# Binomial Expansion – negative & fractional powers

This page details the more advanced use of binomial expansion. You should be familiar with all of the material from the more basic Binomial Expansion page first.

Recall that the first formula provided in the Edexcel formula booklet is:

$(a+b)^n=a^n+\left(\begin{array}{c}n\\1\end{array}\right)a^{n-1}b+\left(\begin{array}{c}n\\2\end{array}\right)a^{n-2}b^2+…+\left(\begin{array}{c}n\\r\end{array}\right)a^{n-r}b^r+…+b^n, \hspace{20pt}\left(n\in{\mathbb N}\right)$

where $\left(\begin{array}{c} n\\r\end{array}\right)=\frac{n!}{r!(n-r)!}$. However, this formula is only valid for positive integer $n$.

In addition to this, the booklet also provides a second formula for negative and fractional powers:

$\left(1+x\right)^n=1+nx+\frac{n(n-1)}{1\times 2}x^2+…+\frac{n(n-1)…(n-r+1)}{1\times 2\times …\times r}x^r+…,\hspace{20pt}\left(\vert x\vert <1, n\in {\mathbb R}\right)$

The first formula is only valid for positive integer $n$ but this formula is valid for all $n$. This includes negative and fractional powers. Note, however, the formula is not valid for all values of $x$. As stated, the $x$ values must be between -1 and 1.

### Range of Validity for Binomial Expansions

As stated above, the second formula for binomial expansion in the Edexcel Formula Booklet is only valid for $\vert x\vert <1$. This is because, unlike for positive integer $n$, these expansions have an infinite number of terms (as indicted by the … in the formula). Subsequently, we require the series to converge as the powers of $x$ become large. For this to happen, we must have $\vert x\vert <1$. Questions may ask you to find the binomial expansion without explicitly stating the value of $n$ and ask you to identify the values for which the expansion is valid. See the Identifying the Power Example.

Also notice that in this second formula there is a very specific format inside the brackets – it must be 1 plus something. Therefore, if there is something other than 1 inside these brackets, the coefficient must be factored out. Do this by first writing $(a+bx)^n=\left(a\left(1+\frac{bx}{a}\right)\right)^n=a^n\left(1+\frac{bx}{a}\right)^n$. Then find the expansion of $\left(1+\frac{bx}{a}\right)^n$ using the formula. Do this by replacing all $x$ with $\frac{bx}{a}$. This inevitably changes the range of validity. It follows that this expansion will be valid for $\left\vert \frac{bx}{a}\right\vert <1$ or $\vert x\vert <\frac{a}{b}$. See the Factoring Out Example.

At more advanced levels, questions may ask you to use partial fractions first. See the Using Partial Fractions question.

## Binomial Expansion Examples

1. Find the first four terms in ascending powers of $x$ of the binomial expansion of $\frac{1}{(1+2x)^2}$.
2. State the range of validity for your expansion.

Find the first three terms, in ascending powers of $x$, of the expansion of $\sqrt{4-3x}$. State the range of validity of your expansion and use it to find an approximation to $\sqrt{3.7}$.

1. Express $f(x)=\frac{3+5x}{(1-x)(1+\frac{1}{2}x)}$ as partial fractions.
2. Show that the quadratic approximation to $f(x)$ is given by $f(x)\approx 3+\frac{13}{2}x+\frac{19}{4}x^2$.
3. State the range of values of $x$ for which this approximation is valid.