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Identity: (a - b)3 = a3 - b3 - 3ab(a - b)
How is this identity obtained? Let's see how. Taking LHS of the identity: (a - b)3 This can also be written as: = (a - b) (a - b) (a - b) Now, 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 orange & 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 (3a - 2b)3 Solution: This proceeds as: Given polynomial (3a - 2b)3 represents the identity (a - b)3 Where a = 3a and b = 2b Now substitute values of a and b in the identity i.e. (a - b)3 = a3 - b3 - 3ab(a - b) and we get: (3a - 2b)3 = (3a)3 - (2b)3 - 3(3a) (2b)(3a - 2b) Expand the exponential forms and we get: = 27a3 - 8b3 - 3(3a) (2b)(3a - 2b) Solve multiplication process and we get: = 27a3 - 8b3 - 18ab(3a - 2b) Hence, (3a - 2b)3 = 27a3 - 8b3 - 18ab(3a - 2b) Example 2: Solve (5x - 4y)3 Solution: This proceeds as: Given polynomial (5x - 4y)3 represents identity i.e. (a - b)3 Where a = 5x and b = 4y Now apply values of a and b on the identity i.e. (a - b)3 = a3 - b3 - 3a2b + 3ab2 and we get: (5x - 4y)3 = (5x)3 - (4y)3 - 3(5x)2 (4y) + 3(5x) (4y)2 Expand the exponential forms and we get: = 125x3 - 64y3 - 3(25x2)(4y) + 3(5x) (16y2) Solve multiplication process and we get: = 125x3 - 64y3 - 300x2y + 240xy2 Hence, (5x - 4y)3 = 125x3 - 64y3 - 300x2y + 240xy2 |