Binomial Expansion

Binomial Expansion is a method of expanding the expression of powers of a binomial term.

We must be aware of the Formula \((a+b)^{2} = a^{2}+b^{2}+2ab\), but where does it come from?

The answer is Binomial Expansion; which is defined as expanding the binomial term raised to any power.

Mathematical Representation of Binomial Expression-

The Binomial Expression is generally represented as:

\((a+b)^{n} = ^{n}C_{0}.a^{n}.b^{0} +^{n} C_{1} . a^{n-1} . b^{1} + …….. + ^{n}C_{n-1}.a^{1}.b^{n-1} + ^{n}C_{n}.a^{0}.b^{n}\)

We can write binomial expansion for negative powers as follows

\((x+a)^{-n} = \sum_{k=0}^{\infty } \left ( _{k} ^{-n} \right ) x^{k} a^{-n-k}\)

\(= \sum_{k}^{\infty } (-1)^{k} \left ( ^{n+k+1} _{k} \right ) x^k a^\left ( -n-k \right )\)

Binomial Expansion for Fraction:

We can write binomial expansion for fraction power as follows

For example : \((x+y)^{\frac{2}{3}} = x^{\frac{2}{3}}\left ( 1+\frac{y}{x} \right )^{\frac{2}{3}}\)

Use the Binomial Theorem to expand \(\left ( 1+\frac{y}{x} \right )^{\frac{2}{3}}\).

This way we can make binomial expansion of fraction power simpler and easier.

The coefficient of Binomial Expansion:

Let us have a look of expansion of Binomial \((a+b)^{n}\), where 0<n<5.

\((a+b)^{0} = 1\)

\((a+b)^{1} = (a+b)\)

\((a+b)^{2} = (a^{2}+2ab+b^{2})\)

\((a+b)^{3} =(a^{3}+3a^{2}b+3ab^{2}+b^{3} \)

\((a+b)^{3} =(a^{3}+3a^{2}b+3ab^{2}+b^{3} \)

\((a+b)^{4} =(a^{4}+4a^{3}b+6a^{2}b^{2}+4ab^{3}+b^{4} \)

\((a+b)^{5} =(a^{5}+5a^{4}b+10a^{3}b^{2}+10a^{2}b^{3}+5ab^{4}+b^{5} \)

Let us write only the coefficients of the exponents

Exponent

Coefficients

Sum

0

1

1

1

1 1

2

2

1 2 1

4

3

1 3 3 1

8

4

1 4 6 4 1

16

5

1 5 10 10 5 1

32

From the table above, we can find the general formula for the sum of the coefficient which is given by:

2n, (where n is the power of the Binomial term).

This can also be re-written as:

\(^{n}C_{0} +^{n}C_{1} +^{n}C_{2} + ……… +^{n} C_{n-2} +^{n} C_{n-1} +^{n} C_{n} = 2^{n}\)

Let’s take an example and expand it using binomial series expansion formula and simplify it.

\((x^{2}+3)^{6} = ^{6}C_{0}(x^{2})^{6} + ^{6}C_{1}(x^{2})^{5}(3)+^{6} C_{2}(x^{2})^{4}(3)^{2}+ ^{6}C_{3}(x^{2})^{3}(3)^{3}+ ^{6}C_{4}(x^{2})^{2}(3)^{4}+ ^{6}C_{5}(x^{2})(3)^{5} + ^{6}C_{6}(3)^{6}\)

\((x^{2}+3)^{6} = (x^{2})^{6} + 6(x^{2})^{5}(3)+ 15(x^{2})^{4}(3)^{2}+20(x^{2})^{3}(3)^{3}+ 15(x^{2})^{2}(3)^{4} + 6(x^{2})(3)^{5} + (3)^{6}\)

\((x^{2}+3)^{6} = x^{12} + 18x^{10}+ 90x^{8} + 540x^{6} + 1215x^{4} + 1458x^{2}+729\)

Now , let’s take an example where we have two variables are involved.

\((2x-5y)^{7}= ^{7}C_{0}(2x)^{7} + ^{7}C_{1} (2x)^{6}(-5y) + ^{7}C_{2}(2x)^{5}(-5y)^{2} + ^{7}C_{3}(2x)^{4}(-5y)^{3} + ^{7}C_{4}(2x)^{3}(-5y)^{4} + ^{7}C_{5} (2x)^{2}(-5y)^{5} + ^{7}C_{6}(2x)^{1}(-5y)^{6} + ^{7}C_{7}(-5y)^{7}\)

\((2x-5)^{7}= 128x^{7} -2240x^{6}y + 16800x^{5}y^{2} – 70000x^{4}y^{3} + 175000x^{3}y^{4} – 262500x^{2}y^{5} + 218750xy^{6} – 78125 y^{7}\)

Let us now take a look when sum of three terms are involved:

\((a+b+c)^{2}=(a+(b+c))^{2}\)

\(=a^{2}+2a(b+c)+(b+c))^{2}\)

\(= a^{2} + 2ab + 2ac + b^{2} + 2bc + c^{2}\)

\(= a^{2}+ b^{2}+ c^{2} + 2ab + 2bc + 2ac\)<

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