# 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:

## 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 $$\frac{5376}{x^3}$$