1
Solved example of binomial theorem
$\left(x+3\right)^5$
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.
$\left(\begin{matrix}5\\0\end{matrix}\right)x^{5}3^{0}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)x^{3}3^{2}+\left(\begin{matrix}5\\3\end{matrix}\right)x^{2}3^{3}+\left(\begin{matrix}5\\4\end{matrix}\right)x^{1}3^{4}+\left(\begin{matrix}5\\5\end{matrix}\right)x^{0}3^{5}$
3
Calculate the power $3^{0}$
$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 1x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)x^{3}3^{2}+\left(\begin{matrix}5\\3\end{matrix}\right)x^{2}3^{3}+\left(\begin{matrix}5\\4\end{matrix}\right)x^{1}3^{4}+\left(\begin{matrix}5\\5\end{matrix}\right)x^{0}3^{5}$
4
Calculate the power $3^{2}$
$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 1x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)x^{2}3^{3}+\left(\begin{matrix}5\\4\end{matrix}\right)x^{1}3^{4}+\left(\begin{matrix}5\\5\end{matrix}\right)x^{0}3^{5}$
5
Calculate the power $3^{3}$
$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 1x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)x^{1}3^{4}+\left(\begin{matrix}5\\5\end{matrix}\right)x^{0}3^{5}$
6
Calculate the power $3^{4}$
$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 1x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 81x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)x^{0}3^{5}$
7
Calculate the power $3^{5}$
$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 1x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 81x^{1}+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243x^{0}$
8
Any expression to the power of $1$ is equal to that same expression
$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot 1x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 81x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243x^{0}$
9
Any expression multiplied by $1$ is equal to itself
$\left(\begin{matrix}5\\0\end{matrix}\right)x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 81x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 243x^{0}$
10
Any expression (except $0$ and $\infty$) to the power of $0$ is equal to $1$
$\left(\begin{matrix}5\\0\end{matrix}\right)x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 81x+\left(\begin{matrix}5\\5\end{matrix}\right)\cdot 1\cdot 243$
11
Multiply $1$ times $243$
$\left(\begin{matrix}5\\0\end{matrix}\right)x^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 81x+243\cdot \left(\begin{matrix}5\\5\end{matrix}\right)$
12
Calculate the binomial coefficient $\left(\begin{matrix}5\\0\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$x^{5}\frac{5!}{\left(0!\right)\left(5!\right)}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot 3x^{4}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 81x+243\cdot \left(\begin{matrix}5\\5\end{matrix}\right)$
13
Calculate the binomial coefficient $\left(\begin{matrix}5\\1\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$x^{5}\frac{5!}{\left(0!\right)\left(5!\right)}+3x^{4}\frac{5!}{\left(1!\right)\left(4!\right)}+\left(\begin{matrix}5\\2\end{matrix}\right)\cdot 9x^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 81x+243\cdot \left(\begin{matrix}5\\5\end{matrix}\right)$
14
Calculate the binomial coefficient $\left(\begin{matrix}5\\2\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$x^{5}\frac{5!}{\left(0!\right)\left(5!\right)}+3x^{4}\frac{5!}{\left(1!\right)\left(4!\right)}+9x^{3}\frac{5!}{\left(2!\right)\left(3!\right)}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot 27x^{2}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 81x+243\cdot \left(\begin{matrix}5\\5\end{matrix}\right)$
15
Calculate the binomial coefficient $\left(\begin{matrix}5\\3\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$x^{5}\frac{5!}{\left(0!\right)\left(5!\right)}+3x^{4}\frac{5!}{\left(1!\right)\left(4!\right)}+9x^{3}\frac{5!}{\left(2!\right)\left(3!\right)}+27x^{2}\frac{5!}{\left(3!\right)\left(2!\right)}+\left(\begin{matrix}5\\4\end{matrix}\right)\cdot 81x+243\cdot \left(\begin{matrix}5\\5\end{matrix}\right)$
16
Calculate the binomial coefficient $\left(\begin{matrix}5\\4\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$x^{5}\frac{5!}{\left(0!\right)\left(5!\right)}+3x^{4}\frac{5!}{\left(1!\right)\left(4!\right)}+9x^{3}\frac{5!}{\left(2!\right)\left(3!\right)}+27x^{2}\frac{5!}{\left(3!\right)\left(2!\right)}+81x\frac{5!}{\left(4!\right)\left(1!\right)}+243\cdot \left(\begin{matrix}5\\5\end{matrix}\right)$
17
Calculate the binomial coefficient $\left(\begin{matrix}5\\5\end{matrix}\right)$ applying the formula: $\left(\begin{matrix}n\\k\end{matrix}\right)=\frac{n!}{k!(n-k)!}$
$x^{5}\frac{5!}{\left(0!\right)\left(5!\right)}+3x^{4}\frac{5!}{\left(1!\right)\left(4!\right)}+9x^{3}\frac{5!}{\left(2!\right)\left(3!\right)}+27x^{2}\frac{5!}{\left(3!\right)\left(2!\right)}+81x\frac{5!}{\left(4!\right)\left(1!\right)}+243\left(\frac{5!}{\left(5!\right)\left(0!\right)}\right)$
18
Simplify the fraction $\frac{5!}{\left(0!\right)\left(5!\right)}$ by $5!$
$x^{5}\frac{1}{0!}+3x^{4}\frac{5!}{\left(1!\right)\left(4!\right)}+9x^{3}\frac{5!}{\left(2!\right)\left(3!\right)}+27x^{2}\frac{5!}{\left(3!\right)\left(2!\right)}+81x\frac{5!}{\left(4!\right)\left(1!\right)}+243\left(\frac{5!}{\left(5!\right)\left(0!\right)}\right)$
19
Simplify the fraction $\frac{5!}{\left(5!\right)\left(0!\right)}$ by $5!$
$x^{5}\frac{1}{0!}+3x^{4}\frac{5!}{\left(1!\right)\left(4!\right)}+9x^{3}\frac{5!}{\left(2!\right)\left(3!\right)}+27x^{2}\frac{5!}{\left(3!\right)\left(2!\right)}+81x\frac{5!}{\left(4!\right)\left(1!\right)}+243\left(\frac{1}{0!}\right)$
20
Multiply the fraction and term
$\frac{x^{5}}{0!}+3x^{4}\frac{5!}{\left(1!\right)\left(4!\right)}+9x^{3}\frac{5!}{\left(2!\right)\left(3!\right)}+27x^{2}\frac{5!}{\left(3!\right)\left(2!\right)}+81x\frac{5!}{\left(4!\right)\left(1!\right)}+243\left(\frac{1}{0!}\right)$
21
Multiply the fraction and term
$\frac{x^{5}}{0!}+3x^{4}\frac{5!}{\left(1!\right)\left(4!\right)}+9x^{3}\frac{5!}{\left(2!\right)\left(3!\right)}+27x^{2}\frac{5!}{\left(3!\right)\left(2!\right)}+81x\frac{5!}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
22
Multiplying the fraction by $3x^{4}$
$\frac{x^{5}}{0!}+\frac{3\left(5!\right)\left(x^{4}\right)}{\left(1!\right)\left(4!\right)}+9x^{3}\frac{5!}{\left(2!\right)\left(3!\right)}+27x^{2}\frac{5!}{\left(3!\right)\left(2!\right)}+81x\frac{5!}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
23
Multiplying the fraction by $9x^{3}$
$\frac{x^{5}}{0!}+\frac{3\left(5!\right)\left(x^{4}\right)}{\left(1!\right)\left(4!\right)}+\frac{9\left(5!\right)\left(x^{3}\right)}{\left(2!\right)\left(3!\right)}+27x^{2}\frac{5!}{\left(3!\right)\left(2!\right)}+81x\frac{5!}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
24
Multiplying the fraction by $27x^{2}$
$\frac{x^{5}}{0!}+\frac{3\left(5!\right)\left(x^{4}\right)}{\left(1!\right)\left(4!\right)}+\frac{9\left(5!\right)\left(x^{3}\right)}{\left(2!\right)\left(3!\right)}+\frac{27\left(5!\right)\left(x^{2}\right)}{\left(3!\right)\left(2!\right)}+81x\frac{5!}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
25
Multiplying the fraction by $81x$
$\frac{x^{5}}{0!}+\frac{3\left(5!\right)\left(x^{4}\right)}{\left(1!\right)\left(4!\right)}+\frac{9\left(5!\right)\left(x^{3}\right)}{\left(2!\right)\left(3!\right)}+\frac{27\left(5!\right)\left(x^{2}\right)}{\left(3!\right)\left(2!\right)}+\frac{81x\left(5!\right)}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
26
The factorial of $0$ is $1$
$\frac{x^{5}}{1}+\frac{3\left(5!\right)\left(x^{4}\right)}{\left(1!\right)\left(4!\right)}+\frac{9\left(5!\right)\left(x^{3}\right)}{\left(2!\right)\left(3!\right)}+\frac{27\left(5!\right)\left(x^{2}\right)}{\left(3!\right)\left(2!\right)}+\frac{81x\left(5!\right)}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
27
The factorial of $1$ is $1$
$\frac{x^{5}}{1}+\frac{3\left(5!\right)\left(x^{4}\right)}{1\left(4!\right)}+\frac{9\left(5!\right)\left(x^{3}\right)}{\left(2!\right)\left(3!\right)}+\frac{27\left(5!\right)\left(x^{2}\right)}{\left(3!\right)\left(2!\right)}+\frac{81x\left(5!\right)}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
28
The factorial of $4$ is $24$
$\frac{x^{5}}{1}+\frac{3\left(5!\right)\left(x^{4}\right)}{1\cdot 24}+\frac{9\left(5!\right)\left(x^{3}\right)}{\left(2!\right)\left(3!\right)}+\frac{27\left(5!\right)\left(x^{2}\right)}{\left(3!\right)\left(2!\right)}+\frac{81x\left(5!\right)}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
29
The factorial of $5$ is $120$
$\frac{x^{5}}{1}+\frac{120\cdot 3x^{4}}{1\cdot 24}+\frac{9\left(5!\right)\left(x^{3}\right)}{\left(2!\right)\left(3!\right)}+\frac{27\left(5!\right)\left(x^{2}\right)}{\left(3!\right)\left(2!\right)}+\frac{81x\left(5!\right)}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
30
The factorial of $2$ is $2$
$\frac{x^{5}}{1}+\frac{120\cdot 3x^{4}}{1\cdot 24}+\frac{9\left(5!\right)\left(x^{3}\right)}{2\left(3!\right)}+\frac{27\left(5!\right)\left(x^{2}\right)}{\left(3!\right)\left(2!\right)}+\frac{81x\left(5!\right)}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
31
The factorial of $3$ is $6$
$\frac{x^{5}}{1}+\frac{120\cdot 3x^{4}}{1\cdot 24}+\frac{9\left(5!\right)\left(x^{3}\right)}{2\cdot 6}+\frac{27\left(5!\right)\left(x^{2}\right)}{\left(3!\right)\left(2!\right)}+\frac{81x\left(5!\right)}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
32
The factorial of $5$ is $120$
$\frac{x^{5}}{1}+\frac{120\cdot 3x^{4}}{1\cdot 24}+\frac{120\cdot 9x^{3}}{2\cdot 6}+\frac{27\left(5!\right)\left(x^{2}\right)}{\left(3!\right)\left(2!\right)}+\frac{81x\left(5!\right)}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
33
The factorial of $3$ is $6$
$\frac{x^{5}}{1}+\frac{120\cdot 3x^{4}}{1\cdot 24}+\frac{120\cdot 9x^{3}}{2\cdot 6}+\frac{27\left(5!\right)\left(x^{2}\right)}{6\left(2!\right)}+\frac{81x\left(5!\right)}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
34
The factorial of $2$ is $2$
$\frac{x^{5}}{1}+\frac{120\cdot 3x^{4}}{1\cdot 24}+\frac{120\cdot 9x^{3}}{2\cdot 6}+\frac{27\left(5!\right)\left(x^{2}\right)}{6\cdot 2}+\frac{81x\left(5!\right)}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
35
The factorial of $5$ is $120$
$\frac{x^{5}}{1}+\frac{120\cdot 3x^{4}}{1\cdot 24}+\frac{120\cdot 9x^{3}}{2\cdot 6}+\frac{120\cdot 27x^{2}}{6\cdot 2}+\frac{81x\left(5!\right)}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
36
Multiply $1$ times $24$
$\frac{x^{5}}{1}+\frac{120\cdot 3x^{4}}{24}+\frac{120\cdot 9x^{3}}{2\cdot 6}+\frac{120\cdot 27x^{2}}{6\cdot 2}+\frac{81x\left(5!\right)}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
37
Multiply $120$ times $3$
$\frac{x^{5}}{1}+\frac{360x^{4}}{24}+\frac{120\cdot 9x^{3}}{2\cdot 6}+\frac{120\cdot 27x^{2}}{6\cdot 2}+\frac{81x\left(5!\right)}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
38
Multiply $2$ times $6$
$\frac{x^{5}}{1}+\frac{360x^{4}}{24}+\frac{120\cdot 9x^{3}}{12}+\frac{120\cdot 27x^{2}}{6\cdot 2}+\frac{81x\left(5!\right)}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
39
Multiply $120$ times $9$
$\frac{x^{5}}{1}+\frac{360x^{4}}{24}+\frac{1080x^{3}}{12}+\frac{120\cdot 27x^{2}}{6\cdot 2}+\frac{81x\left(5!\right)}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
40
Multiply $6$ times $2$
$\frac{x^{5}}{1}+\frac{360x^{4}}{24}+\frac{1080x^{3}}{12}+\frac{120\cdot 27x^{2}}{12}+\frac{81x\left(5!\right)}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
41
Multiply $120$ times $27$
$\frac{x^{5}}{1}+\frac{360x^{4}}{24}+\frac{1080x^{3}}{12}+\frac{3240x^{2}}{12}+\frac{81x\left(5!\right)}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
42
Any expression divided by one ($1$) is equal to that same expression
$x^{5}+15x^{4}+90x^{3}+270x^{2}+\frac{81x\left(5!\right)}{\left(4!\right)\left(1!\right)}+\frac{243}{0!}$
43
The factorial of $4$ is $24$
$x^{5}+15x^{4}+90x^{3}+270x^{2}+\frac{81x\left(5!\right)}{24\left(1!\right)}+\frac{243}{0!}$
44
The factorial of $1$ is $1$
$x^{5}+15x^{4}+90x^{3}+270x^{2}+\frac{81x\left(5!\right)}{24\cdot 1}+\frac{243}{0!}$
45
The factorial of $5$ is $120$
$x^{5}+15x^{4}+90x^{3}+270x^{2}+\frac{120\cdot 81x}{24\cdot 1}+\frac{243}{0!}$
46
The factorial of $0$ is $1$
$x^{5}+15x^{4}+90x^{3}+270x^{2}+\frac{120\cdot 81x}{24\cdot 1}+\frac{243}{1}$
47
Multiply $24$ times $1$
$x^{5}+15x^{4}+90x^{3}+270x^{2}+\frac{120\cdot 81x}{24}+\frac{243}{1}$
48
Multiply $120$ times $81$
$x^{5}+15x^{4}+90x^{3}+270x^{2}+\frac{9720x}{24}+\frac{243}{1}$
$x^{5}+15x^{4}+90x^{3}+270x^{2}+\frac{9720x}{24}+243$
50
Take $\frac{9720}{24}$ out of the fraction
$x^{5}+15x^{4}+90x^{3}+270x^{2}+405x+243$
Final Answer
$x^{5}+15x^{4}+90x^{3}+270x^{2}+405x+243$