Binomial Expansion (AQA GCSE Further Maths): Revision Note

Pascal's Triangle

What is Pascal's Triangle?

• Pascal's Triangle is a triangle of number patterns (shown below)

  • where each number is the sum of the two numbers above it
    

    • It can help to imagine zeros outside the triangle

  • The triangle is symmetric

Binomial expansion

What is a binomial?

• A binomial is the sum or difference of two different terms

  • e.g. $3+4x$ or $p-q$

• A binomial can be raised to a power, $n$

  • This has the form ${\left(a+b\right)}^n$

How do I expand a binomial?

• You can expand a binomial by multiplying out brackets, but the bigger the power, the longer this takes

  • ${\left(a+b\right)}^1=a+b$

  • ${\left(a+b\right)}^2=\left(a+b\right)\left(a+b\right)$

    • This gives $a^2+2ab+b^2$

  • ${\left(a+b\right)}^3=\left(a+b\right)\left(a+b\right)\left(a+b\right)$

    • This gives $a^3+3a^{2}b+3a{b}^2+b^3$

  • ${\left(a+b\right)}^4=\left(a+b\right)\left(a+b\right)\left(a+b\right)\left(a+b\right)$

    • This eventually gives $a^4+4a^{3}b+6a^2{b}^2+4a{b}^3+b^4$

How are binomials related to Pascal's triangle?

• The coefficients of the terms in the expansion of ${\left(a+b\right)}^n$ correspond to a row in Pascal's triangle

  • ${\left(a+b\right)}^0=\mathbf{1}$

  • ${\left(a+b\right)}^1=\mathbf{1}a+\mathbf{1}b$

  • ${\left(a+b\right)}^2=\mathbf{1}{a}^2+\mathbf{2}ab+\mathbf{1}{b}^2$

  • ${\left(a+b\right)}^3=\mathbf{1}{a}^3+\mathbf{3}{a}^{2}b+\mathbf{3}a{b}^2+\mathbf{1}{b}^3$

  • ${\left(a+b\right)}^4=\mathbf{1}{a}^4+\mathbf{4}{a}^{3}b+\mathbf{6}{a}^2{b}^2+\mathbf{4}a{b}^3+\mathbf{1}{b}^4$

How do I expand a binomial using Pascal's triangle?

• Using $a^0=b^0=1$, you can write down rules to expand ${\left(a+b\right)}^n$ without having to multiply out brackets

  • It is the sum of terms in the form $\mathrm{Pascal}\mathrm{coefficient}\times a^{\left(...\right)}\times b^{\left(...\right)}$ where

    • the powers of $a$ decrease from $a^n$ to $a^0$

    • the powers of $b$ increase from $b^0$ to $b^n$

    • the Pascal coefficients come from the row starting with $1,n,...$

  • There should be $\left(n+1\right)$ terms in total

• For example, to expand ${\left(a+b\right)}^4$, there will be 4 + 1 = 5 terms as follows:

• This simplifies to

  • ${\left(a+b\right)}^4=a^4+4a^{3}b+6a^2{b}^2+4a{b}^3+b^4$

How do I expand binomials with harder terms?

• You need to be familiar with index laws, e.g.:

  • ${\left(2x\right)}^3=2^3{x}^3=8x^3$

  • ${\left(\frac{x}{2}\right)}^3=\frac{x^3}{2^3}=\frac{x^3}{8}$

• For example

  • To expand ${\left(2x+3\right)}^4$

    • Imagine ${\left(a+b\right)}^4$ where $a=2x$ and $b=3$

    • Put brackets around $\left(2x\right)$ and $\left(3\right)$

    • Then use the rules above

  • ${\left(2x+3\right)}^4=1{\left(2x\right)}^4{\left(3\right)}^0+4{\left(2x\right)}^3{\left(3\right)}^1+6{\left(2x\right)}^2{\left(3\right)}^2+4{\left(2x\right)}^1{\left(3\right)}^3+1{\left(2x\right)}^0{\left(3\right)}^4$

  • Apply the index laws carefully

    • $=2^4{x}^4\times 1+4\times 2^3{x}^3\times 3+6\times 2^2{x}^2\times 3^2+4\times 2x\times 3^3+3^4$

    • $=16x^4+96x^3+216x^2+216x+81$

• Note that the final coefficients are not symmetric

  • even though the Pascal coefficients used in the working are

How do I expand binomials with negative terms?

• You need to be familiar with powers of negatives: 

  • ${\left(-2\right)}^2=4$

    • Even powers are positive

  • ${\left(-2\right)}^3=-8$

    • Odd powers are negative

• For example, ${\left(2x-3\right)}^4$ is ${\left(a+b\right)}^4$ with $a=\left(2x\right)$ and $b=\left(-3\right)$

  • ${\left(2x-3\right)}^4=1{\left(2x\right)}^4{\left(-3\right)}^0+4{\left(2x\right)}^3{\left(-3\right)}^1+6{\left(2x\right)}^2{\left(-3\right)}^2+4{\left(2x\right)}^1{\left(-3\right)}^3+1{\left(2x\right)}^0{\left(-3\right)}^4$

    • $=2^4{x}^4\times 1+4\times 2^3{x}^3\times \left(-3\right)+6\times 2^2{x}^2\times 9+4\times 2x\times \left(-27\right)+81$

    • $=16x^4-96x^3+216x^2-216x+81$

  • The signs alternate between positive and negative