Binomial Expansions

(how to find a specific term, or a specific power of \(x\), in a binomial expansion)


We now learn how to find a specific term, or a specific power of \(x\), in a binomial expansion, without writing all of the terms in the expansion.


Typical Question

Find the \(x^4\) term in the expansion of: \[\begin{pmatrix}2x^2 - x\end{pmatrix}^5\]


The Approach

The idea for answering such questions is to work with the general term of the binomial expansion. For instance, looking at \(\begin{pmatrix}2x^2 - x\end{pmatrix}^5\), we know from the binomial expansions formula that we can write: \[\begin{pmatrix}2x^2 - x\end{pmatrix}^5 = \sum_{r=0}^5\begin{pmatrix}5\\r \end{pmatrix}.\begin{pmatrix}2x^2\end{pmatrix}^{5-r}.(-x)^r\] In this case, the general term would be: \[t_r = \begin{pmatrix}5\\r \end{pmatrix}.\begin{pmatrix}2x^2\end{pmatrix}^{5-r}.(-x)^r\] Locating a specific power of \(x\), such as the \(x^4\), in the binomial expansion therefore consists of determining the value of \(r\) at which \(t_r\) corresponds to that power of \(x\). For \(x^4\) that would mean determining the value of \(r\) at which \(t_r = \begin{pmatrix}5\\r \end{pmatrix}.\begin{pmatrix}2x^2\end{pmatrix}^{5-r}.(-x)^r\) is an \(x^4\) term.

We further explain the method for answering such questions through a series of tutorials. These tutorials deal with more and more complicated expansions. Start working through the exercise, further down, as soon as you feel ready.


Tutorial 1

To learn the method, let's see how to find the \(x^8\) term in the expansion of: \[\begin{pmatrix}x^2+2\end{pmatrix}^7\] This is explained in the following tutorial.


Tutorial 2

In this second tutorial we work through another example, in which we find the \(x^7\) term in the expansion of \(\begin{pmatrix}x^2-3x\end{pmatrix}^5\).


Tutorial 3

In this third tutorial we work through another example, in which we find the \(x^8\) term in the expansion of \(\begin{pmatrix}2x^3+\frac{1}{x}\end{pmatrix}^8\).


Tutorial 4

In this fourth tutorial we work through another example, in which we find the constant term in the expansion of \(\begin{pmatrix}2x-\frac{3}{x^2}\end{pmatrix}^9\).


Exercise 1

Using the method described in the tutorials above, answer each of the following:

  1. Find the \(x^6\) term in the expansion of \(\begin{pmatrix}x^3+3\end{pmatrix}^5\).
  2. Find the \(x^{10}\) term in the expansion of \(\begin{pmatrix}x^3+2x\end{pmatrix}^6\).
  3. Find the constant term in the expansion of \(\begin{pmatrix}x^2 + \frac{1}{x}\end{pmatrix}^6\).
    Note: the constant term corresponds to the \(x^0\) term.
  4. Find the \(x\) term in the expansion of \(\begin{pmatrix}2x^2-\frac{1}{x}\end{pmatrix}^5\).
  5. Find the \(x^{-2}\) term in the expansion of \(\begin{pmatrix}2x^5 - \frac{1}{x^2}\end{pmatrix}^8\).
  6. Find the \(x^2\) term in the expansion of \(\begin{pmatrix}2x^3 + \frac{3}{x}\end{pmatrix}^6\).
  7. Find the \(x^{33}\) term in the expansion of \(\begin{pmatrix}x^3 - 2x^5\end{pmatrix}^7\).
  8. Find the \(x^{-3}\) term in the expansion of \(\begin{pmatrix}x^5 + \frac{2}{x^3}\end{pmatrix}^9\).

Note: this exercise can be downloaded as a worksheet to practice with: Worksheet 1

Solution Without Working

We find the following results:

  1. The general term in the expansion of \(\begin{pmatrix}x^3+3\end{pmatrix}^5\) is: \[t_r = \begin{pmatrix}5 \\ r \end{pmatrix}.3^r.x^{15-3r}\] The \(x^6\) term occurs when \(r=3\) and we find: \[t_3 = 270x^6\] So the \(x^6\) term is \(270x^6\).

  2. The general term in the expansion of \(\begin{pmatrix}x^3+2x\end{pmatrix}^6\) is: \[t_r = \begin{pmatrix}6 \\ r\end{pmatrix}2^rx^{18-2r}\] The \(x^{10}\) term occurs when \(r = 4\) and we find: \[t_4 = 240x^{10}\] So the \(x^{10}\) term is \(240x^{10}\).

  3. The general term in the expansion of \(\begin{pmatrix}x^2 + \frac{1}{x}\end{pmatrix}^6\) is: \[t_r = \begin{pmatrix} 6 \\ r \end{pmatrix}.x^{12-3r}\] The constant term, or \(x^0\) term, occurs when \(r = 4\) and we find: \[t_4 = 15\] So the constant term is \(15\).

  4. The general term in the expansion of \(\begin{pmatrix}2x^2-\frac{1}{x}\end{pmatrix}^5\) is: \[t_r = \begin{pmatrix} 5 \\ r \end{pmatrix}2^{5-r}.(-1)^r.x^{10-3r} \] The \(x\) term occurs when \(r = 3\) and we find: \[t_3 = -40x\] So the \(x\) term is \(-40x\).

  5. The general term in the expansion of \(\begin{pmatrix}2x^5 - \frac{1}{x^2}\end{pmatrix}^8\) is: \[t_r = \begin{pmatrix}8 \\ r \end{pmatrix}.2^{8-r}.(-1)^r.x^{40-7r}\] The \(x^{-2}\) occurs when \(r = 6\) and we find: \[t_6 = 112.x^{-2}\] So the \(x^{-2}\) term is \(112.x^{-2}\), which can also be written \(\frac{112}{x^2}\).

  6. The general term in the expansion of \(\begin{pmatrix}2x^3 + \frac{3}{x}\end{pmatrix}^6\) is: \[t_r = \begin{pmatrix}6 \\ r \end{pmatrix}2^{6-r}.3^r.x^{18-4r}\] The \(x^2\) term occurs when \(r = 4\) and we find: \[t_4 = 4860x^2\] So the \(x^2\) term is \(4860x^2\).

  7. The general term in the expansion of \(\begin{pmatrix}x^3 - 2x^5\end{pmatrix}^7\) is: \[t_r = \begin{pmatrix}7\\ r \end{pmatrix}.\begin{pmatrix}-2\end{pmatrix}^r.x^{21+2r}\] The \(x^{33}\) terms occurs when \(r=6\) and we find: \[t_6 = 448x^{33}\] So the \(x^{33}\) term is \(448x^{33}\).

  8. The general term in the expansion of \(\begin{pmatrix}x^5 + \frac{2}{x^3}\end{pmatrix}^9\) is: \[t_r = \begin{pmatrix} 9 \\ r \end{pmatrix}.2^r.x^{45-8r}\] The \(x^{-3}\) term occurs when \(r = 6\) and we find: \[t_6 = 5376x^{-3}\] So the \(x^{-3}\) term is \(5376x^{-3}\), which can also be written \(\frac{5376}{x^3}\)


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