(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) Add above like terms, highlighted in red and we get: = { 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 - 2a2b + 2ab2 Rearrange the terms and we get: = a3 - b3 - a2b - 2a2b + ab2 + 2ab2 Add like terms, highlighted in green & red and we get: = a3 - b3 - 3a2b + 3ab2 Or we can further solve it: Take 3ab common from the above blue highlighted terms and we get: = a3 - b3 - 3ab(a - b) 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 example of 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)26y + 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 using (A - B) Cube formula A) (2x - 3y)3 B) (5x - 4y)3 C) (9x - 5y)3 D) (13x - 7y)3 E) (15x - 9y)3 F) (17x - 11y)3 G) (19x - 13y)3