Binomial expression :
It is the expansion of the powers of the algebraic function. the algebraic function (x+y)n can be expanded using the binomial theorem. the coefficient will be positive integer.
The binomial coefficients:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
Expansion of binomial expression:
(x+y)n=axnyn-b+axn-1yn-(b+1)+...............
Examples of Binomial expression:
Binomial expressions with variables :
The expression which has variables inside the binomial function is known as binomial expression with variables.
1) ( x + y ) 2 = 1x2y0 + 2xy + 1x0y2
= 1x2 + 2xy + 1y2
= x2 + 2xy + y2
2) (x+y)5 = 1x5y0 + 5x4y1 + 10x3y2 + 10x2y3 + 5x1y4 + 1x0y5
= x5y0 + 5x4y1 + 10x3y2 + 10x2y3 + 5x1y4 + x0y5
= x5 + 5x4y1 + 10x3y2 + 10x2y3 + 5x1y4 + y5
Binomial expressions with constants:
The expression which has variables and constant inside the binomial function is known as binomial expression with constants.
1) (x + 7)2 = x2*70 + 2*x1*71 + x0*72
= x2 + 14x + 49
2) (x + 1)3 = 1*x3*y0 + 3*x2*y1 + 3*x1*y2 + 1*x0*y3
= x3 + 3x2y + 3xy2 + y3
Binomial expression involving subtraction:
This method can be applied until the second term can be negotiated.
Expansion of binomial expression involving subtraction:
(x-y)n = axnyn-0 - axn-1y1 + ...............
Example on binomial expression:
1) ( x - y ) 2 = 1x2y0 - 2xy + 1x0y2
= 1x2 - 2xy + 1y2
= x2 - 2xy + y2
2) (x-y)5 = 1x5y0 - 5x4y1 + 10x3y2 - 10x2y3 + 5x1y4 - 1x0y5
= x5y0 - 5x4y1 + 10x3y2 - 10x2y3 + 5x1y4 - x0y5
= x5 - 5x4y1 + 10x3y2 - 10x2y3 + 5x1y4 - y5
3) (x - 1)3 = 1*x3*y0 - 3*x2*y1 + 3*x1*y2 - 1*x0*y3
= x3 - 3x2y + 3xy2 - y3
4) (x - 7)2 = x2*70 - 2*x1*71 + x0*72
= x2 - 14x + 49
