The Binomial Expansions Formula will allow us to quickly find all of the terms in the expansion of any binomial raised to the power of \(n\): \[\begin{pmatrix} a + b \end{pmatrix}^n \] Where \(n\) is a positive integer.
By the end of this section we'll know how to write all the terms in the expansions of binomials like:
We start by learning the binomial expansion formula, we then watch a tutorial to learn how the binomial expansion formula works.
We'll then work our way through a written, detailed, example as well consolidate our knowledge with several exercises.
All of the terms of \(\begin{pmatrix} a + b \end{pmatrix}^n\) can be written using the binomial expansions formula, which states: \[\begin{pmatrix} a + b \end{pmatrix}^n = \sum_{r=0}^n \begin{pmatrix} n \\ r \end{pmatrix}a^{n-r}.b^r\] Where \(\begin{pmatrix} n \\ r \end{pmatrix}\) is the binomial coefficient, sometimes witten \( ^nC_r\), and is calculated as: \[\begin{pmatrix} n \\ r \end{pmatrix} = \frac{n!}{(n-r)!r!}\]
In the following tutorials we learn how the binomial expansions formula works and how to use it to write all the terms in any binomial raised to the power of \(n\), \(\begin{pmatrix} a + b \end{pmatrix}^n\).
Watch these now before working through exercise 1.
In this first tutorial, we show how to use the binomial expansions formula to write all of the terms in the expansion of: \[\begin{pmatrix}a+b\end{pmatrix}^3\] Comparing \(\begin{pmatrix}a+b\end{pmatrix}^3\) to \(\begin{pmatrix}a+b\end{pmatrix}^n\), it is clear that the only difference is that we replaced \(n\) by \(3\). Consequently all we have to do is replace \(n\) in the formula \(\begin{pmatrix} a + b \end{pmatrix}^n = \sum_{r=0}^n \begin{pmatrix} n \\ r \end{pmatrix}a^{n-r}.b^r\) by \(3\); leading us to: \[\begin{pmatrix} a + b \end{pmatrix}^3 = \sum_{r=0}^3 \begin{pmatrix} 3 \\ r \end{pmatrix}a^{3-r}.b^r\] The tutorial then shows us how to write all of the terms.
In this second tutorial, we show how to use the binomial expansions formula to write all of the terms in the expansion of: \[\begin{pmatrix}a+b\end{pmatrix}^4\] Comparing \(\begin{pmatrix}a+b\end{pmatrix}^4\) to \(\begin{pmatrix}a+b\end{pmatrix}^n\), it is clear that the only difference is that we replaced \(n\) by \(4\). Al we have to do is replace \(n\) in the formula \(\begin{pmatrix} a + b \end{pmatrix}^n = \sum_{r=0}^n \begin{pmatrix} n \\ r \end{pmatrix}a^{n-r}.b^r\) by \(4\); leading us to: \[\begin{pmatrix} a + b \end{pmatrix}^4 = \sum_{r=0}^4 \begin{pmatrix} 4 \\ r \end{pmatrix}a^{4-r}.b^r\] Again, the tutorial then shows us how to write all of the terms.
In this tutorial we spend a few minutes to show, in more detail, than the two tutorials above, how to calculate the binomial coefficients in the binomial expansion.
Note: the second tutorial on binomial coefficients (mentionned at the end of this tutorial) is further down the page.
Write all the terms in the expansion of each of the following binomials:
In this first example, we show how to write all of the terms in the expansion of: \[\begin{pmatrix}x + 2 \end{pmatrix}^5\] We start by using the binomial theorem formula to write: \[\begin{pmatrix}x + 2 \end{pmatrix}^5 = \begin{pmatrix} 5 \\ r \end{pmatrix}x^{5-r}.2^r\] We then use the general term: \[t_r = \begin{pmatrix} 5 \\ r \end{pmatrix}x^{5-r}.2^r\] to write all the terms of the expansion.
In this second example, we show how to write all of the terms in the expansion of: \[\begin{pmatrix}x - 2 \end{pmatrix}^4\] We start by re-writing the subtraction as an addition, that is we use the fact that: \[\begin{pmatrix}x - 2 \end{pmatrix}^4 = \begin{pmatrix}x + (-2)\end{pmatrix}^4\] to then use the binomial theorem formula to write: \[\begin{pmatrix}x + (-2) \end{pmatrix}^4 = \begin{pmatrix} 4 \\ r \end{pmatrix}x^{4-r}.(-2)^r\] We then use the general term: \[t_r = \begin{pmatrix} 4 \\ r \end{pmatrix}x^{4-r}.(-2)^r\] to write all the terms of the expansion.
In this first example, we show how to write all of the terms in the expansion of: \[\begin{pmatrix}x + 2 \end{pmatrix}^5\] We start by using the binomial theorem formula to write: \[\begin{pmatrix}x + 2 \end{pmatrix}^5 = \begin{pmatrix} 5 \\ r \end{pmatrix}x^{5-r}.2^r\] We then use the general term: \[t_r = \begin{pmatrix} 5 \\ r \end{pmatrix}x^{5-r}.2^r\] to write all the terms of the expansion.
Using the binomial expansion formula and the method for writing all the terms of any expansion \(\begin{pmatrix}a+b \end{pmatrix}^n\), we learn how to write all the terms in the expansion of binomials looking like \(\begin{pmatrix}a+x\end{pmatrix}^n\) and \(\begin{pmatrix}x+b\end{pmatrix}^n\).
The method for doing this is shown in tutorial 3, below.
Write all the terms in the expansion of each of the following binomials:
To write all the terms in the expansion of binomials in which the \(x\) term is either \(ax\), a power \(x^m\), or even a combination of both \(ax^m\), such as:
In the following tutorials we work through examples showing how to write all the terms of expansions of the type \(\begin{pmatrix}ax+b\end{pmatrix}^n\) and \(\begin{pmatrix}x^m+b\end{pmatrix}^n\).
Watch these tutorials before working through exercise 3.
In the following tutorial we show, in detail, how to write all the terms in the expansion of: \[\begin{pmatrix} 2 + 3x\end{pmatrix}^4\] We do this using the binomial expansion formula and using the fact that: \[\begin{pmatrix} 2 + 3x\end{pmatrix}^4 = \begin{pmatrix} 2 + \begin{pmatrix} 3x \end{pmatrix} \end{pmatrix}^4\] Along with the following rule for exponents: \[\begin{pmatrix}ax\end{pmatrix}^n = a^n.x^n\]
In the following tutorial we show, in detail, how to write all the terms in the expansion of: \[\begin{pmatrix} 2 + x^2\end{pmatrix}^4\] We do this using the binomial expansion formula and using the fact that: \[\begin{pmatrix} 2 + x^2\end{pmatrix}^4 = \begin{pmatrix} 2 + \begin{pmatrix} x^2 \end{pmatrix} \end{pmatrix}^4\] Along with the following rule for exponents: \[\begin{pmatrix}x^m\end{pmatrix}^n = x^{m\times n}\]
Write all the terms in the expansion of each of the following binomials:
In the following tutorial we learn how to calculate the binomial coefficient with a calculator.
The calculator used here is the TI NSpire CX