The Binomial Expansion

Summary

  • Pascal’s Triangle can be used to multiply out a bracket.
  • When we have large powers, we can use combination and factorial notation to help expand binomial expressions.
  • { C }_{ r }^{ n }\quad =\quad (\overset { n }{ r } )\quad =\quad \frac { n! }{ (n\quad -\quad r)!\quad \times \quad r! }
  • { (x+y) }^{ n }={ x }^{ n }+n{ x }^{ n-1 }{ y }+\frac { n }{ 2! } (n-1){ x }^{ n-2 }{ y }^{ 2 }+\frac { n }{ 3! } (n-1){ (n-2)x }^{ n-3 }{ y }^{ 3 }+........{ y }^{ n }
  • { (x+y) }^{ n }={ x }^{ n }+{ C }_{ 1 }^{ n }{ x }^{ n-1 }{ y }+{ C }_{ 2 }^{ n }{ x }^{ n-2 }{ y }^{ 2 }+\quad { C }_{ 3 }^{ n }{ x }^{ n-3 }{ y }^{ 3 }+........{ y }^{ n }

What is a Binomial?

A binomial is an expression which consists of two terms only i.e 2x + 3y and 4p – 7q are both binomials.

What is Binomial Expansion and Binomial coefficients?

We know that  { (x+y) }^{ 2 }={ x }^{ 2 }+2xy+{ y }^{ 2 }.  The expansion of  { (x+y) }^{ n }  is known as Binomial expansion and the coefficients in the binomial expansion are called binomial coefficients.

Listed below are the binomial expansion of  { (x+y) }^{ n }  for n = 1, 2, 3, 4 & 5.

{ (x+y) }^{ 1 }=x+y

{ (x+y) }^{ 2 }={ x }^{ 2 }+2xy+{ y }^{ 2 }

{ (x+y) }^{ 3 }={ x }^{ 3 }+3{ x }^{ 2 }y+{ 3xy }^{ 2 }+{ y }^{ 3 }

{ (x+y) }^{ 4 }={ x }^{ 4 }+4{ x }^{ 3 }y+{ 6{ x }^{ 2 }y }^{ 2 }+{ 4xy }^{ 3 }+{ y }^{ 4 }

{ (x+y) }^{ 5 }={ x }^{ 5 }+5{ x }^{ 4 }y+{ 10{ x }^{ 3 }y }^{ 2 }+{ 10{ x }^{ 2 }y }^{ 3 }+{ 5xy }^{ 4 }+{ y }^{ 5 }

Some important features in these expansions are:

  • If the power of the binomial expansion is n, then there are (n+1) terms.
  • The sum of the powers of x and y in each term is equal to the power of the binomial i.e equal to n.
  • The powers of x in the expansion of  { (x+y) }^{ n }  are in descending order while the powers of y are in ascending order.
  • All the binomial coefficients follow a particular pattern which is known as Pascal’s Triangle.
BinomialCoefficients
{ (x+y) }^{ 1 } 1+1
{ (x+y) }^{ 2 } 1+2+1
{ (x+y) }^{ 3 }1+3+3+1
{ (x+y) }^{ 4 }1+4+6+4+1
{ (x+y) }^{ 5 }1+5+10+10+5+1

 

Pascals Triangle gives us a very good method of finding the binomial coefficients but there are certain problems in this method:

1.      If n is very large, then it is very difficult to find the coefficients.

2.      To find any binomial coefficient, we need the two coefficients just above it.

To solve the above problems we can use combinations and factorial notation to help us expand binomial expressions. For larger indices, it is quicker than using the Pascal’s Triangle.

{ (x+y) }^{ n }={ x }^{ n }+{ C }_{ 1 }^{ n }{ x }^{ n-1 }{ y }+{ C }_{ 2 }^{ n }{ x }^{ n-2 }{ y }^{ 2 }+\quad { C }_{ 3 }^{ n }{ x }^{ n-3 }{ y }^{ 3 }+........{ y }^{ n } { (x+y) }^{ n }={ x }^{ n }+n{ x }^{ n-1 }{ y }+\frac { n }{ 2! } (n-1){ x }^{ n-2 }{ y }^{ 2 }+\frac { n }{ 3! } (n-1){ (n-2)x }^{ n-3 }{ y }^{ 3 }+........{ y }^{ n }

We calculate the value of  { C }_{ r }^{ n }  by the following formula  { C }_{ r }^{ n }\quad =\quad \frac { n! }{ (n\quad -\quad r)!\quad \times \quad r! } ,  it can also be written as  (\overset { n }{ r } ).

This is known as the binomial theorem.

Example #1

Q Use the Pascal’s Triangle to find the expansion of  { (x+2y) }^{ 3 }

Solution:

As the power of the expression is 3, we look at the 3rd line in Pascal’s Triangle to find the coefficients.

1+1
1+2+1
1+3+3+1 \longleftarrow

Therefore, the coefficients are 1, 3, 3, 1 so:

{ (x+2y) }^{ 3 }={ x }^{ 3 }+3{ x }^{ 2 }(2y)+{ 3x(2y) }^{ 2 }+1{ (2y) }^{ 3 }

Ans:      { =x }^{ 3 }+6{ x }^{ 2 }y+12xy^{ 2 }+8{ y }^{ 3 }

Example #2

Q Use the binomial theorem to find the expansion of  { (3-2x) }^{ 5 }

Solution:

We know as n = 5 there will be 6 terms. Hence:

{ (3-2x) }^{ 5 }=3{ x }^{ 5 }+(\overset { 5 }{ 1 } ){ 3 }^{ 4 }(-2x)+(\overset { 5 }{ 2 } ){ 3 }^{ 3 }{ (-2x) }^{ 2 }+

+(\overset { 5 }{ 3 } ){ 3 }^{ 2 }{ (-2x) }^{ 3 }+(\overset { 5 }{ 4 } ){ 3 }^{ 1 }{ (-2x) }^{ 4 }+{ (-2x) }^{ 5 }

{ =243-810x+1080{ x }^{ 2 }-720{ x }^{ 3 }+240{ x }^{ 4 }-32{ x }^{ 5 } }

Reference
  1. Edexcel AS and A Level Modular Mathematics C2
Categories Algebra