(A + B + C) Whole Square

(a + b)2 = a2 + b2 + 2ab	(a - b)2 = a2 + b2 - 2ab	a2 - b2 = (a + b) (a - b)	(x + a) (x + b) = x2 + x(a + b) + ab	(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
(a + b)3 = a3 + b3 + 3ab(a + b)	(a - b)3 = a3 - b3 - 3ab(a - b)	a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - bc - ca)

Before you understand (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca, you are advised to read: How to multiply Variables ? How to multiply Constant and Variable ? Multiplication of Polynomials ? What is Exponential Form and Laws of Exponents ? How identity of (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca is obtained Taking LHS of the identity: (a + b +c)2 This can also be written as: = (a + b + c) (a + b + c) Multiply as we do multiplication of trinomials and we get: = a(a + b + c) + b(a + b + c) + c(a + b + c) = a2 + ab + ac + ab + b2 + bc + ac + bc + c2 Rearrange the terms and we get: = a2 + b2 + c2 + ab + ab + bc + bc + ac + ac Add like terms and we get: = a2 + b2+ c2 + 2ab + 2bc + 2ca Hence, in this way we obtain the identity i.e. (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca Following are the few examples of this identity Example 1: Solve (4p + 5q + 3r)2 Solution: This proceeds as: Given polynomial (4p + 5q + 3r)2 represents identity first i.e. (a + b+ c)2 Where a = 4p, b = 5q and c = 3r Now apply values of a, b and c on the identity i.e. (a + b +c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca and we get: (4p + 5q + 3r)2 = (4p)2 + (5q)2 + (3r)2 + 2(4p)(5q) + 2(5q)(3r) + 2(3r)(4p) Expand the exponential forms and we get: = 16p2 + 25q2 + 9r2 + 2(4p)(5q) + 2(5q)(3r) + 2(3r)(4p) Solve brackets and we get: = 16p2 + 25q2 + 9r2 + 40pq + 30qr + 24rp Hence, (4p + 5q + 3r)2 = 16p2 + 25q2 + 9r2 + 40pq + 30qr + 24rp Example 2: Solve (8x + 4y + 7z)2 Solution: This proceeds as: Given polynomial (8x + 4y+ 7z)2 represents identity first i.e. (a + b + c)2 Where a = 8x, b = 4y and c = 7z Now apply values of a, b and c on the identity i.e. (a + b +c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca and we get: (8x + 4y+ 7z)2 = (8x)2 + (4y)2 + (7z)2 + 2(8x)(4y) + 2(4y)(7z) + 2(7z)(8x) Expand the exponential forms and we get: = 64x2 + 16y2 + 49z2 + 2(8x)(4y) + 2(4y)(7z) + 2(7z)(8x) Solve multiplication process and we get: = 64x2 + 16y2 + 49z2 + 64xy + 56yz + 112zx Hence, (8x + 4y + 7z)2 = = 64x2 + 16y2 + 49z2 + 64xy + 56yz + 112zx