left(5!\right)}{24\cdot 1}x+\frac{243}{0!}$
130
The factorial of $5$ is
$x^{5}+15x^{4}+90x^{3}+\frac{27\cdot 120}{6\cdot 2}x^{2}+\frac{81\cdot 120}{24\cdot 1}x+\frac{243}{0!}$
131
The factorial of $0$ is
$x^{5}+15x^{4}+90x^{3}+\frac{27\cdot 120}{6\cdot 2}x^{2}+\frac{81\cdot 120}{24\cdot 1}x+\frac{243}{1}$
132
Multiply $6$ times $2$
$x^{5}+15x^{4}+90x^{3}+\frac{27\cdot 120}{12}x^{2}+\frac{81\cdot 120}{24\cdot 1}x+\frac{243}{1}$
133
Multiply $27$ times $120$
$x^{5}+15x^{4}+90x^{3}+\frac{3240}{12}x^{2}+\frac{81\cdot 120}{24\cdot 1}x+\frac{243}{1}$
134
Multiply $24$ times $1$
$x^{5}+15x^{4}+90x^{3}+\frac{3240}{12}x^{2}+\frac{81\cdot 120}{24}x+\frac{243}{1}$
135
Multiply $81$ times $120$
$x^{5}+15x^{4}+90x^{3}+\frac{3240}{12}x^{2}+\frac{9720}{24}x+\frac{243}{1}$
136
Divide $3240$ by $12$
$x^{5}+15x^{4}+90x^{3}+270x^{2}+\frac{9720}{24}x+\frac{243}{1}$
137
Divide $9720$ by $24$
$x^{5}+15x^{4}+90x^{3}+270x^{2}+405x+\frac{243}{1}$
$x^{5}+15x^{4}+90x^{3}+270x^{2}+405x+243$
Final answer to the problem
$x^{5}+15x^{4}+90x^{3}+270x^{2}+405x+243$