Binomial Expansion (Edexcel IGCSE Further Pure Maths): Revision Note

Binomial Expansion

What is the Binomial Expansion?

• The binomial expansion gives a method for expanding a two-term expression in a bracket raised to a power

  • For example  ${\left(a+b\right)}^n$

  • You may also see it referred to as the binomial theorem
    

  • In this note $n$ will be a positive integer

    • See the 'General Binomial Expansion' revision note for the general case

• To expand a bracket with a two-term expression in it:

  • Determine what $a$ and $b$ are for your example

  • Then use the formula for the binomial expansion

    • ${\left(a+b\right)}^n=a^n+\left(\begin{array}{c}n\\ 1\end{array}\right)a^{n-1}b+\dots +\left(\begin{array}{c}n\\ r\end{array}\right)a^{n-r}{b}^r+\dots +b^n$

  • $\left(\begin{array}{c}n\\ r\end{array}\right)$  in the formula is known as the binomial coefficient

    • $\left(\begin{array}{c}n\\ r\end{array}\right)=\frac{n!}{r!\left(n-r\right)!}=\frac{n\left(n-1\right)...\left(n-r+1\right)}{r!}$

    • $n!$ ("$n$ factorial") is defined by $n!=1\times 2\times 3\times ...\times n$

    • You may also see $\left(\begin{array}{c}n\\ r\end{array}\right)$  written as ${}_{ }{}^n{\mathrm{C}}_r$

    • Your calculator should be able to calculate $\left(\begin{array}{c}n\\ r\end{array}\right)$ for you

    • Or you can use Pascal's triangle (see the next section)

• To get all the terms

  • Start with $r=0$

  • Then use  $r=1$, $r=2$,...  until you get up to $r=n$

  • So there will always be $\mathit{n}\mathbf{+}\mathbf{1}$ terms in the full expansion

• This version of the binomial expansion formula is not on the exam formula sheet
  

  • But it is a special case of the Binomial Series formula for ${\left(1+x\right)}^n$ which is on the formula sheet

  • See the 'General Binomial Expansion' revision note

• When expanding something like ${\left(p+qx\right)}^n$ you may only be asked to find the first few terms of an expansion

  • Check whether the question wants ascending or descending powers of x

    • For ascending powers start with the constant term, $p^n$

    • For descending powers start with the term with $x$, ${\left(qx\right)}^n$

    • Choosing $a$ and $b$ appropriately will make it easier to follow the formula above

• If you are not writing the full expansion you can either

  • show that the series continues by putting an ellipsis (…) after your final term

  • or show that the terms you have found are an approximation of the full series by using the 'approximately equals' sign (≈)

Finding binomial coefficients using Pascal's triangle

• Pascal’s triangle is a way of arranging (and finding!) the binomial coefficients

  • The first row has just the number 1

  • Each row begins and ends with a 1

  • Starting in the third row

    • Each other terms is the sum of the two terms immediately above it

• Pascal’s triangle is an alternative way of finding the binomial coefficients $\left(\begin{array}{c}n\\ r\end{array}\right)$  (also written ${}_{ }{}^n{\mathrm{C}}_r$ )

  • It can be useful for finding the values of the coefficients without a calculator

    • Most useful for smaller values of $n$

    • For larger values of $n$ it is slow and prone to arithmetic errors

• Taking the first row as corresponding to $n=0$,

  • each row gives the binomial coefficient values for the corresponding value of $n$

  • within a row the values run from $r=0$ to $r=n$

  • e.g. from the 6^th row of the table ($n=5$):

    • ${\left(a+b\right)}^5=a^5+\mathbf{5}{a}^{4}b+\mathbf{10}{a}^3{b}^2+\mathbf{10}{a}^2{b}^3+\mathbf{5}a{b}^4+b^5$

How do I find the coefficient of a single term?

• You may just be asked to find the coefficient of a single term, rather than the whole expansion

• Use the formula for the general term

  • $\left(\begin{array}{c}n\\ r\end{array}\right)a^{n-r}{b}^r$

• To find a particular power of $\mathit{x}$ term in an expansion

  • Choose which value of $\mathit{r}$  you will need to use in the formula

  • The laws of indices can help you decide which value of $r$  to use:

    • For ${(p+qx)}^n$,  to find the coefficient of $x^2$ let  $a=p,b=qx$ and use $r=2$

    • For ${(p+q{x}^2)}^n$,  to find the coefficient of $x^2$ let  $a=p,b=q{x}^2$ and use $r=1$

    • For something like ${\left(px+\frac{q}{x}\right)}^n$, you need to consider how the powers will cancel each other

      • E.g. for ${\left(px+\frac{q}{x}\right)}^6$ , to find the coefficient of $x^2$ let $a=px,b=\frac{q}{x}$ and user $r=2$

      • Because then $\begin{array}{l}{x}^{n-r}{\left(\frac{1}{x}\right)}^r=x^{6-2}{\left(\frac{1}{x}\right)}^2=x^4\left(\frac{1}{x^2}\right)=x^2\\ \end{array}$

    • There are a lot of variations, so practice is better than trying to memorise formulae for $r$!

• If you know the coefficient of a particular term, you can use it to find an unknown in the brackets

  • Use the laws of indices to choose the correct term

  • Then use the general term formula to form and solve an equation