(A + B) Cube

(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)3 = a3 + b3 + 3ab(a + b), 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)3 = a3 + b3 + 3ab(a + b) is obtained Taking LHS of the identity: (a + b)3 This can also be written as: = (a + b) (a + b) (a + b) Multiply first two binomials as shown below: = { a(a + b) + b(a + b) } (a + b) = { a2 + ab + ab + b2 } (a + b) Rearrange the terms in curly braces and we get: = { a2 + b2 + ab + ab } (a + b) = { a2 + b2 + 2ab } (a + b) Multiply trinomial with binomial as shown below: = a2(a + b) + b2(a + b) + 2ab(a + b) = a3 + a2b + ab2 + b3 + 2ab(a + b) = a3 + b3 + a2b + ab2 + 2ab(a + b) = a3 + b3 + a2b + ab2 + 2ab(a + b) Take ab common from the above red highlighted terms and we get: = a3 + b3 + ab(a + b) + 2ab(a + b) Add like terms and we get: = a3 + b3 + 3ab(a + b) Or we can further solve this to get: a3 + b3 + 3a2b + 3ab2 Hence, in this way we obtain the identity i.e. (a + b)3 = a3 + b3 + 3ab(a + b) = a3 + b3 + 3a2b + 3ab2 Let's try some examples on this identity Example 1: Solve (2a + 3b)3 Solution: This proceeds as: Given polynomial (2a + 3b)3 represents identity i.e. (a + b)3 Where a = 2a and b = 3b Now apply values of a and b on the identity i.e. (a + b)3 = a3 + b3 + 3ab(a + b) and we get: (2a + 3b)3 = (2a)3 + (3b)3 + 3(2a) (3b)(2a + 3b) Expand the exponential forms and we get: = 8a3 + 27b3 + 3(2a) (3b)(2a + 3b) Solve multiplication process and we get: = 8a3 + 27b3 + 18ab(2a + 3b) Hence, (2a + 3b)3 = 8a3 + 27b3 + 18ab(2a + 3b) Example 2: Solve (5x + 6y)3 Solution: This proceeds as: Given polynomial (5x + 6y)3 represents identity i.e. (a + b)3 Where a = 5x and b = 6y Now apply values of a and b on the identity i.e. (a + b)3 = a3 + b3 + 3a2b + 3ab2 and we get: (5x + 6y)3 = (5x)3 + (6y)3 + 3(5x)2 (6y) + 3(5x) (6y)2 Expand the exponential forms and we get: = 125x3 + 216y3 + 3(25x2)(6y) + 3(5x) (36y2) Solve multiplication process and we get: = 125x3 + 216y3 + 450x2y + 540xy2 Hence, (5x + 6y)3 = 125x3 + 216y3 + 450x2y + 540xy2 Study More Solved Questions / Examples Solve the following equations by using (A + B) Cube formula: A) (15x + 9y)3 B) (19x + 13y)3 C) (13x + 7y)3 D) (12x + 8y)3 Solve the following equations by using (A + B) Cube formula E) (23x + 6y)3 F) (16x + 9y)3 G) (9x + 3y)3 H) (9x + 7y)3