Basic Technical Mathematics, 11th Edition

19.4 The Binomial Theorem

• Properties of ${\mathbf{(\hspace{0.17em}}\mathit{a}\mathbf{+\hspace{0.17em}}\mathit{b}\mathbf{)\hspace{0.17em}}}^{\mathit{n}}$ • Factorial Notation • Binomial Theorem • Pascal’s Triangle • Binomial Series

To expand ${(x\hspace{0.17em}+\hspace{0.17em}2)}^5$ would require a number of multiplications that would be a tedious operation. We now develop the binomial theorem, by which it is possible to expand binomials to any given power without direct multiplication. Using this type of expansion, it is also possible to expand some expressions for which direct multiplication is not possible. The binomial theorem is used to develop expressions needed in certain mathematics topics and in technical applications.

By direct multiplication, we may obtain the following expansions of the binomial $a\hspace{0.17em}+\hspace{0.17em}b\hspace{0.17em}:\hspace{0.17em}$

$$\begin{array}{l}{(a\hspace{0.17em}+\hspace{0.17em}b)}^0\hspace{0.17em}=\hspace{0.17em}1\hfill \\ {(a\hspace{0.17em}+\hspace{0.17em}b)}^1\hspace{0.17em}=\hspace{0.17em}a\hspace{0.17em}+\hspace{0.17em}b\hfill \\ {(a\hspace{0.17em}+\hspace{0.17em}b)}^2\hspace{0.17em}=\hspace{0.17em}{a}^2\hspace{0.17em}+\hspace{0.17em}2ab\hspace{0.17em}+\hspace{0.17em}{b}^2\hfill \\ {(a\hspace{0.17em}+\hspace{0.17em}\; ...}^{}\hfill \end{array}$$