Binomial Expression:
Def: An expression consisting of two terms is called a binomial expression, e.g, x+a, 2x+3y, 5x^{2}6y^{2}, 2x1/3x are all binomial expressions.
Binomial Theorem
Binomial theorem helps us to expand any power of a given binomial expression. In this chapter, we propose to find the expansion of (x+a)^{n} where n € ^{+}I. Here n is called the power or the index of the binomial expression.
Development of binomial expansion
By actual multiplication, we may obtain the following expansion.
(x+y)^{1} = x+y
(x+y)^{2} = x^{2} + 2xy + y^{2} = ^{2}C_{0}x^{2} + ^{2}C_{1}xy + ^{2}C_{2}y ^{2}
(x+y)^{3} = x^{3}+ 3x^{2}y + 3xy^{2} + y^{3} = ^{3}C_{0}.x^{3} + ^{3}C_{1}. x^{2}y + ^{3}C_{2} .xy^{2} + ^{3} C_{2}.y^{3}
(x+y)^{4} = x^{4} + 4x^{3}y + 6x^{2}y^{2} + 4xy^{3} + y^{4}
= ^{4}C_{0}x^{4} + ^{4}C_{1}x^{3}y + ^{4}C_{2}x^{2}y^{2} + ^{4}C^{3}xy^{3} + ^{4}C^{}y^{4}
(x+y)^{5} = x^{5} + 5x^{4}y + 10x^{3}y^{2} + 10x^{2}y^{3} + 5xy^{4} + y^{5}
= ^{5}C_{0}x^{5} + ^{5}C_{1}x^{4}y + ^{5}C_{2}x^{3}y^{2} + ^{5}C_{3}x^{2}y^{3} + ^{5}C_{4}xy^{4} + ^{5}C_{4}xy^{4} + ^{5}C_{}xy_{4}xy^{4} + ^{5}C_{5}y^{5}
A careful observation of these expansions shows that (x+y)^{n} where (n = 1, 2, 3, 4, 5) when expanded has the following properties.
The coefficients form a certain pattern as shown below:
Pascal's Triangle
Inspection will show that each term in the table is derived by adding together the two terms in the line above, which lie on either side of it. Thus in the line n = 5, the term 10 is found by adding together the terms 4 and 6 in the line n = 4
The coefficients in combinatorial form called binomial coefficients may be rewritten as
 First term = ^{n}C_{0}x^{n} = x^{n}
 Second term = ^{n}C_{1}x^{n1}y = nx^{n1}y
 Third term = ^{n}C_{2}x^{n2}y^{2} = [n(n1)]/(1x2)x^{n2}y^{2}
 Fourth term = ^{n}C_{3}x^{n3}y^{3} = [n(n1)(n=2)]/(1x2x3)x^{n3}y^{3}
 Fifth term = ^{n}C_{5}x^{n}y^{4} = [n(n1)(n2)(n3)]/(1x2x3x4)x^{n4}y^{4}
Continuing the above process, it is easily seen that last but one term [i.e, n^{th} term] of the exapansion = ^{n}C_{n1} = nxy^{n1}
Last term = [i.e., (n+1)^{th} term] = ^{n}C_{n}y^{n} = y^{n}
(x+y)^{n} = ^{n}C_{0}x^{n} + ^{n}C_{1}x^{n1}y + ^{n}C_{2}x^{n2}y^{2} + ............ + ^{n}C_{r}x^{nr}y^{r} + .............. + ^{n}C_{n1}xy^{n1} + ^{n}C_{n}y^{n}..........1
= x^{n} + nx^{n1}y + [n(n1)]/(1x2)x^{n2}y^{2} + [n(n1)(n2)]/(1x2x3)x^{n3}y^{3} + ............... +y^{n}.
Example: Expand (1+4x)^{5}
Solution: (1+4x)^{5} = 1 + ^{5}C_{1}4x + ^{5}C_{2}(x)^{2} + ^{5}C_{3}(4x)^{3} + ^{5}C_{4}(4x)^{4} + (4x)^{5}
= 1 + ^{5}C_{1}4x + ^{5}C_{2}(4x)^{2} + ^{5}C_{2}(4x)^{3}
+ ^{5}C_{1}(4x)^{} + (4x)^{5} [because ^{5}C_{3} = ^{5}C_{3}; ^{5}C_{4} = ^{5}C_{1}]
= 1+5x4x + (5x4)/(1x2)x16x^{2} + (5x)/(1x2) x6x^{2} + 5x256x^{4} + 1024x^{5}
Directions: Expand the following.
