Binomial Theorem:
[katex]\boxed{(a+b)^n=\begin{pmatrix}n\\0\end{pmatrix}a^nb^0+\begin{pmatrix}n\\1\end{pmatrix}a^{n-1}b^1+\begin{pmatrix}n\\2\end{pmatrix}a^{n-2}b^2+\begin{pmatrix}n\\3\end{pmatrix}a^{n-4}b^4+……..\begin{pmatrix}n\\n-1\end{pmatrix}a^1b^{n-1}+\begin{pmatrix}n\\n\end{pmatrix}a^0b^n}[/katex]
where [katex]a[/katex] and [katex]b[/katex] are real numbers and [katex]n\geq1[/katex]
- [katex]\begin{pmatrix}n\\0\end{pmatrix},\begin{pmatrix}n\\1\end{pmatrix},\begin{pmatrix}n\\2\end{pmatrix},……….,\begin{pmatrix}n\\n\end{pmatrix}[/katex] are binomial cofficient.
- [katex]a[/katex] and [katex]b[/katex] are exponents and [katex]n[/katex] is called index.
- The exponent of [katex]a[/katex] decreases from index to zero.
- The exponent of [katex]b[/katex] increases from zero to index.
- The degree of each terms is equal to its index.
- The numbers of terms in the expansion is one greater than its index.
Example
Using Binomial Theorem:
[katex](a+2b)^5[/katex]
[katex]\boxed{(a+b)^n=\begin{pmatrix}n\\0\end{pmatrix}a^nb^0+\begin{pmatrix}n\\1\end{pmatrix}a^{n-1}b^1+\begin{pmatrix}n\\2\end{pmatrix}a^{n-2}b^2+\begin{pmatrix}n\\3\end{pmatrix}a^{n-4}b^4+……..\begin{pmatrix}n\\n-1\end{pmatrix}a^1b^{n-1}+\begin{pmatrix}n\\n\end{pmatrix}a^0b^n}[/katex]
HERE
[katex]a=a[/katex]
[katex]b=2b[/katex]
[katex]n=5[/katex]
put in above expansion.
[katex](a+2b)^5=\begin{pmatrix}5\\0\end{pmatrix}\left(a\right)^5.\left(2b\right)^0+\begin{pmatrix}5\\1\end{pmatrix}\left(a\right)^4.\left(2b\right)^1+\begin{pmatrix}5\\2\end{pmatrix}\left(a\right)^3.\left(2b\right)^2+\begin{pmatrix}5\\3\end{pmatrix}\left(a\right)^2.\left(2b\right)^3+\begin{pmatrix}5\\4\end{pmatrix}\left(a\right)^1.\left(2b\right)^4+\begin{pmatrix}5\\5\end{pmatrix}\left(a\right)^0.\left(2b\right)^5[/katex]
[katex](a+2b)^5=(1).(a)^5.(1)+(5).(a)^4.(2b)+(10).(a)^3.(4b)^2+(10).(a)^2.(8b)^3+(5).(a).(16b)^4+(1).(1).(32b)^5[/katex]
[katex](a+2b)^5=a^5+10a^4b+40a^3b^2+80a^2b^3+80ab^4+32b^5[/katex]
you can also see roles theorem