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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

\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}

HERE

a=a

b=2b

n=5
Binomial Theorem

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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