,
Title: The Binomial Theorem
 

[Pascal's Tri.][Gen. 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 gives 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 first line represents the coefficients for n=0.

(a+b)0= 1

The second line represents the coefficients for n=1.

(a+b)1= a + b

The third line represents the coefficients for n=2.

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

The sixth 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.

back to top

 

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
and '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)

 

back to top

 

The Particular Binomial Expansion

This is for (1+x)n , where n can take any value positive or negative, and x is a fraction ( -1<x<1 ).

binomial expansion particular solution

Example

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

particular solution#3

 

back to top

 

your stop for the best in math, science & programming tutorials on the Net revision help to get a better result incremental success advanced physics for secondary/high school, including much in-depth content common to first year university courses your one stop for the best in math, science and programming tuition revision help for a better result incremental success advanced physics for high school/secondary and 1st year university fast-track learning for everyone

[ PURE MATHS ][ MECHANICS ][ STATISTICS ]

VIDEO

parametric differen.
equation of a tangent
equation of a normal
rate of change prob.1
rate of change prob.2
the Chain Rule
Chain Rule probs. #1
Chain Rule probs. #2
Chain Rule probs. #3
the Product Rule
parametric eqs.prob#1
parametric eqs.prob#2
intro. to integration
integration by parts 1
integration by parts 2
area under a curve
volumes of revolution
area between curves
Binomial Theorem
Bin. Theorem problems
Trig. Identities
Half Angle Formula
Double Angle Formula
Vectors,mod,resultant
Vector problems
Vector problems in 3D
MORE . . .
 

INTERACTIVE

 
derivative formula
tangents & normals
inverse functions
MORE . . .
 

EXAM PAPERS(.pdf)

 
Edxl C1 Pure specimen
Edxl C1 Pure answers
Edxl C2 Pure specimen
Edxl C2 Pure answers
Edxl C3 Pure specimen
Edxl C3 Pure answers
Edxl C4 Pure specimen
Edxl C4 Pure answers
MORE . . .

TOPIC NOTES(.pdf)

the derivative formula
tangents & normals
maxima & minima
chain rule
diffn.exponentials,logs
diffn.trigonometric fns.
product rule
quotient rule
parametric equations
implicit equations
differential equations
integration formula
int.'by substitution'
int.'by parts'
algebraic fractions
definite integrals
areas under curves
volumes of revolution
Trapezium Rule
integ. diff. equations
radians
sine, cosine, tangent
sec, cosec, cotan
Sine & Cosine Rules
Pythagorean ID's
compound angles
Sigma Notation
arithmetic progression
geometic progression
line between points
more straight lines
parametric equations
circles & ellipses
MORE . . .
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Google