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Binomial Expansions Formula

how it works & how to use it

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:

\(\begin{pmatrix} 2 + x \end{pmatrix}^4\), \(\begin{pmatrix} 2x - 3 \end{pmatrix}^5\), \(\begin{pmatrix} 4 + x^2 \end{pmatrix}^4\)...

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.

Binomial Expansions - Formula

All of the terms of \(\begin{pmatrix} a + b \end{pmatrix}^n\) can be written using the binomial expansions formula, which states: \[\begin{pmatrix} a + b \end{pmatrix}^n = \begin{pmatrix} n \\ r \end{pmatrix}a^{n-r}.b^r\] Where \(\begin{pmatrix} n \\ r \end{pmatrix}\) is the binomial coefficient, sometimes witten \( ^nC_r\), and is calculated as: \[\begin{pmatrix} n \\ r \end{pmatrix} = \frac{n!}{(n-r)!r!}\]

Tutorials

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.

Tutorial 1

In this first tutorial, we learn how to read the binomial expansion formula and how it works to write the terms in the expansion of \(\begin{pmatrix} a + b \end{pmatrix}^n\).



Tutorial 2

In tutorial 2, we learn how to use the binomial expansion formula to write all the terms in any binomial expansion \(\begin{pmatrix}a+b\end{pmatrix}^n\).

We show this by working through detailed examples. In particular, we show how to write all of the terms in the expansions of:

\(\begin{pmatrix}a+b\end{pmatrix}^3\) and \(\begin{pmatrix} a + b \end{pmatrix}^4\)



Exercise 1

Write all the terms in the expansion of each of the following binomials:

  1. \(\begin{pmatrix}a+b \end{pmatrix}^3\)
  2. \(\begin{pmatrix}a+b \end{pmatrix}^4\)
  3. \(\begin{pmatrix}a+b \end{pmatrix}^5\)
  4. \(\begin{pmatrix}a+b \end{pmatrix}^6\)

Answers Without Working

  1. \(\begin{pmatrix}a+b \end{pmatrix}^3 = a^3 + 3a^2b + 3ab^2 + b^3\)
  2. \(\begin{pmatrix}a+b \end{pmatrix}^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4\)
  3. \(\begin{pmatrix}a+b \end{pmatrix}^5 = a^5 + 5a^4b+10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5\)
  4. \(\begin{pmatrix}a+b \end{pmatrix}^6 = a^6+6a^5b+15a^4b^2+20a^3b^3+15a^2b^4+6ab^5+b^6\)

Writing all the terms of \(\begin{pmatrix}x+b\end{pmatrix}^n\) and \(\begin{pmatrix}a+x\end{pmatrix}^n\)

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.

Tutorial 3

Exercise 2

Write all the terms in the expansion of each of the following binomials:

  1. \(\begin{pmatrix}x + 2 \end{pmatrix}^3\)
  2. \(\begin{pmatrix}x - 1 \end{pmatrix}^4\)
  3. \(\begin{pmatrix}x + 2 \end{pmatrix}^5\)
  4. \(\begin{pmatrix} 3 - x \end{pmatrix}^5\)
  5. \(\begin{pmatrix}1 + x \end{pmatrix}^6\)

Answers Without Working

  1. \(\begin{pmatrix} x + 2 \end{pmatrix}^3 = x^3 + 6x^2 + 12x + 8\)
  2. \(\begin{pmatrix} x - 1 \end{pmatrix}^4 = x^4 - 4x^3 + 6x^2 - 4x + 1 \)
  3. \(\begin{pmatrix} x + 2\end{pmatrix}^5 = x^5 + 10x^4 + 40 x^3 + 80x^2 + 80x + 32\)
  4. \(\begin{pmatrix} 3 - x \end{pmatrix}^5 = 243 - 81x + 27x^2 - 9x^3 + 3x^4 - x^5\)
  5. \(\begin{pmatrix} 1 + x \end{pmatrix}^6 = 1 + 6x + 15x^2 + 20x^3 + 15x^2 + 6x + 1\)

Writing all the terms of \(\begin{pmatrix}ax+b\end{pmatrix}^n\) and \(\begin{pmatrix}x^m+b\end{pmatrix}^n\)

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:

\(\begin{pmatrix}2x+5\end{pmatrix}^4\), \(\begin{pmatrix}x^2+2\end{pmatrix}^5\), \(\begin{pmatrix}2x^3-1\end{pmatrix}^3\), ... .
It is essential to write any \(x\) term in paretheses and use the following laws of exponents:
  • products raised to a power: \( \begin{pmatrix}ax\end{pmatrix}^n = a^n.x^n \)
  • powers raised to a power: \(\begin{pmatrix}x^m \end{pmatrix}^n = x^{m\times n}\)
  • combinations of both: \(\begin{pmatrix}ax^m \end{pmatrix}^n = a^n.x^{m\times n}\)
Using these laws, as well as the fact that: \[\begin{pmatrix}ax+b\end{pmatrix}^n = \begin{pmatrix} \begin{pmatrix} ax \end{pmatrix}+b\end{pmatrix}^n \]
and
\[\begin{pmatrix}x^m+b\end{pmatrix}^n = \begin{pmatrix} \begin{pmatrix}x^m \end{pmatrix}+b\end{pmatrix}^n\] we can write all the terms in such expansions.

Tutorials 4 & 5

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.

Tutorial 4

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: \[\begin{pmatrix}ax\end{pmatrix}^n = a^n.x^n\]

watch tutorial 4


Tutorial 5

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: \[\begin{pmatrix}x^m\end{pmatrix}^n = x^{m\times n}\]

watch tutorial 5


Exercise 3

Write all the terms in the expansion of each of the following binomials:

  1. \(\begin{pmatrix}2x + 3 \end{pmatrix}^3\)
  2. \(\begin{pmatrix} 4x - 3 \end{pmatrix}^4\)
  3. \(\begin{pmatrix}1 - 3x \end{pmatrix}^5\)
  4. \(\begin{pmatrix} 5 + 3x \end{pmatrix}^4\)
  5. \(\begin{pmatrix} 2x + 1 \end{pmatrix}^6\)
  6. \(\begin{pmatrix} 2+x^2 \end{pmatrix}^4\)
  7. \(\begin{pmatrix} x^3 - 1 \end{pmatrix}^6\)
  8. \(\begin{pmatrix}2x^2 + 3 \end{pmatrix}^5\)

Answers Without Working

  1. \(\begin{pmatrix}2x + 3 \end{pmatrix}^3 = 8x^3 + 36x^2 + 54x + 27\)
  2. \(\begin{pmatrix} 4x - 3 \end{pmatrix}^4 = 256x^4 - 768x^3 + 864x^2 - 432x + 81 \)
  3. \(\begin{pmatrix}1 - 3x \end{pmatrix}^5 = 1 - 15x + 90x^2 - 270x^3 + 405x^4 + 243x^5\)
  4. \(\begin{pmatrix} 5 + 3x \end{pmatrix}^4 = 625 + 1500x + 1350x^2 540x^3 + 81x^4 \)
  5. \(\begin{pmatrix} 2x + 1 \end{pmatrix}^6 = 64x^6 + 192x^5 + 240x^4 + 160x^3 + 60x^2 + 12x + 1\)

Calculator Technique

In the following tutorial we learn how to calculate the binomial coefficient with a calculator.
The calculator used here is the TI NSpire CX

Tutorial