Algebra and Trigonometry, 5th Edition

11.7 The Binomial Theorem

• Expand a power of a binomial using Pascal’s triangle or factorial notation.

• Find a specific term of a binomial expansion.

• Find the total number of subsets of a set of n objects.

In this section, we consider ways of expanding a binomial ${(a+b)}^n$.

Binomial Expansion Using Pascal’s Triangle

Consider the following expanded powers of ${(a+b)}^n$, where $a+b$ is any binomial and n is a whole number. Look for patterns.

$$\begin{array}{rcc}{(a+b)}^0& =& 1\\ {(a+b)}^1& =& a+b\\ {(a+b)}^2& =& a^2+2ab+b^2\\ {(a+b)}^3& =& a^3+3a^{2}b+3a{b}^2+b^3\\ {(a+b)}^4& =& a^4+4a^{3}b+6a^2{b}^2+4a{b}^3+b^4\\ {(a+b)}^5& =& a^5+5a^{4}b+10a^3{b}^2+10a^2{b}^3+5a{b}^4+b^5\end{array}$$

Each expansion is a polynomial. There are some patterns to be noted.

1. There is one more term than the power of the exponent, n. That is, there are $n+1$ terms in the expansion of ${(a+b)}^n$.

2. In each ...