Binomial Theorem

Introduction:

A binomial is an algebraic expression of two terms which are connected by the operation + or -.

Consider the following algebraic identity:

(x+y) = x+y

(x + y)² = x² + 2xy + y²

(x +y)³ =  x³+3x²y + 3xy² + y³

(x + y)⁴ = (x + y)²(x + y)² = x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴

From the above just identify the pattern

(x+y)² is a sum of terms of  x² , x^2-1y and y²

(x+y)³ is a sum of terms of  x³ , x^3-1 y, x^3-2 y²  and y³

(x+y)⁴is a sum  of  terms of x⁴, x^4-1y, x^4-2 y² , x^4-3y³ and  y⁴

In general,

(x+y)ⁿ is a sum of terms of xⁿ, x^n-1 y, x^n-2y², x^n-3y³,…, x¹y^n-1,yⁿ with some integer coefficients.

 (x+y)ⁿ = nC₀ xⁿ y⁰ +nC₁x^n-1y¹ + nC₂x^n-2y² + nC₃x^n-3y³ +… + nC_n-1x¹y^n-1 + nC_n x⁰yⁿ.

Note:

• From the above expansion, the general term is nC_r x^n-r y^r. this is nothing but the (r+1)th  term  and it is denoted by T_r+1 .i.e.  T_r+1 = nC_r x^n-ry^r
• The (n+1)th term is T_n+1=nC_nx^n-nyⁿ =nC_n yⁿ, the last term .Thus there are n+1 terms in the expansion (x+y)ⁿ
• The degree of x in each term decreases while that of  y increases such that the sum of the powers in each term is equal to n.We can write (x + y)ⁿ =
• nC₀, nC₁ ,nC₂,…,nC_r, …nC_n are called binomial coefficients.
• From the relation nC_r = nC_n-r, we see that the coefficients of terms equidistant from the beginning and the end are equal.
• In the expansion nC₀=1; nC₁=n;nC₂=; nC₃= and so on where n!= n(n-1)(n-2)…2.1.
• The binomial coefficients of the various terms of the expansion of (x+y)ⁿ for n= 1, 2, 3,… form a pattern.

Binomials			Binomial Coefficients
(x+y)0					    1	
(x+y)1				       1   	  1
(x+y)2			           1        2        1
(x+y)3			       1         3         3	    1	

Example: Expand (x+2)⁶

In the binomial expansion put x=x, y=2, n=6

(x+2)⁶=6C₀ x⁶ + 6C₁ x⁵ 2¹ + 6C₂ x⁴ 2² + 6C₃ x³ 2³ + 6C₄ x² 2⁴ + 6C₅ x 2⁵ + 6C₆  2⁶

=6x⁶ +12x⁵+60x⁴+120x³+240 x² +192x+64

Binomial series:

From the binomial formula, if we let x=1 and y=x, we can also obtain the binomial series which is valid for any real number n if |x|<1.