PURE MATHEMATICS - Algebra

 

The Binomial Theorem

 

 

Pascal's Triangle

The General Expansion n≥1

Expansion for -1<n<1>

 

 

 

Introduction

 

This section of work is to do with the expansion of (a+b)n and (1+x)n .

 

Pascal's Triangle and the Binomial Theorem give us a way of expressing the expansion as a sum of ordered terms.

 

 

Pascal's Triangle

 

This is a method of predicting the coefficients of the binomial series.

 

Coefficients are the constants(1, 2, 3, 4, 5, 6 etc.) that multiply each variable, or group of variables.

 

Consider (a+b)n variables a, b .

 

 

pascal's Triangle

 

 

The 1st. line represents the coefficients for n=0.

 

(a+b)0= 1

 

The 2nd, line represents the coefficients for n=1.

 

(a+b)1= a + b

 

The 3rd. line represents the coefficients for n=2.

 

(a+b)2= a2 + 2ab + b2

 

Hence for the 6th. line . . .

 

The 6th. line represents the coefficients for n=5.

 

(a+b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5

 

 

The Binomial Theorem builds on Pascal's Triangle in practical terms, since writing out triangles of numbers has its limits.

 

 

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The General Binomial Expansion ( n ≥ 1 )

 

This is a way of finding all the terms of the series, the coefficients and the powers of the variables.

 

The coefficients, represented by nCr , are calculated using probability theory.

 

For a deeper understanding you may wish to look at where nCr comes from; but for now you must accept that:

 

nCr equation

 

where,

 

'n' is the power/index of the original expression

 

'r' is the number order of the term minus one

 

 

If n is a positive integer, then:

 

the binomial expansion

 

 

Example #1

 

binomial expansion example

 

 

Example #2

 

binomial problem

 

 

It is suggested that the reader try making similar questions, working through the calculations and checking the answer here (max. value of n=8)

 

 

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The Particular Binomial Expansion

 

This is for (1+x)n ,

 

where,

 

n can take any value (positive or negative)

 

x is a fraction in the range -1 < x < 1

 

 

binomial expansion particular solution

 

 

 

Example

 

Find the first 4 terms of the expression (x+3)1/2 .

 

 

particular solution#3

 

 

 

 

 

 

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