• A-Level Maths

The Binomial Expansion

Pascal's Triangle is used to expand binomial expressions like (a+b)ⁿ. It is created by summing adjacent pairs to find the number beneath this pair, starting from 1. Here are the first five rows:

The (n + 1)th row of Pascal's triangle gives the coefficients in the expansion of (a+b)ⁿ

Factorial Notation

Parts of Pascal's triangle can be calculated quickly using factorial notation, ⁿCr (spoken "n choose r"):

Expanding (a+b)ⁿ

When n ∈ ℕ (when n is a positive integer) the binomial expansion is in its simplest form:

The general term in an expansion of (a+b)ⁿ is given as:

Expanding (1+x)ⁿ

If n is a fraction or a negative number, you need to use this form of the binomial expansion:

It is valid when |x| < 1 and when n ∈

The general term in this expansion is given as:

x-term Coefficient

When the x term has a coefficient, so the binomial is in the form (1+bx)ⁿ, treat (bx) as x, and follow the standard expansion for (1+x)ⁿ

The expansion for (1+bx)ⁿ is valid for |bx| < 1, or |x| < 1/|b|

Double Coefficients

If the binomial is in the form (a+bx)ⁿ, you have to take a factor of aⁿ out of each term:

The expansion for (a+bx)ⁿ, where n is negative or a fraction, is valid when |bx/a| < 1, or |x| < |a/b|
  • Often, complex expressions can be simplified first by splitting them into partial fractions (see notes sheet on algebraic methods), then by using a binomial expansion.

Rate these notesNot a fanNot so goodGoodVery goodBrillRate these notes
White Logo.png

Do not download, print, reproduce, or modify these notes in any way. To share, only use the share buttons for each notes sheet or a direct link to a notes sheet or any page on engineeringnotes.net. Always credit such sharing with a link that reads 'EngineeringNotes.net'

What do you rate our notes?

© 2020 EngineeringNotes.net, All Rights Reserved     •     Privacy Policy

If you would like to use these notes in any other way or for any other purpose, please contact us using the form directly above. We will be happy to help accommodate your  requests in a way that protects our work and rights.