Question

What is the binomial theorem?

Answer 1

Binomial theorem:

• Binomial theorem describes the algebraic expansion of the power of a binomial.
• It is possible to expand any non-negative intergers powers of a binomial.
• The binomial theorem is: ${\left(a+b\right)}^{\mathbf{n}}\mathbf{=}{{}^{\mathbf{n}}\mathbf{C}}_{\mathbf{0}}{\mathbf{a}}^{\mathbf{n}}{\mathbf{b}}^{\mathbf{0}}\mathbf{+}{{}^{\mathbf{n}}\mathbf{C}}_{\mathbf{1}}{\mathbf{a}}^{\mathbf{n}\mathbf{-}\mathbf{1}}{\mathbf{b}}^{\mathbf{1}}\mathbf{+}{{}^{\mathbf{n}}\mathbf{C}}_{\mathbf{2}}{\mathbf{a}}^{\mathbf{n}\mathbf{-}\mathbf{2}}{\mathbf{b}}^{\mathbf{2}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}{{}^{\mathbf{n}}\mathbf{C}}_{\mathbf{n}}{\mathbf{a}}^{\mathbf{0}}{\mathbf{b}}^{\mathbf{n}\mathbf{}}$.
• Using the summation notation, it can be written as ${\left(a+b\right)}^{\mathbf{n}}\mathbf{=}\mathbf{\sum }_{\mathbf{r}\mathbf{=}\mathbf{0}}^{\mathbf{n}}{{}^{\mathbf{n}}\mathbf{C}}_{\mathbf{r}}{\mathbf{a}}^{\mathbf{n}\mathbf{-}\mathbf{r}}{\mathbf{b}}^{\mathbf{r}}$, where $n\ge 0,{{}^{n}C}_r$is always a positive integer.
• A Pascal's triangle shows the coefficients for each power of the binomial

Hence, binomial theorem is explained.