(a-b)^3 Formula

(a-b)^3 Formula: Solution With Proof

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Mathematics is one of the most interesting subject which can make your mind 2 times more faster. Some student used to hate this because it’s complicated to solve some problems. Most complicated topic is “Algebra” because this contains variables as well as polynomials too with some exponential powers. We used to solve them by some given formulas.

But it’s very important to know that how these formulas made before? How our great mathmeticians used to prove that? Here we are going to talk about this topic here. One of the most well common formula “(a-b)^3 formula”. We are going to prove (a-b)^3 formula below for better understanding for students.

Contents hide
1 (a-b)^3 Formula
2 Proof Of (a-b)^3 Formula
3 Example On (a-b)^3 Formula

(a-b)^3 Formula

But first we sould know the formula of (a-b)^3:

(a-b)^3 = a^3 – 3a^2b + 3ab^2 – b^3

Proof Of (a-b)^3 Formula

Let’s prove the identity for
(a – b)^3 = a^3 – 3a^2b + 3ab^2 – b^3

Step 1: Expand (a – b)^3

By definition,
(a – b)^3 = (a – b)(a – b)(a – b)

First, expand the first two terms:
(a – b)(a – b) = a^2 – 2ab + b^2

Now multiply that result with the third (a – b):
(a^2 – 2ab + b^2)(a – b)

Step 2: Multiply each term in the first bracket by (a – b)

= a^2(a – b) – 2ab(a – b) + b^2(a – b)

Now expand each of those:

  1. a^2(a – b) = a^3 – a^2b
  2. -2ab(a – b) = -2a^2b + 2ab^2
  3. b^2(a – b) = ab^2 – b^3

Step 3: Combine all the terms

= a^3 – a^2b – 2a^2b + 2ab^2 + ab^2 – b^3
= a^3 – 3a^2b + 3ab^2 – b^3

✅ Proven:
(a – b)^3 = a^3 – 3a^2b + 3ab^2 – b^3

Example On (a-b)^3 Formula

Alright, let’s prove (a – b)^3 = a^3 – 3a^2b + 3ab^2 – b^3 with an actual example.

Let’s take:
a = 4, b = 1

Left Side:

(a – b)^3 = (4 – 1)^3 = 3^3 = 27

Right Side:

a^3 – 3a^2b + 3ab^2 – b^3

Now plug in the values:

  • a^3 = 4^3 = 64
  • a^2 = 4^2 = 16
  • b^2 = 1^2 = 1
  • b^3 = 1^3 = 1

Now compute:

= 64 – 3×16×1 + 3×4×1 – 1
= 64 – 48 + 12 – 1
= (64 + 12) – (48 + 1)
= 76 – 49
= 27

By expanding the expression (a – b)^3 and simplifying it step by step, we proved that it equals a^3 – 3a^2b + 3ab^2 – b^3. Using real numbers (a = 4, b = 1), both sides of the equation resulted in the same value, confirming the identity is correct. This shows that the formula is valid both algebraically and numerically.

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