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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Practise This Question

The average  marks  of  the students in Mathematics of class 8th is 45. But most of the students were given 60 out of 100.The Median of the class would be: