Investigation 9.4. Powers of Binomials.
In this investigation we will look for patterns in the expansion of \((a+b)^n\text{.}\) We begin by computing several such powers.
Expand each power and fill in the blanks. Arrange the terms in each expansion in descending powers of \(a\text{.}\)
 \((a+b)^0 = \)
 \((a+b)^1 = \)
 \((a+b)^2 = \)

\((a+b)^3 = \)Hint: Start by writing\begin{equation*} (a+b)^3 = (a+b)(a+b)^2 \end{equation*}and use your answer to #3.

\((a+b)^4 = \)Hint: Start by writing\begin{equation*} (a+b)^4 = (a+b)(a+b)^4 \end{equation*}and use your answer to #4.
 \((a+b)^5 = \)

Do you see a relationship between the exponent \(n\) and the number of terms in the expansion of \((a+b)^n\text{?}\) (Notice that for \(n=0\) we have \((a+b)^0=1\text{,}\) which has one term.) Fill in the table below.
\(n\) Number of terms
in \((a+b)^n\)\(0\) \(\hphantom{0000}\) \(1\) \(\hphantom{0000}\) \(2\) \(\hphantom{0000}\) \(3\) \(\hphantom{0000}\) \(4\) \(\hphantom{0000}\) \(5\) \(\hphantom{0000}\)  First observation: In general, the expansion of \((a+b)^n\) has terms.

Next we’ll consider the exponents on \(a\) and \(b\) in each term of the expansions. Refer to your expanded powers in parts 15, and fill in the next table.
\(n\) First term of
\((a+b)^n\)Last term of
\((a+b)^n\)Sum of exponents
on \(a\) and \(b\)
in each term\(0\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(1\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(2\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(3\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(4\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(5\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\)  Second observation: In any term of the expansion of \((a+b)^n\text{,}\) the sum of the exponents on \(a\) and on \(b\) is

In fact, we can be more specific in describing the exponents in the expansions. We will use \(k\) to label the terms in the expansion of \((a+b)^n\text{,}\) starting with \(k=0\text{.}\) For example, for \(n=2\) we label the terms as follows: .\begin{equation*} \begin{aligned}[t] (a+b)^2 = \amp ~~a^2~~~~+~~~~2ab~~~~+~~~~b^2\\ \amp k=0~~~~~~~~k=1~~~~~~~~k=2 \\ \end{aligned} \end{equation*}We can make a table showing the exponents on \(a\) and \(b\) in each term of \((a+b)^2\text{:}\)Case \(n=2\text{:}\)
\(k\) \(0\) \(1\) \(2\) Exponent on \(a\) \(2\) \(1\) \(0\) Exponent on \(b\) \(0\) \(1\) \(2\) Complete the tables shown below for the cases \(n=3,~ n=4\) and \(n=5\text{.}\)Case \(n=3\text{:}\)\(k\) \(0\) \(1\) \(2\) \(3\) Exponent on \(a\) \(3\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) Exponent on \(b\) \(0\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) Case \(n=4\text{:}\)\(k\) \(0\) \(1\) \(2\) \(3\) \(4\) Exponent on \(a\) \(4\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) Exponent on \(b\) \(0\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) Case \(n=5\text{:}\)\(k\) \(0\) \(1\) \(2\) \(3\) \(4\) \(5\) Exponent on \(a\) \(4\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) Exponent on \(b\) \(0\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\)  Third observation: The variable factors of the \(k^{\text{th}}\) term in the expansion of \((a+b)^n\) may be expressed as . (Fill in the correct powers in terms of \(k\) and \(n\) for \(a\) and \(b\text{.}\))
In the next investigation we will look for patterns in the coefficients of the terms of the expansions.