Toolkit 9Cube of a difference(a−b)3=a3−3a2b+3ab2−b3(a-b)^3=a^3-3a^2b+3ab^2-b^3(a−b)3=a3−3a2b+3ab2−b3ProofUsing Toolkit 7 — Square of a difference,(a−b)2=a2−2ab+b2.(a-b)^2 = a^2 - 2ab + b^2.(a−b)2=a2−2ab+b2.Therefore,(a−b)3=(a−b)(a−b)2=(a−b)(a2−2ab+b2)=a3−2a2b+ab2−a2b+2ab2−b3=a3−3a2b+3ab2−b3.□\begin{aligned} (a-b)^3 &= (a-b)(a-b)^2 \\ &= (a-b)(a^2 - 2ab + b^2) \\ &= a^3 - 2a^2b + ab^2 - a^2b + 2ab^2 - b^3 \\ &= a^3 - 3a^2b + 3ab^2 - b^3. \quad\square \end{aligned}(a−b)3=(a−b)(a−b)2=(a−b)(a2−2ab+b2)=a3−2a2b+ab2−a2b+2ab2−b3=a3−3a2b+3ab2−b3.□Related ProblemsMajorAMC 12A 2022 (Problem 14)MinorAMC 10A 2025 (Problem 9)