Binomial Theorem with Examples

Binomial Theorem:

$\boxed{(a+b)^n=\begin{pmatrix}n\\0\end{pmatrix}a^nb^0+\begin{pmatrix}n\\1\end{pmatrix}a^{n-1}b^1+\begin{pmatrix}n\\2\end{pmatrix}a^{n-2}b^2+\begin{pmatrix}n\\3\end{pmatrix}a^{n-4}b^4+……..\begin{pmatrix}n\\n-1\end{pmatrix}a^1b^{n-1}+\begin{pmatrix}n\\n\end{pmatrix}a^0b^n}$

where $a$ and $b$ are real numbers and $n\geq1$

$\begin{pmatrix}n\\0\end{pmatrix},\begin{pmatrix}n\\1\end{pmatrix},\begin{pmatrix}n\\2\end{pmatrix},……….,\begin{pmatrix}n\\n\end{pmatrix}$ are binomial cofficient.

$a$ and $b$ are exponents and $n$ is called index.

The exponent of $a$ decreases from index to zero.

The exponent of $b$ 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:

$(a+2b)^5$

HERE

$a=a$

$b=2b$

$n=5$

put in above expansion.

$(a+2b)^5=\begin{pmatrix}5\\0\end{pmatrix}\left(a\right)^5.\left(2b\right)^0+\begin{pmatrix}5\\1\end{pmatrix}\left(a\right)^4.\left(2b\right)^1+\begin{pmatrix}5\\2\end{pmatrix}\left(a\right)^3.\left(2b\right)^2+\begin{pmatrix}5\\3\end{pmatrix}\left(a\right)^2.\left(2b\right)^3+\begin{pmatrix}5\\4\end{pmatrix}\left(a\right)^1.\left(2b\right)^4+\begin{pmatrix}5\\5\end{pmatrix}\left(a\right)^0.\left(2b\right)^5$

$(a+2b)^5=(1).(a)^5.(1)+(5).(a)^4.(2b)+(10).(a)^3.(4b)^2+(10).(a)^2.(8b)^3+(5).(a).(16b)^4+(1).(1).(32b)^5$

$(a+2b)^5=a^5+10a^4b+40a^3b^2+80a^2b^3+80ab^4+32b^5$

you can also see roles theorem

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