Binomial Theorem or Binomial Expansion
1. What is Binomial Expression?

Any expression that has two terms is called a binomial expression.
Eg: x + y, x + 2, 2x + y, 3x + 4y, 3x² + 4y³ .

2. Binomial Expansion for any positive integral index:
Consider the following algebraic formulas to finally arrive at the Binomial Theorem
( x + y )^{1} = x + y
( x + y )^{2} = x^{2} + 2xy + y^{2}
( x + y )^{3} = x^{3} + 3x^{2} y + 3xy^{2} + y^{3}
(x + y )^{4} = x^{4} + 4x^{3} y + 6x^{2} y^{2} + 4xy^{3} + y^{4}
(x + y )^{5} = x^{5} + 5x^{4} y + 10x^{3} y^{2} + 10x^{2} y^{3} + 5xy^{3} + y^{5}

Now,let us make the following observations based on the above five algebraic formulas:

1. The coefficients in each of the above binomial expansions follow a pattern. The pattern can be understood with Pascal’s triangle as follows:

2. First let us become familiar with the values of the binomial coefficients
(see point no. 4 in red below for a detailed discussion of binomial coefficients)
Remember the Combinations Formula ^{n} c_{r} ?
It is

and use it to write the following sample values in the binomial expansion no.5 above:

^{5} c_{0} = 1, ^{5} c_{1} = 5, ^{5} c_{2} = 10,^{5} c_{3} = 10, ^{5} c_{4} = 5, ^{5} c_{5} = 5

3. The first term in each of the above binomial expansions is x^{n} or ^{n} C_{0} x^{n}
The second term in the above expansions is nx^{n-1}

4. As the expansion proceeds the power of x decreases by one, while the power of y increases by one.

5. The number of terms in each of the expansion is (n+1)

6. Note: the last term is (n + 1)^{ th} term, not n^{th} term.

7. Generalizing the above properties in the above four algebraic expansions, we can write the following general binomial theorem:

Let x and y be two real numbers and index n be a positive integer. Then, the Binomial Theorem or Binomial Expansion of (x + y)^{n} is

(x+y)^{n} = ^{n} c_{0} .x^{n} + ^{n} c_{1} .x^{n-1} y + ^{n} c_{2} .x^{n-2} y^{2} +^{n} c_{3} .x^{n-3} y^{3} +……. +^{n} c_{r} .x^{n-r} y^{r} +…. +^{n} c_{n} .y^{n}

3. Number of terms in the binomial expansion: (x + y)^{n}

1. The number of terms in the binomial expansion is always one more than the index n ,

i.e., in a binomial expansion, number of terms = n + 1 , where n is the index.

2. As the terms are written, the power of x decreases by one, while the power of y increases by one.