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a minus b whole cube formula

The a minus b whole cube formula is:

(a−b)3=a3−3a2b+3ab2−b3(a-b)^3=a^3-3a^2b+3ab^2-b^3(a−b)3=a3−3a2b+3ab2−b3

Quick Scoop: The Core Formula

  • Standard algebraic identity:

(a−b)3=a3−3a2b+3ab2−b3(a-b)^3=a^3-3a^2b+3ab^2-b^3(a−b)3=a3−3a2b+3ab2−b3

  • In words:
    “Cube of first term, minus three times square of first times second, plus three times first times square of second, minus cube of second.”

How the Formula Is Derived (Short Story Style)

Think of (a−b)3(a-b)^3(a−b)3 as (a−b)(a−b)(a−b)(a-b)(a-b)(a-b)(a−b)(a−b)(a−b).

First multiply (a−b)(a−b)(a-b)(a-b)(a−b)(a−b) to get (a2−2ab+b2)(a^2-2ab+b^2)(a2−2ab+b2).

Then multiply this result by (a−b)(a-b)(a−b):

(a2−2ab+b2)(a−b)(a^2-2ab+b^2)(a-b)(a2−2ab+b2)(a−b)

Expanding term by term and combining like terms gives:

  • a2⋅a=a3a^2\cdot a=a^3a2⋅a=a3
  • a2⋅(−b)=−a2ba^2\cdot (-b)=-a^2ba2⋅(−b)=−a2b
  • −2ab⋅a=−2a2b-2ab\cdot a=-2a^2b−2ab⋅a=−2a2b
  • −2ab⋅(−b)=2ab2-2ab\cdot (-b)=2ab^2−2ab⋅(−b)=2ab2
  • b2⋅a=ab2b^2\cdot a=ab^2b2⋅a=ab2
  • b2⋅(−b)=−b3b^2\cdot (-b)=-b^3b2⋅(−b)=−b3

Adding them:

a3−a2b−2a2b+2ab2+ab2−b3=a3−3a2b+3ab2−b3a^3-a^2b-2a^2b+2ab^2+ab^2-b^3 =a^3-3a^2b+3ab^2-b^3a3−a2b−2a2b+2ab2+ab2−b3=a3−3a2b+3ab2−b3

Handy Pattern to Remember

For both (a+b)3(a+b)^3(a+b)3 and (a−b)3(a-b)^3(a−b)3, the coefficients are the same: 1:3:3:11:3:3:11:3:3:1.

  • (a+b)3=a3+3a2b+3ab2+b3(a+b)^3=a^3+3a^2b+3ab^2+b^3(a+b)3=a3+3a2b+3ab2+b3
  • (a−b)3=a3−3a2b+3ab2−b3(a-b)^3=a^3-3a^2b+3ab^2-b^3(a−b)3=a3−3a2b+3ab2−b3

So you can:

  1. Write the “plus” version.
  2. Change the signs alternatingly for the “minus” version.

Example Use

Compute (2x−3y)3(2x-3y)^3(2x−3y)3 using the identity:

(2x−3y)3=(2x)3−3(2x)2(3y)+3(2x)(3y)2−(3y)3(2x-3y)^3=(2x)^3-3(2x)^2(3y)+3(2x)(3y)^2-(3y)^3(2x−3y)3=(2x)3−3(2x)2(3y)+3(2x)(3y)2−(3y)3

That becomes:

8x3−36x2y+54xy2−27y38x^3-36x^2y+54xy^2-27y^38x3−36x2y+54xy2−27y3

Extra Note

This identity is heavily used in:

  • Simplifying algebraic expressions.
  • Mental math shortcuts when numbers fit the pattern a−ba-ba−b.
  • Competitive exams and board-level problems.

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