Title: The Binomial Theorem

[Pascal's Tri.][Gen. Expansion n≥1 ][Expansion for -1<n<1 ]






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.

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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
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)


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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


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

particular solution#3


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