Binomischer Lehrsatz Rechner

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 $3^{0}$

1\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

Calculate the power $3^{1}$

1\left(\begin{matrix}5\\0\end{matrix}\right)x^{5}+3\left(\begin{matrix}5\\1\end{matrix}\right)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}

5

Calculate the power $3^{2}$

1\left(\begin{matrix}5\\0\end{matrix}\right)x^{5}+3\left(\begin{matrix}5\\1\end{matrix}\right)x^{4}+9\left(\begin{matrix}5\\2\end{matrix}\right)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}

6

Calculate the power $3^{3}$

1\left(\begin{matrix}5\\0\end{matrix}\right)x^{5}+3\left(\begin{matrix}5\\1\end{matrix}\right)x^{4}+9\left(\begin{matrix}5\\2\end{matrix}\right)x^{3}+27\left(\begin{matrix}5\\3\end{matrix}\right)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}

7

Calculate the power $3^{4}$

1\left(\begin{matrix}5\\0\end{matrix}\right)x^{5}+3\left(\begin{matrix}5\\1\end{matrix}\right)x^{4}+9\left(\begin{matrix}5\\2\end{matrix}\right)x^{3}+27\left(\begin{matrix}5\\3\end{matrix}\right)x^{2}+81\left(\begin{matrix}5\\4\end{matrix}\right)x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 3^{5}x^{0}

8

Calculate the power $3^{5}$

1\left(\begin{matrix}5\\0\end{matrix}\right)x^{5}+3\left(\begin{matrix}5\\1\end{matrix}\right)x^{4}+9\left(\begin{matrix}5\\2\end{matrix}\right)x^{3}+27\left(\begin{matrix}5\\3\end{matrix}\right)x^{2}+81\left(\begin{matrix}5\\4\end{matrix}\right)x^{1}+243\left(\begin{matrix}5\\5\end{matrix}\right)x^{0}

9

Any expression to the power of $1$ is equal to that same expression

1\left(\begin{matrix}5\\0\end{matrix}\right)x^{5}+3\left(\begin{matrix}5\\1\end{matrix}\right)x^{4}+9\left(\begin{matrix}5\\2\end{matrix}\right)x^{3}+27\left(\begin{matrix}5\\3\end{matrix}\right)x^{2}+81\left(\begin{matrix}5\\4\end{matrix}\right)x+243\left(\begin{matrix}5\\5\end{matrix}\right)x^{0}

10

Any expression multiplied by $1$ is equal to itself

\left(\begin{matrix}5\\0\end{matrix}\right)x^{5}+3\left(\begin{matrix}5\\1\end{matrix}\right)x^{4}+9\left(\begin{matrix}5\\2\end{matrix}\right)x^{3}+27\left(\begin{matrix}5\\3\end{matrix}\right)x^{2}+81\left(\begin{matrix}5\\4\end{matrix}\right)x+243\left(\begin{matrix}5\\5\end{matrix}\right)x^{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}+3\left(\begin{matrix}5\\1\end{matrix}\right)x^{4}+9\left(\begin{matrix}5\\2\end{matrix}\right)x^{3}+27\left(\begin{matrix}5\\3\end{matrix}\right)x^{2}+81\left(\begin{matrix}5\\4\end{matrix}\right)x+243\left(\begin{matrix}5\\5\end{matrix}\right)

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 $1$

\frac{5!}{1\cdot 1}x^{5}

14

The factorial of $5$ is $120$

\frac{120}{1\cdot 1}x^{5}

15

Any expression multiplied by $1$ is equal to itself

120x^{5}

16

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}

17

The factorial of $0$ is $1$

\frac{5!}{1\cdot 1}x^{5}

18

The factorial of $5$ is $120$

\frac{120}{1\cdot 1}x^{5}

19

Any expression multiplied by $1$ is equal to itself

120x^{5}

20

Calculate the binomial coefficient $\left(\begin{matrix}5\\1\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

3\frac{5!}{\left(1!\right)\left(5-1\right)!}x^{4}

21

The factorial of $1$ is $1$

3\frac{5!}{1\cdot 1}x^{4}

22

The factorial of $5$ is $120$

3\frac{120}{1\cdot 1}x^{4}

23

Any expression multiplied by $1$ is equal to itself

120\cdot 3x^{4}

24

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}

25

The factorial of $0$ is $1$

\frac{5!}{1\cdot 1}x^{5}

26

The factorial of $5$ is $120$

\frac{120}{1\cdot 1}x^{5}

27

Any expression multiplied by $1$ is equal to itself

120x^{5}

28

Calculate the binomial coefficient $\left(\begin{matrix}5\\1\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

3\frac{5!}{\left(1!\right)\left(5-1\right)!}x^{4}

29

The factorial of $1$ is $1$

3\frac{5!}{1\cdot 1}x^{4}

30

The factorial of $5$ is $120$

3\frac{120}{1\cdot 1}x^{4}

31

Any expression multiplied by $1$ is equal to itself

120\cdot 3x^{4}

32

Calculate the binomial coefficient $\left(\begin{matrix}5\\2\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

9\frac{5!}{\left(2!\right)\left(5-2\right)!}x^{3}

33

The factorial of $2$ is $2$

9\frac{5!}{2\cdot 1}x^{3}

34

The factorial of $5$ is $120$

9\frac{120}{2\cdot 1}x^{3}

35

Any expression multiplied by $1$ is equal to itself

9\frac{120}{2}x^{3}

36

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}

37

The factorial of $0$ is $1$

\frac{5!}{1\cdot 1}x^{5}

38

The factorial of $5$ is $120$

\frac{120}{1\cdot 1}x^{5}

39

Any expression multiplied by $1$ is equal to itself

120x^{5}

40

Calculate the binomial coefficient $\left(\begin{matrix}5\\1\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

3\frac{5!}{\left(1!\right)\left(5-1\right)!}x^{4}

41

The factorial of $1$ is $1$

3\frac{5!}{1\cdot 1}x^{4}

42

The factorial of $5$ is $120$

3\frac{120}{1\cdot 1}x^{4}

43

Any expression multiplied by $1$ is equal to itself

120\cdot 3x^{4}

44

Calculate the binomial coefficient $\left(\begin{matrix}5\\2\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

9\frac{5!}{\left(2!\right)\left(5-2\right)!}x^{3}

45

The factorial of $2$ is $2$

9\frac{5!}{2\cdot 1}x^{3}

46

The factorial of $5$ is $120$

9\frac{120}{2\cdot 1}x^{3}

47

Any expression multiplied by $1$ is equal to itself

9\frac{120}{2}x^{3}

48

Calculate the binomial coefficient $\left(\begin{matrix}5\\3\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

27\frac{5!}{\left(3!\right)\left(5-3\right)!}x^{2}

49

The factorial of $3$ is $6$

27\frac{5!}{6\cdot 1}x^{2}

50

The factorial of $5$ is $120$

27\frac{120}{6\cdot 1}x^{2}

51

Any expression multiplied by $1$ is equal to itself

27\frac{120}{6}x^{2}

52

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}

53

The factorial of $0$ is $1$

\frac{5!}{1\cdot 1}x^{5}

54

The factorial of $5$ is $120$

\frac{120}{1\cdot 1}x^{5}

55

Any expression multiplied by $1$ is equal to itself

120x^{5}

56

Calculate the binomial coefficient $\left(\begin{matrix}5\\1\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

3\frac{5!}{\left(1!\right)\left(5-1\right)!}x^{4}

57

The factorial of $1$ is $1$

3\frac{5!}{1\cdot 1}x^{4}

58

The factorial of $5$ is $120$

3\frac{120}{1\cdot 1}x^{4}

59

Any expression multiplied by $1$ is equal to itself

120\cdot 3x^{4}

60

Calculate the binomial coefficient $\left(\begin{matrix}5\\2\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

9\frac{5!}{\left(2!\right)\left(5-2\right)!}x^{3}

61

The factorial of $2$ is $2$

9\frac{5!}{2\cdot 1}x^{3}

62

The factorial of $5$ is $120$

9\frac{120}{2\cdot 1}x^{3}

63

Any expression multiplied by $1$ is equal to itself

9\frac{120}{2}x^{3}

64

Calculate the binomial coefficient $\left(\begin{matrix}5\\3\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

27\frac{5!}{\left(3!\right)\left(5-3\right)!}x^{2}

65

The factorial of $3$ is $6$

27\frac{5!}{6\cdot 1}x^{2}

66

The factorial of $5$ is $120$

27\frac{120}{6\cdot 1}x^{2}

67

Any expression multiplied by $1$ is equal to itself

27\frac{120}{6}x^{2}

68

Calculate the binomial coefficient $\left(\begin{matrix}5\\4\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

81\frac{5!}{\left(4!\right)\left(5-4\right)!}x

69

The factorial of $4$ is $24$

81\frac{5!}{24\cdot 1}x

70

The factorial of $5$ is $120$

81\frac{120}{24\cdot 1}x

71

Any expression multiplied by $1$ is equal to itself

81\frac{120}{24}x

72

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+243\left(\begin{matrix}5\\5\end{matrix}\right)

73

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+243\left(\begin{matrix}5\\5\end{matrix}\right)

74

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+243\left(\begin{matrix}5\\5\end{matrix}\right)

75

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+243\left(\begin{matrix}5\\5\end{matrix}\right)

76

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+243\left(\begin{matrix}5\\5\end{matrix}\right)

77

Simplify the fraction $\frac{5!}{\left(0!\right)\left(5!\right)}$ by $5!$

\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+243\left(\begin{matrix}5\\5\end{matrix}\right)

78

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+243\left(\begin{matrix}5\\5\end{matrix}\right)

79

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+243\left(\begin{matrix}5\\5\end{matrix}\right)

80

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}

81

The factorial of $0$ is $1$

\frac{5!}{1\cdot 1}x^{5}

82

The factorial of $5$ is $120$

\frac{120}{1\cdot 1}x^{5}

83

Any expression multiplied by $1$ is equal to itself

120x^{5}

84

Calculate the binomial coefficient $\left(\begin{matrix}5\\1\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

3\frac{5!}{\left(1!\right)\left(5-1\right)!}x^{4}

85

The factorial of $1$ is $1$

3\frac{5!}{1\cdot 1}x^{4}

86

The factorial of $5$ is $120$

3\frac{120}{1\cdot 1}x^{4}

87

Any expression multiplied by $1$ is equal to itself

120\cdot 3x^{4}

88

Calculate the binomial coefficient $\left(\begin{matrix}5\\2\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

9\frac{5!}{\left(2!\right)\left(5-2\right)!}x^{3}

89

The factorial of $2$ is $2$

9\frac{5!}{2\cdot 1}x^{3}

90

The factorial of $5$ is $120$

9\frac{120}{2\cdot 1}x^{3}

91

Any expression multiplied by $1$ is equal to itself

9\frac{120}{2}x^{3}

92

Calculate the binomial coefficient $\left(\begin{matrix}5\\3\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

27\frac{5!}{\left(3!\right)\left(5-3\right)!}x^{2}

93

The factorial of $3$ is $6$

27\frac{5!}{6\cdot 1}x^{2}

94

The factorial of $5$ is $120$

27\frac{120}{6\cdot 1}x^{2}

95

Any expression multiplied by $1$ is equal to itself

27\frac{120}{6}x^{2}

96

Calculate the binomial coefficient $\left(\begin{matrix}5\\4\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

81\frac{5!}{\left(4!\right)\left(5-4\right)!}x

97

The factorial of $4$ is $24$

81\frac{5!}{24\cdot 1}x

98

The factorial of $5$ is $120$

81\frac{120}{24\cdot 1}x

99

Any expression multiplied by $1$ is equal to itself

81\frac{120}{24}x

100

Calculate the binomial coefficient $\left(\begin{matrix}5\\5\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$

243\left(\frac{5!}{\left(5!\right)\left(5-5\right)!}\right)

101

Simplify the fraction $\frac{5!}{\left(5!\right)\left(5-5\right)!}$ by $5!$

243\left(\frac{1}{\left(5-5\right)!}\right)

102

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)}

103

Simplify the fraction $\frac{243\left(5!\right)}{\left(5!\right)\left(0!\right)}$ by $5!$

\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}{0!}

104

The factorial of $0$ is $1$

\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!}

105

The factorial of $1$ is $1$

\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!}

106

The factorial of $5$ is $120$

\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!}

107

The factorial of $2$ is $2$

\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!}

108

The factorial of $5$ is $120$

\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!}

109

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!}

110

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!}

111

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!}

112

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!}

113

Divide $360$ by $24$

\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!}

114

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!}

115

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!}

116

The factorial of $3$ is $6$

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!}

117

The factorial of $5$ is $120$

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!}

118

The factorial of $4$ is $24$

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!}

119

The factorial of $5$ is $120$

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!}

120

The factorial of $0$ is $1$

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}

121

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}

122

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}

123

Multiply $24$ times $1$

x^{5}+15x^{4}+90x^{3}+\frac{3240}{12}x^{2}+\frac{81\cdot 120}{24}x+\frac{243}{1}

124

Multiply $81$ times $120$

x^{5}+15x^{4}+90x^{3}+\frac{3240}{12}x^{2}+\frac{9720}{24}x+\frac{243}{1}

125

Divide $3240$ by $12$

x^{5}+15x^{4}+90x^{3}+270x^{2}+\frac{9720}{24}x+\frac{243}{1}

126

Divide $9720$ by $24$

x^{5}+15x^{4}+90x^{3}+270x^{2}+405x+\frac{243}{1}

127

Divide $243$ by $1$

x^{5}+15x^{4}+90x^{3}+270x^{2}+405x+243

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